Proof of Theorem 1. Let \((\phi_t^{\mathcal{M}})_{t \in \mathbb{R}}\) be a locally Hamiltonian flow and let \(\mathcal{M}' \subset \mathcal{M}\) be the minimal component as in the statement. By the classical representation theorem recalled in Section 2.2, the restriction \((\phi_t^{\mathcal{M}}|_{\mathcal{M}'})\) is measure-theoretically isomorphic to a suspension flow \((\phi_t, X)\) over an IET \(T\) under a roof function \(\varphi\) as in 5 , via a measure-preserving isomorphism \(\Phi: X \to \mathcal{M}'\). By [16], the set of flows for which \(T\) is typical has full measure with respect to Katok’s fundamental class.

Fix \(\nu > 0\) and \(C' > 0\). By Theorem 1 (applied with exponent \(\nu/4\)), there exist an increasing sequence of times \((t_m)\), rectangles \((A_m)\), functions \((g_m)\) with \(g_m \in C^1_c(A_m)\), and a constant \(C>0\), all independent of \(\nu\), such that for any \(m\) \[|\mathrm{Cor}_{t_m}(g_m, g_m)| \geqslant C,\] and, for \(m\) large enough, \(g_m\) satisfies 10 –11 for \(\nu/4\).

For each \(m\), define \(\tilde{g} _m = g_m \circ q^{-1}\circ \Phi^{-1}\). Since \(\Phi\) is measure-preserving and intertwines \((\phi_t)\) with \((\phi_t^{\mathcal{M}})\): \(\text{Cor}_{t_m}(\tilde{g}_m,\tilde{g}_m) =\text{Cor}_{t_m}(g_m,g_m).\)

By Proposition 2, for \(m\) (hence \(t_m\)) sufficiently large \[\|\tilde{g} _m\|_{C^1(\mathcal{M}')}^2 \lesssim (\log t_m)^{2+\nu/2}\leqslant\frac{C}{C'}(\log t_m)^{2+\nu}.\] Therefore, for any prescribed \(C' > 0\), if \(m\) is large enough \[\begin{align} |\mathrm{Cor}_{t_m}(\tilde{g} _m, \tilde{g} _m)| &\geqslant C= C\cdot\frac{\|\tilde{g} _m\|_{C^1(\mathcal{M}')}^2}{\|\tilde{g} _m\|_{C^1(\mathcal{M}')}^2} \geqslant C'\|\tilde{g} _m\|_{C^1(\mathcal{M}')}^2 (\log t_m)^{-2-\nu}. \end{align}\] ◻

Proof. It suffices to prove the statement for a fixed Rauzy class \(\mathfrak{R}\), with respect to the product measure \(\nu_{\mathfrak{R}} = \mathrm{Leb}_{\Delta_{d}} \otimes \mu_{\mathfrak{R}}\) on \(\mathcal{M}_d(\mathfrak{R})\), where \(\mu_{\mathfrak{R}}\) is the uniform measure on \(\mathfrak{R}\). We treat the conditions in order, based on the results of Viana [19] and Marmi-Moussa-Yoccoz [20].

The Keane condition. For any fixed \(\pi \in \mathfrak{R}\), the set of \(\lambda \in \Delta_{d}\) violating the Keane condition is a countable union of hyperplanes defined by rational affine relations among the \(\lambda_i\). These are \(\mathrm{Leb}_{\Delta_{d}}\)-null, so the Keane condition holds for \(\nu_{\mathfrak{R}}\)-almost every \((\underline\lambda,\pi) \in \mathcal{M}_d(\mathfrak{R})\).

Condition 15 . Condition 15 is precisely the Roth-type condition on the Zorich induction introduced in Section 1.3 of [20], where it is shown to hold for \(\nu_{\mathfrak{R}}\)-almost every \((\underline\lambda,\pi) \in \mathcal{M}_d(\mathfrak{R})\).

Condition 16 . To verify this condition, we present an argument analogous to [21], leveraging the topological openness of parameters in a natural extension of the space \(\mathcal{M}_d(\mathfrak{R})\). We carry this out in Appendix 11. ◻

Proof. Two distinct points \((x,s), (x',s') \in A\) are identified in \(X\) only if \((x',s') = \phi_u(x,s)\) for some \(u \neq 0\), which requires \(s' = s + u - S_k(\varphi)(x)\) and \(x' = T^k x\) for some \(k \geqslant 1\). In particular, \(x' \in T^k I \cap I\), which by ?? cannot happen for \(1 \leqslant k \leqslant h-1\). It remains to rule out \(k \geqslant h\). Since \(s, s' \in [0, \gamma h/4]\) and \(S_k(\varphi)(x) \geqslant\gamma k \geqslant\gamma h\) for \(k \geqslant h\), we have \[s' = s + u - S_k(\varphi)(x) \leqslant\frac{\gamma h}{4} -\gamma h < 0,\] contradicting \(s' \geqslant 0\). Hence \(q|_A\) is injective. ◻

Proof. For \(r\geqslant e^2\) let \(f\colon[0,1]^2\to\mathbb{R}\) be a smooth non-negative function, \(f\) compactly supported on \((0,1)^2\), satisfying \[\begin{align} & \cdot f=1 \quad \text{ on } \quad[1/\log r,1-1/\log r]^2, \\ &\cdot 0\leqslant f\leqslant 1, \\ &\cdot \|Df\|_\infty\leqslant 2 \log r. \end{align}\] Such an \(f\) exists by a standard smooth cutoff construction. Define \(g: A \to \mathbb{R}\) by \[g(x,s) := f\bigl(\tfrac{x-a}{b-a},\, \tfrac{s}{t}\bigr).\] Clearly \(g\in C^1(A)\) and, since \(f\) is compactly supported in \((0,1)^2\), \(g\) is compactly supported in \(A^\circ\). Hence \(g\in C^1_c(A)\). The bound \(0\leqslant g\leqslant 1\) follows immediately from \(0\leqslant f\leqslant 1\).

By the chain rule, \[\|\partial_x g\|_\infty = (b-a)^{-1}\|\partial_1 f\|_\infty \leqslant 2(b-a)^{-1}\log r, \quad \|\partial_s g\|_\infty = t^{-1}\|\partial_2 f\|_\infty \leqslant 2t^{-1}\log r.\]

Finally, since \(f = 1\) on \([1/\log r, 1-1/\log r]^2\) and \(0 \leqslant f \leqslant 1\) elsewhere, \[\int_A g = t(b-a)\int_{[0,1]^2} f \geqslant t(b-a)\left(1 - \frac{2}{\log r}\right)^2 \geqslant t(b-a)\left(1 - \frac{4}{\log r}\right).\] ◻

Proof. Injectivity follows from Lemma 1. Apply Lemma 2 with \(t = \gamma h/4\) and \(r=h\), so that \((b-a)^{-1} \leqslant Ch\) by ?? , giving the bounds on \(\|Dg\|_\infty\). ◻

Proof. Recall the definition of \(\varphi'\) in 18 . Let \(1\leqslant l\leqslant h_{i}^{(n)}\) and consider \(S_l (\varphi')(x)\). The term \(\psi_1'(x)+\psi_2'(x)\) is uniformly bounded, so \[|S_l (\psi_1'(x)+\psi_2'(x))(x)|\leqslant l\left| \left| \psi_1' \right| \right|_{L^{\infty}}+l\left| \left| \psi_2' \right| \right|_{L^{\infty}},\qquad i=1,2.\]

On \(F_{n,i}\), the orbit segment \(\{T^k x\}_{k=0}^{h_{i}^{(n)}-1}\) avoids intervals of length \(2\delta/q_{i}^{(n)}\) around the singularities, and \(\{T^k I_i^{(n)}\}_{k=0}^{h_{i}^{(n)}-1}\) are disjoint intervals of \(\mathbb{T}\), so the smallest gap between consecutive points when ordered in \((0,1)\) is \(\lambda_{i}^{(n)}\). The worst-case sum is therefore comparable to the harmonic series \[\sum_{k=0}^{l-1}\left( \frac{\max c_y^{\pm}}{\|T^k x\|}+\max |b_y^{\pm}|(1-\log \|T^k x\|)\right)\lesssim^{\varphi} \sum_{j=0}^{l-1} \frac{q_{i}^{(n)}}{\delta+j} \lesssim^\delta q_{i}^{(n)}{(\log l+ 1)}.\] Thus there exists a constant \(C_1 = C_1(\delta, \varphi, T) > 0\) such that \[{|S_{l} (\varphi')(x)| \leqslant C_1 q_{i}^{(n)} \log h_{i}^{(n)} \qquad \forall\, x \in F_{n,i},\qquad 0\leqslant l\leqslant h_{i}^{(n)}.}\] ◻

Proof. By 16 , \[\left. \begin{align} {\lambda_{j^*}^{(r_m)}} \geqslant 3{\lambda^{(r_m)}}/4, \\ T^{r_m}I_{j^*}^{(r_m)}\subset I^{(r_m)} \end{align} \right\}\] \[\implies \lambda\left(F_{r_m}\cap T^{h_{r_m}}F_{r_m}\right)\geqslant\lambda^{(r_m)}/2-2\delta/q_{r_m}\geqslant\left(1/2-{2\delta}\right)\lambda^{(r_m)}>0.\] Therefore \(F_m'\neq\emptyset\).

\(E_m\) is formed of \(h_{r_m}\) disjoint intervals, each corresponding to an iterate under \(T\) of \(F_{r_m}\). If \(x, y\) lie in the same connected component of \(E_{m}\), \[\left|S_{h_{r_m}} (\varphi)(x) - S_{h_{r_m}} (\varphi)(y)\right| \leqslant|x-y| \cdot \sup_{\xi \in [x,y]} |S_{h_{r_m}}( \varphi')(\xi)|,\] so, provided \(m\) large enough, 19 gives \[\sup_{x,y \in J} \left|S_{h_{r_m}} (\varphi)(x) - S_{h_{r_m}} (\varphi)(y)\right| \leqslant C_1 q_{r_m} \log h_{r_m} \lambda(F_{r_m} ) \leqslant C_1\log h_{r_m}.\]

Consider the first \(q_{r_m}\) iterates of the orbit of \(x_0\in F'_m\). Each of them lie in a different interval of \(E_m\) and \(x_0,T^{q_{r_m}}x_0\in F_{r_m}\). Hence \[\begin{align} \sup_{0< l < h_{r_m}} \left|S_{h_{r_m}} (\varphi)(T^lx_0) - S_{h_{r_m}} (\varphi)(x_0)\right| &\leqslant\sup_{0< l < h_{r_m}} \left|S_{l} (\varphi)(x_0) - S_{l} (\varphi)(T^{h_{r_m}}x_0)\right| \\&\leqslant C_1 \log h_{r_m}. \end{align}\]

To pass from a single connected component to all of \(E_m\), let \(x, y \in E_m\) be arbitrary. If they lie in the same component, the bound \(C_1 \log h_{r_m}\) applies directly. If not, pick any \(x_0 \in F'_m\) and \(l_1, l_2\) such that \(T^{l_1} x_0\) and \(T^{l_2} x_0\) lie in the same components as \(x\) and \(y\) respectively. Then the triangular inequality gives the result for \(C_2 = 4C_1\). ◻

Proof. It is clear by their definitions that \(T^k\) is a translation on \(I\) for any \(0\leqslant k<h\), and \[\lambda(I)=\lambda(F_{n,i})= \frac{{1-2\delta}}{q_i^{(n)}}\geqslant\frac{1}{2h_i^{(n)}d}=:C^{-1}h^{-1}.\] ◻

Proof. Given \({m}\), for each induction step \(k\), let \(S_k := \left\{j \in \{1,\dots,d\} : h_j^{(k)}\lambda_j^{(k)} \geqslant\frac{1}{d}\right\}\) be the set of dominant components. Since \(\sum_{j=1}^d h_j^{(k)}\lambda_j^{(k)} = 1\), \(S_k \neq \emptyset\) for all \(k\). We define \[n:=\min\left\{k:\min_{j\in S_k} h_j^{(k)}\geqslant\frac{2C_2}{\gamma}\log h_{r_m}\right\}.\] Clearly \(n\to\infty\) as \(m\to\infty\). Pick \(m\) large enough so that \(n\geqslant 2\).

By definition of \(n\), at step \(n\) we can choose an index \(i \in S_n\) such that \[\frac{1}{d}\leqslant h_i^{(n)}\lambda_i^{(n)}\leqslant 1.\] Fixing this \(i\), we automatically have \(h_i^{(n)} \geqslant\min_{j \in S_n} h_j^{(n)} \geqslant\frac{2C_2}{\gamma}\log h_{r_m}\).

Furthermore, by the minimality of \(n\), the condition must fail at step \(n-1\), meaning there exists an index \(j \in S_{n-1}\) such that \(h_{j}^{(n-1)} < \frac{2C_2}{\gamma}\log h_{r_m}\). Since \(j \in S_{n-1}\), it satisfies \(h_{j}^{(n-1)} \geqslant\frac{1}{d\lambda_{j}^{(n-1)}} \geqslant\frac{1}{d\max_j\lambda_j^{(n-1)}}\). Combined with 15 , we obtain \[\begin{align} h_i^{(n)}&\geqslant\frac{2C_2}{\gamma}\log h_{r_m} \geqslant h_{j}^{(n-1)}\geqslant\frac{1}{d\max_j\lambda_j^{(n-1)}}\geqslant\frac{1}{d C_\theta\left(\min_j \lambda_j^{(n-1)}\right)^{\frac{1}{1+\theta}}}\\ &\geqslant\frac{1}{d C_\theta\left(\max_j \lambda_j^{(n)}\right)^{\frac{1}{1+\theta}}} \geqslant\frac{1}{d C_\theta^{\frac{2+\theta}{1+\theta}}\left(\min_j \lambda_j^{(n)}\right)^{\frac{1}{(1+\theta)^2}}}\geqslant\frac{1}{d C_\theta^{\frac{2+\theta}{1+\theta}} \left(\lambda_i^{(n)}\right)^{\frac{1}{(1+\theta)^2}}}\\ &\geqslant\frac{1}{d C_\theta^{\frac{2+\theta}{1+\theta}}}\left(h_i^{(n)}\right)^{\frac{1}{(1+\theta)^2}}=:\hat{C}_\theta \left(h_i^{(n)}\right)^{\frac{1}{(1+\theta)^2}}, \end{align}\] where the final inequality follows from the total area constraint, implying \(\lambda_i^{(n)} \leqslant 1/h_i^{(n)}\). This completes the proof. ◻

Proof. The existence of \(g_m\) for \(m\) large enough is a direct application of Lemma 5, and it follows from 20 and 22 that \[t_m - \inf_{F_{r_m}} S_{h_{r_m}} (\varphi)-3{\gamma h_n}/{4}\leqslant C_2\log h_{r_m}-\gamma h_n/2\leqslant 0.\]

To prove 25 , first note that it is enough to prove it for \(\sup_{F_{r_m}} S_{h_{r_m}}(\varphi)\) instead of \(t_m\), since 22 implies that the quotient \[\frac{h_n}{h_{r_m}}\lesssim\frac{(\log h_{r_m})^{(1+\theta)^2}}{h_{r_m}}\to 0\] as \(m\to \infty\).

Now, by definition of \(\gamma\), \[S_{h_{r_m}}(\varphi)(x) \geqslant\gamma h_{r_m} , \qquad \forall x\in \mathbb{T},\] so it is only left to show the upper bound.

To show the upper bound we will use the same argument as in the proof of Lemma 3. The term \(\psi(x)\) is uniformly bounded, so \[S_{h_{r_m}}(\psi)(x)\lesssim h_{r_m},\qquad \forall x \in \mathbb{T}.\] On \(F_{r_m}\), the orbit segment \(\{T^k x\}_{k=0}^{h_{r_m}-1}\) avoids intervals of length \(2\delta/q_{r_m}\) around the singularities, and the smallest gap between consecutive points is \(q_{r_m}^{-1}\). The worst-case sum is therefore comparable to the series \[-\sum_{k=0}^{h_{r_m}-1} \log{\|T^k x\|}\leqslant\sum_{j=0}^{h_{r_m}-1} \log \frac{q_{r_m}}{\delta+j} \leqslant\log \frac{q_{r_m}}{\delta}+\int_0^{h_{r_m}}\log\frac{q_{r_m}}{\delta+x}dx\lesssim^\delta h_{r_m},\] for any \(x\in F_{r_m}\), where we have used that 16 implies \(h_{r_m}\geqslant q_{r_m}/2\). ◻

Proof. Define \(I'_n=\left(a+\delta/q_n+1/q_n^2,b-\delta/q_n-1/q_n^2\right)\) and recall \(A_m=I_n\times[0,\gamma h_n/4].\) This choice of \(I'_n\) is made so that \(T^{h_{r_m}}I_n'\subset I_n\), for \(m\) large enough, while its length is asymptotically close to \(\lambda(I_n)=\lambda(I_n')+2/q_n^2\). Indeed, for \(m\) large enough, 16 and 22 imply \[\text{dist}(x,T^{h_{r_m}}x)\leqslant\frac{\lambda^{(r_m)}}{4}\leqslant\frac{1}{3q_{r_m}}\leqslant\frac{1}{q_n^2}, \qquad x\in F_{r_m}.\]

Set \[\begin{align} \begin{cases} J_m &:= I'_n\cap E_m, \\ B_m &:= J_m\times\left[0,\frac{\gamma h_n}{4}\right]\subset A_m,\\ B'_m &:= \left\{(x,t):x\in T^{h_{r_m}}J_m, t\in\left(\frac{\gamma h_n}{4},S_{q_n}(\varphi)(x)\right)\right\}\subset A_m^c. \end{cases} \end{align}\] By definition of \(E_m\) and 16 , we have \[\mu(B_m)=\lambda(I_n'\cap E_m)\frac{\gamma h_n}{4}\geqslant\left(\mu(I_n)-\frac{2}{q_n^2}\right)\frac{\gamma h_n}{4}-\int_{\left(\mathcal{T}_{j^*}^{(r_m)}\right)^c}\varphi\geqslant(1-\delta)\mu(A_m),\] for \(\varepsilon(\delta,\varphi)>0\) small enough and \(n\) large enough. Here we have used that \(h_n\leqslant q_n\) and that \(\varphi \in L^1(\mathbb{T})\) and \(\lambda\left(\mathcal{T}_{j^*}^{(r_m)}\right)\geqslant 1-\varepsilon\).

The rectangle \(A_m\) has base length \(\frac{1-2\delta}{q_n}\) and height \(\gamma h_n/4\), so \(\mu(A_m)\geqslant\frac{(1-2\delta)\gamma}{4d}.\)

The conditions 23 –24 on \(t_m\) forces \(\phi_{t_m}(B_m)\cap A_n\subset B'_m\cap A_m=\varnothing\), hence \[\mu(\phi_{t_m}(A_m)\cap A_m) \leqslant\mu(A_m\setminus B_m) \leqslant\delta\,\mu(A_m).\]

Recall that \(g_m\) is defined using Corollary 1, and therefore satisfies the estimates 17 . Consequently, \[\left|\mu(g_m)^2-\langle g_m,\, g_m\circ\phi_{t_m}\rangle\right| \geqslant\left(\mu\left(A_m\right)\left(1-\frac{4}{\log h_n}\right)\right)^2-\mu\left(A_m\setminus B_m\right)\] \[\geqslant\mu\left(A_m\right)^2-\delta\mu\left(A_m\right)-O\left({\frac{1}{\log h_{n}}}\right) \geqslant\frac{(1-2\delta)^2\gamma^2}{16d^2}-\delta\frac{(1-2\delta)\gamma}{4d}-O\left({\frac{1}{\log h_{n}}}\right).\] So one can fix \(\delta(\varphi)>0\) small enough so that, for \(m\) large enough such that the last term is \(\leqslant\frac{\gamma^2}{64d^2}\), it holds \[\left|\mu(g_m)^2-\langle g_m,\, g_m\circ\phi_{t_m}\rangle\right|\geqslant\frac{\gamma^2}{32d^2}.\]

Finally, using that \(g_m\) satisfies the estimates 17 and \(t_m\) relates to \(h_{n}\) by 22 and 25 , we get \[\|\partial_xg\|_{\infty}\lesssim h_{n} \log h_{n}\lesssim (\log t_m)^{{(1+\theta)^2}}\log\log t_m,\] and \[\|\partial_sg\|_{\infty}\lesssim h_{n}^{-1} \log h_{n}\lesssim (\log t_m)^{-1}\log\log t_m,\] where the constants depend on \(\delta\), \(\varphi\), \(T\) and \(\theta\).

As a consequence of 17 , \(\|g_m\|_\infty=1\). This completes the proof. ◻

Proof of Theorem 1. Choose \(\theta > 0\) small enough that \(((1+\theta)^2 - 1) < \nu\) and \(\delta\) such that \[\frac{(1-2\delta)^2}{16}-\frac{\delta(1-2\delta)d}{4\gamma}\geqslant\frac{3}{64}.\] Then pick \(\varepsilon\) so that \(\int_{\left(\mathcal{T}_{j^*}^{(r_m)}\right)^c}\varphi\leqslant\frac{\delta (1-2\delta)\gamma}{8d}\).

We proof the result holds for all IETs satisfying 15 –16 , which is a full measure condition by Proposition 3. For such IETs and our choice of \((\theta,\varepsilon)\), Lemma 8 produces a constant \(C=\gamma ^2/32d^2\), a sequence \(t_m \to \infty\), rectangles \(A_m\), functions \(g_m \in C^1_c(A_m)\), and \(M>0\) such that, for every \(m\geqslant M\) \[|\text{Cor}_{t_m}(g_m,g_m)|=|\langle g_m\circ\phi_{t_m},g_m\rangle-\mu(g_m)^2|\geqslant C,\] and \(\|g_m\|_\infty=1\), \(\|\partial_xg\|_{\infty}=o((\log t_m)^{1+\nu})\), \(\|\partial_sg\|_{\infty}=o((\log t_m)^{-1+\nu})\).

Note that, for \(\varphi\) and \(T\) fixed, so are \(\varepsilon\) and \(\delta\), so we recall from Remark 5 that the sequences \((t_m),(A_m),(g_m)\) are independent of the choice of \(\nu\). However, \(M\) is not. Without loss of generality, we can assume \(M(\nu=1)=1\), so that 9 holds for any \(m\).

If \(m\geqslant M(\nu)\), by definition of \(A_m\) (the support of \(g_m\)) and using that \(h_n\) and \(t_m\) are related via 22 and 25 , for every \((x, s)\in A_m\) \[\mathrm{dist}(T^k x, D_0)> \frac{\delta}{q_n}\geqslant\hat{\delta}\,(\log t_{m})^{-(1+\theta)^2}>(\log t_{m})^{-1-\nu},\] for \(0\leqslant k< h_n\), and \[N(x,s)\leqslant\tfrac{1}{4}h_n\leqslant\frac{1}{\hat{\delta}}(\log t_m)^{(1+\theta)^2}<(\log t_{m})^{1+\nu},\] for \(m\) large enough, where \(\hat{\delta}=\hat{\delta}(\delta,\varphi,T,\theta) > 0\). This concludes the proof. ◻

Proof. We will first prove the following: for every \(s\in (0,\varepsilon)\), \(\exists \theta_1(s),\theta_2(s)\in(\alpha,\beta)\) such that \[\begin{align} &(f^{-1}(s+\varepsilon)-f^{-1}(s))-(f^{-1}(s)-f^{-1}(s-\varepsilon))=-\sum_{i=1}^2\frac{f''(f^{-1}(\theta_i(s)))}{2f'(f^{-1}(\theta_i(s)))^3}\varepsilon^2. \label{eq:taylor95crossings} \end{align}\tag{27}\]

Let \(g=f^{-1}\). Then \(g\) is \(C^2\) on \([-\varepsilon,2\varepsilon]\). Fix \(s\in(0,\varepsilon)\).

We apply the Taylor expansion of \(g\) at \(s\). For \(\varepsilon>0\), we have \[g(s+(-1)^i\varepsilon) = g(s)+(-1)^i g'(s)\varepsilon+\frac{g''(\theta_i(s))}{2}\varepsilon^2,\] for \(i=1,2\), and some \(\theta_1(s)\in(s-\varepsilon,s)\) and \(\theta_2(s)\in(s,s+\varepsilon)\).

Subtracting gives \[(g(s+\varepsilon)-g(s))-(g(s)-g(s-\varepsilon))= \frac{g''(\theta_1(s))}{2}\varepsilon^2+\frac{g''(\theta_2(s))}{2}\varepsilon^2.\]

It remains to express the derivatives of \(g\) in terms of those of \(f\). Using the identities for derivatives of the inverse function, we have \[g'(s)=\frac{1}{f'(f^{-1}(s))}, \qquad g''(s)= -\frac{f''(f^{-1}(s))}{f'(f^{-1}(s))^3}.\] Substituting these identities yields 27 .

To prove 26 , notice that, by hypothesis, \(\frac{-f''(x)}{f'(x)^3}\geqslant\frac{1}{\sqrt{k}MC^4},\) on \([\alpha,\beta]\). Therefore \[(g(s+\varepsilon)-g(s))-(g(s)-g(s-\varepsilon))\geqslant\frac{2}{\sqrt{k}MC^4}\varepsilon^2 \geqslant\frac{\varepsilon^3}{\sqrt{k}M},\] for \(\varepsilon<\varepsilon^*\) small enough. ◻

Proof. Observe that, with \(g=f^{-1}\), the difference becomes \[\mu(E\cap D^+)- \mu(E\cap D^-) = \int_{0}^{\varepsilon} \Bigl[(g(s+\varepsilon)-g(s))-(g(s)-g(s-\varepsilon))\Bigr]\,ds.\]

By Lemma 9, for every \(s\in(0,\varepsilon)\), the term on the integral is lower bounded by \(\frac{\varepsilon^3}{\sqrt{k}M}\). Integrating yields the result. ◻

Proof. Let \(\eta(q) = ({2q \log q\log\log q})^{-1}\). Then \(q\eta(q) = ({2 \log q\log\log q})^{-1}\) is decreasing and \[\sum_{q=2}^\infty \eta(q) = \sum_{q=2}^\infty \frac{1}{2q \log q\log\log q} = \infty.\] By Khinchin’s theorem [24], for Lebesgue almost every \(\alpha\), there exist infinitely many coprime pairs \((p,q)\) such that \(\left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2 \log q\log \log q}.\)

For each \(q\), there exists an \(n\) such that \(q_n \leqslant q < q_{n+1},\) so the best approximation property of continued fractions implies \[\frac{1}{2 q_{n+1}} \leqslant\left|q_n\alpha - p_n\right| \leqslant\left|q\alpha - p\right|< \frac{1}{2q \log q\log \log q} \leqslant\frac{1}{2q_n \log q_n\log \log q_n}.\] Hence, for infinitely many \(n\) for which a Khinchin-good rational \(p/q\) lies in \((q_n,\) \(q_{n+1})\), we obtain \(q_{n+1} \geqslant\, q_n \log q_n\log \log q_n.\) ◻

Proof. Fix \(1\leqslant l\leqslant q_{n}\). The orbit \(\{R_\alpha^k x\}_{k=0}^{l-1}\) is well-separated. That is, by the three-distance theorem, consecutive points in this orbit are separated by at least \(\|q_{n-1}\alpha\|\), and the \((2m+1)\)-st and \((2m+2)\)-nd closest iterates to the singularities lie at distance at least \(x_{\min,l} + m/(2q_n)\) from \(\{0,1\}\). Ordering by distance and using the expression of \(\varphi'\) in 30 gives \[\begin{align} | S_{l}(\varphi'-\psi')(x)| \leqslant 2c_0(x_{\min,l})^{-1}+\sum_{m=1}^{l-1}2c_0\tfrac{2q_n}{m}, \end{align}\] and \[\begin{align} | S_{l}(\psi')(x)| \leqslant-2|b_0|\log x_{\min,l}-\sum_{m=1}^{l-1}2|b_0|\log \tfrac{m}{2q_n} +2l|b_0|+ l\left| \left| \psi_1' \right| \right|_{L^\infty}+ l\left| \left| \psi_2' \right| \right|_{L^\infty}. \end{align}\] Hence, for \(n\) large enough, the existence of \(C_1\) follows.

The proof of 33 is analogous to that of 32 , the main difference being that (see 31 ) \[\begin{align} | S_{l}(\psi'')(x)| &\leqslant 3|b_0|x_{\min,l}^{-1}-6|a_0|\log x_{\min,l}+\sum_{m=1}^{l-1}\left[3|b_0|\tfrac{2q_n}{m}+3|a_0|(3-2\log \tfrac{m}{2q_n})\right]\\ &+ l\left| \left| \psi_2'' \right| \right|_{L^\infty}\lesssim x_{\min,l}^{-1}+q_n\log l, \end{align}\] and \[\frac{1}{q_n^2}\left[\sum_{m=1}^{l-1}3c_0(\tfrac{2q_n}{m})^2 +\sum_{m=1}^{l-1}3|b_0|\tfrac{2q_n}{m}+\sum_{m=1}^{l-1}3|a_0|(3-2\log \tfrac{m}{2q_n})+ l\left| \left| \psi_2'' \right| \right|_{L^\infty}\right],\] is uniformly bounded on \(n\) (because \(\sum m^{-2}<\infty\)), and the dominant contribution comes from the single iterate closest to the singularity.

Finally, we have \(\varphi''(x)\ge{c_0}{\left| \left| x \right| \right|}^{-2}+\psi''(x),\) for any \(x\). Hence, at every continuity point \(x\), \[S_{l}(\varphi)''(x)\geqslant c_0 x_{\min,l}^{-2}-| S_{l}(\psi'')(x)|= c_0 x_{\min,l}^{-2}-O(x_{\min,l}^{-1}+q_n\log l).\] ◻

Proof. The function \(S_{q_{n}}(\varphi)(x)\) is continuous on \(\mathbb{T}\setminus\cup_{k=0}^{q_{n}-1}\{-k\alpha\}\), and on each of the \(q_{n}\) intervals of continuity, the function is \(C^2\).

Note \(x_{\min,q_n}\leqslant 1/q_n\) for any \(x\), so by 34 , taking \(q_n/\log q_n>C_3^{-2}\) we get \(S_{q_{n}}(\varphi)\) is strictly convex on each interval of continuity, and \(\to\infty\) at the singularities. In particular, \(S_{q_{n}}(\varphi')\) is strictly monotone there and has exactly one zero. Hence \(\Gamma_{n}=\{x:S_{q_{n}}(\varphi')(x)=0\}\) consists of exactly \(q_{n}\) points, one in each interval.

Fix \(1>\hat{\delta}>0\) and let \(\tilde{\varphi}'\) denote the restriction of \(\varphi'\) to \([\frac{1}{\hat{\delta} q_{n}\log q_{n}},1-\frac{1}{\hat{\delta} q_{n}\log q_{n}}]\). Applying the Denjoy–Koksma inequality to \(\tilde{\varphi}'\), we get that there exists \(C_4=C_4(\varphi)>{4}/{c_0}\) such that \[\left|S_{q_{n}}(\tilde{\varphi}')(x)-q_{n}\int_\mathbb{T}\tilde{\varphi}'\right|\leqslant C_4\hat{\delta} q_{n}\log q_{n}\leqslant C_4 q_{n}\log q_{n},\] for \(n\) large enough. Hence \[\begin{align} \left|S_{q_{n}}(\tilde{\varphi}')(x)\right|&\leqslant c_0 q_{n}\log q_{n}+2q_n \left| \left| \psi \right| \right|_{L^{\infty}}+C_4 q_{n}\log q_{n}\leqslant(2c_0+C_4)q_{n}\log q_{n}, \end{align}\] and \[-S_{q_{n}}(\tilde{\varphi}')(x)\geqslant c_0 q_{n}\log q_{n}-2q_n \left| \left| \psi \right| \right|_{L^{\infty}}-C_4 \hat{\delta} q_{n}\log q_{n}\geqslant(c_0/2-C_4\hat{\delta})q_{n}\log q_{n},\] for \(n\) large enough.

Let \(n\) be large enough such that \(\frac{1}{\hat{\delta} q_{n}\log q_{n}}\leqslant\frac{1}{2q_{n}}.\)

For the upper bound of 35 and 36 , consider \(x\) such that \(x_{\min,q_{n}}>({\hat{\delta} q_{n}\log q_{n}})^{-1}\). Then, by the previous inequality, it is enough to take \(\hat{\delta}\) such that \(\hat{\delta}<c_0/(4C_4)\) to get \(S_{q_{n}}(\varphi')(x)=S_{q_{n}}(\tilde{\varphi}')(x)\leqslant-C_4^{-1}\, q_{n}\log q_{n}<0\).

The lower bound of 35 follows from noting that \(x_{\min,q_n}^{+}+x_{\min,q_n}^-\geqslant\left| \left| q_{n-1}\alpha \right| \right|\geqslant\frac{1}{2q_n}\) and \(|{\psi'(x)}|\lesssim|\log x|\) for any \(x\), so that for \(x_{\min,q_{n}}<{\hat \delta}({ q_{n}\log q_{n}})^{-1}\), \[\begin{align} |S_{q_{n}}(\varphi')(x)|&\geqslant\varphi'({x_{\min,q_{n}}})-|S_{q_{n}}(\tilde{\varphi}')(x)|\geqslant\frac{c_0}{x_{\min,q_{n}}}-|{\psi'(x_{\min,q_{n}})}|-8q_n{c_0}-|S_{q_{n}}(\tilde{\varphi}')(x)| \\&\geqslant(c_0/2\hat{\delta}-2c_0-C_4)\, q_{n}\log q_{n}>0, \end{align}\] for \(n\) large enough and taking \(\hat{\delta}\) such that \(\hat{\delta}<c_0(4c_0+2C_4)^{-1}\). ◻

Proof. Let \(y\in\Gamma_{r_n}\) and \(l<q_{r_n}\). Since \(S_{q_{r_n}}(\varphi')(y)=0\), we write \[S_{q_{r_n}}(\varphi')(y-l\alpha)= S_{q_{r_n}}(\varphi')(y-l\alpha)-S_{q_{r_n}}(\varphi')(y).\]

Combining the bound 33 with 28 by applying the mean value theorem, we obtain \[|S_{q_{r_n}}(\varphi')(y-l\alpha)|\leqslant|S_{l}(\varphi'')(\theta)|\left| \left| q_{r_n}\alpha \right| \right| \leqslant\frac{C_2\theta_{\min,q_{r_n}}^{-2}}{q_{r_n+1}}\leqslant\frac{C_2\theta_{\min,q_{r_n}}^{-2}}{q_{r_n}\log q_{r_n}\log\log q_{r_n}},\] for some \(\theta\in(y-l\alpha,y+(q_{r_n}-l)\alpha)\) (or \((y+(q_{r_n}-l)\alpha,y-l\alpha)\)). To show that \(S_{q_{r_n}}(\varphi')\) is continuous on this interval, let \(\hat{\delta}\) be as in Lemma 13. For \(n\) large enough, \[\left| \left| q_{r_n}\alpha \right| \right|\leqslant\frac{1}{q_{r_n}\log q_{r_n}\log\log q_{r_n}}\leqslant\frac{\hat{\delta}}{2q_{r_n}\log q_{r_n}}.\] Therefore \[\frac{\hat{\delta} q_{r_n}\log q_{r_n}}{2}\leqslant q_{r_n}\log q_{r_n}\left(\frac{1}{\hat{\delta}} +\frac{\hat{\delta}}{2}\right)^{-1}\leqslant\theta_{\min,q_{r_n}}^{-1}\leqslant\frac{2q_{r_n}\log q_{r_n}}{\hat{\delta}}.\]

Let \(y\in\Gamma_{r_n}\) be the rightmost point when ordered in \((0,1)\), such that for any \(0\leqslant l<q_{r_n}\), \[y_{\min,q_{r_n}}=y_{\min,q_{r_n}}^-=(y-l\alpha)_{\min,q_{r_n}}^-=(y-l\alpha)_{\min,q_{r_n}}.\] By the convexity result 34 , provided \(q_{r_n}/\log q_{r_n}>2C_3^{-2}\) \[S_{q_{r_n}}(\varphi'')(x)\geqslant C_3\, x_{\min,{l}}^{-2}-C_3^{-1}q_{r_n}\geqslant\tfrac{C_3}{2}\, x_{\min,{l}}^{-2}.\] Therefore if \(y'\in\Gamma_{r_n}\) lies in the same continuity interval as \(y-l\alpha\), then \[\begin{align} |y'-y+l\alpha|&\leqslant\frac{|S_{q_{r_n}}(\varphi')(y-l\alpha)|}{\min_{(y'\wedge(y-l\alpha),y'\vee(y-l\alpha))} S_{q_{r_n}}(\varphi'')}\\ &\leqslant\frac{C_2}{q_{r_n}\log q_{r_n}\log\log q_{r_n}}\left(\frac{2q_{r_n}\log q_{r_n}}{\hat{\delta}}\right)^2 \frac{2}{C_3}\left(\frac{1}{\hat{\delta} q_{r_n}\log q_{r_n}}\right)^2\\ &\leqslant\tilde{C} \left(q_{r_n}\log q_{r_n}\log\log q_{r_n}\right)^{-1}, \end{align}\] for some \(\tilde{C}=\tilde{C}(\varphi,\hat{\delta})>0\).

Let \(y'\in\Gamma_{r_n}\). Using the previous step, \[\begin{align} &|S_{q_{r_n}}(\varphi)(y)-S_{q_{r_n}}(\varphi)(y')| \\&\leqslant|S_{q_{r_n}}(\varphi)(y)-S_{q_{r_n}}(\varphi)(y-l\alpha)|+ |S_{q_{r_n}}(\varphi)(y-l\alpha)-S_{q_{r_n}}(\varphi)(y')| \\&= |S_{l}(\varphi)(y-l\alpha+q_{r_n}\alpha)-S_{l}(\varphi)(y-l\alpha)|+ |S_{q_{r_n}}(\varphi)(y-l\alpha)-S_{q_{r_n}}(\varphi)(y')|. \end{align}\] Both terms are bounded by the mean value theorem, using the estimate 32 and the small distance between \(y-l\alpha\) and both \(y-l\alpha+{q_{r_n}\alpha}\) and \(y'\): \[|S_{l}(\varphi)(y-l\alpha+q_{r_n}\alpha)-S_{l}(\varphi)(y-l\alpha)|\leqslant\left| \left| q_{r_n}\alpha \right| \right|2C_1q_{r_n}\log q_{r_n}/\hat{\delta}=O((\log\log q_{r_n})^{-1}),\] \[|S_{q_{r_n}}(\varphi)(y-l\alpha)-S_{q_{r_n}}(\varphi)(y')|\leqslant|y'-y+l\alpha|\,2C_1q_{r_n}\log q_{r_n}/\hat{\delta}=O((\log\log q_{r_n})^{-1}).\]

Altogether, \(|S_{q_{r_n}}(\varphi)(y)-S_{q_{r_n}}(\varphi)(y')|\) decays asymptotically as \(({\log\log q_{r_n}})^{-1}\) with \(n\), for any \(y,y'\in\Gamma_{r_n}\). Since \(S_{q_{r_n}}(\varphi)\) is asymptotically constant on \(\Gamma_{r_n}\), by definition of \(t_n\) in 37 , for \(n\) large enough we obtain \[t_n < S_{q_{r_n}}(\varphi)(y)+2\gamma/3\quad \text{for all } y\in\Gamma_{r_n}.\] This gives 38 .

To prove 39 , pick \(y\in\Gamma_{r_n}\). Using \(\int_{\mathbb{T}}\varphi=1\), Denjoy–Koksma estimates on the restriction of \(\varphi\) to \([\frac{\hat{\delta}}{q_{r_n}\log q_{r_n}},1-\frac{\hat{\delta}}{q_{r_n}\log q_{r_n}}]\) give \[\begin{align} &\left|S_{q_{r_n}}(\varphi)(y)-q_n\int_\mathbb{T}\varphi\chi_{\{\left| \left| x \right| \right|\geqslant\hat{\delta}(q_{r_n}\log{q_{r_n}})^{-1}\}}\right| \leqslant\text{Var}\left(\varphi\chi_{\{\left| \left| x \right| \right|\geqslant\hat{\delta}(q_{r_n}\log{q_{r_n}})^{-1}\}}\right) \\&\leqslant\varphi(\tfrac{\hat{\delta}}{q_{r_n}\log q_{r_n}})+\varphi(1-\tfrac{\hat{\delta}}{q_{r_n}\log q_{r_n}})+\int_{\hat{\delta}(q_{r_n}\log{q_{r_n}})^{-1}}^{1-\hat{\delta}(q_{r_n}\log{q_{r_n}})^{-1}}\frac{3c_0}{x}dx+\text{Var}(\psi)+ 2\left| \left| \psi \right| \right|_{L^\infty}\\ &=O(\log q_{r_n}), \end{align}\] where we have used that \(\psi\) has bounded variation. Note that the integral on the left hand side \(\to 1\) as \(n\to \infty\), and recall \(\hat{\delta}=\hat{\delta}(\varphi)\), so by definition of \(t_n\), this yields \[t_n\asymp^{\varphi} q_{r_n}.\] ◻

Proof. Conditions 37 –38 together imply that \[\Pi_s\phi_{t_n+s}(\{y\}\times[c,d])\subset[2\gamma/3-\varepsilon+c+s,2\gamma/3+d+s]\subset[\gamma/3+d,\gamma],\] placing the image above \(A\) entirely. ◻

Proof of Theorem 4. We can pick \(\varepsilon>0\) small enough such that for each \(t_n\), with \(n\) large enough, Proposition 9 holds, yielding \[\begin{align} \mu(\phi_{t_n}A\cap \phi_\varepsilon A)-\mu(\phi_{t_n}A\cap A)&=\sum_{i,j=1}^{N_0}\mu(\phi_{t_n}A_i\cap \phi_\varepsilon A_j)-\mu(\phi_{t_n}A_i\cap A_j)\\&\geqslant N_0^2\frac{C_5\varepsilon^4}{\log q_{r_n}}=\frac{C_5(d-c)^2\varepsilon^2}{\log q_{r_n}}. \end{align}\] The result follows from the estimate 39 . ◻

Proof. First, note that \(N(y,t_n+\hat{t}+\varepsilon)=q_{r_n}\) \(\forall y\in\Gamma_{r_n}\), and it is the maximum on \(\mathbb{T}\) by Lemma 15. By the definition of \(N(\cdot,\cdot)\) and 32 , for any \(x\) in the same component of \(F_n'\) as \(y\) \[\begin{align} N(x,t_n+\hat{t}+\varepsilon)&\geqslant N(y,t_n+\hat{t}+\varepsilon)-N(y,(S_{N(x,t_n)}(\varphi)(x)-S_{N(x,t_n+\hat{t}+\varepsilon)}(\varphi)(y)))-1 \\&\geqslant N(y,t_n+\hat{t}+\varepsilon)-\frac{1}{\gamma}(S_{N(x,t_n+\hat{t}+\varepsilon)}(\varphi)(x)-S_{N(x,t_n+\hat{t}+\varepsilon)}(\varphi)(y))-1 \\&\geqslant q_{r_n}-\frac{1}{\gamma}\max_{F_n'}S_{N(x,t_n+\hat{t}+\varepsilon)}(\varphi'){|y-x|}-1\geqslant q_{r_n}-\frac{C_1 q_{r_n}\log q_{r_n}}{\delta}\frac{2}{q_{r_n}}-1 \\&\geqslant q_{r_n}-\frac{2C_1 }{\delta}\log q_{r_n}-1. \end{align}\]

Consider each of the \(q_{r_n}\) segments of \(\mathbb{T}\setminus\cup_{m=0}^{q_{r_n}-1}\{R_\alpha^{-m}0\}\). For each of them, there exist \(k_1\), \(k_2\) such that the interval is \((R_\alpha^{-k_1}0,R_\alpha^{-k_2}0)\). Hence \[x_{\min,q_{r_n}}^+=R_\alpha^{k_1} x, \qquad \text{and} \qquad x_{\min,q_{r_n}}^-=1-R_\alpha^{k_2} x,\] for \(x\) on the segment. In particular, \(k_1\) and \(k_2\) each take every value in \(\{0,\dots,q_{r_n}-1\}\) exactly once. This means that all but at most \(\frac{4C_1 }{\delta}\log q_{r_n}+2\) of the maximal components \(I\subset F_n'\) satisfy \[(x_{\min,q_{n}}^-,x_{\min,q_{n}}^+)=\left(x_{\min,N(x,t_n+ \hat{t})}^+,x_{\min,N(x,t_n+ \hat{t})}^-\right)\] for any \(x\in I\). Hence, the measure of the intervals of \(F_n'\) for which it is not true is at most \[\left(\frac{4C_1 }{\delta}\log q_{r_n}+1\right)\frac{2}{q_{r_n}}\leqslant\frac{\delta}{\log q_{r_n}},\] for \(n\) large enough.

On the other hand, noting that each interval of continuity of \(S_{q_{r_n}}(\varphi)\) has length at least \((2q_{r_n})^{-1}\), we get \[\lambda((a,b))-\lambda((a,b)\cap F_n')\leqslant\left[(b-a){2q_{r_n}}+1\right]\tfrac{2\delta}{q_{r_n}\log q_{r_n}}.\]

The result follows from combining the two inequalities and noting that \[(a,b)\cap F_n'\cap\left\{x\in G_{n,{N(x,t_n+\hat{t}+\varepsilon)}}\right\}\subset F_n( \hat{t}).\] ◻

Proof. i) Fix \(l \leqslant q_{r_n}\). By definition, \(G_{n,l}\) is the union of some of the maximal intervals of \(\mathbb{T} \setminus \bigcup_{m=0}^{q_{r_n}-1} \{R_\alpha^{-m}0\}.\) The bound of the length of the intervals then follows from the three-distance theorem.

ii) Now, for any \(x \in G_{n,l}\), we have \(x_{\min,l} \leqslant\frac{1}{q_{r_n}}\). Hence, if \(q_{r_n}/\log q_{r_n} > C_3^{-2}\), then by 34 , the function \(S_l(\varphi)\) is convex on each interval contained in \(G_{n,l}\). Moreover, by the definition of \(t_n\) (see 37 ), the minimum of \(S_{q_{r_n}}(\varphi)(x)\) on each interval of \(\mathbb{T} \setminus \bigcup_{m=0}^{q_{r_n}-1} \{R_\alpha^{-m}0\}\) is bounded above by \[t_n - \frac{2\gamma}{3} + \varepsilon \leqslant t_n + \hat{t}_1 - \hat{t}_2 - 2\varepsilon,\] and \(S_{q_{r_n}}(\varphi)(x)\geqslant S_{l}(\varphi)(x)\). Therefore, this minimum is not attained in \(F_n(\hat{t}_1,l)\) for any \(l \leqslant q_{r_n}\), so \(S_l(\varphi')\) has constant sign.

iii) As we just showed, in the intervals in \(\{t_n+\hat{t_1}- S_l(\varphi)(x)\in[\hat{t}_2-\varepsilon,\hat{t}_2+2\varepsilon]\}\), \(S_l(\varphi)(x)\) is monotone and attains the maximum \(t_2+2\varepsilon\). However, note that we are only considering points in \((a,b)\cap R_\alpha^{-l}(a,b)\), so that the leftmost and rightmost intervals of \((a,b)\cap R_\alpha^{-l}(a,b)\cap\{t_n+\hat{t_1}- S_l(\varphi)(x)\in[\hat{t}_2-\varepsilon,\hat{t}_2+2\varepsilon]\}\) might get cut (see Figure 2). ◻

Proof. By Remark 8, every \(y \in \Gamma_n\) satisfies \(y^-_{\min,q_n} = y_{\min,q_n}\), i.e. the nearest singularity lies to the right of \(y\). We use this throughout the proof.

Let \(x\in H_n\). Let \(y\) be the zero of \(S_{q_{r_n}}(\varphi')\) on the same interval of continuity of \(x\). Since \(S_{q_{r_n}}(\varphi')(z)<0\) if and only if \(z<y\), we have \(y_{\min,q_{r_n}}^-\leqslant x_{\min,q_{r_n}}^-.\)

Recall that, by 38 , \(t_n<S_{q_{r_n}}(\varphi)(y)+2\gamma/3.\) Hence \[S_{q_{r_n}}(\varphi)(x)-S_{q_{r_n}}(\varphi)(y) <d-c+2\gamma/3< \gamma.\]

From 36 we know that, for \(x_{\min,q_{r_n}}^->(\hat{\delta} q_{r_n}\log q_{r_n})^{-1}\), \(\exists C_4>0\) such that \[S_{q_{r_n}}(\varphi')(x)\leqslant-{C_4}\, q_{r_n}\log q_{r_n}.\] Let \(z<y\) be the point in the same continuity interval of \(x\) and \(y\) such that \(z_{\min,q_{r_n}}^-=(\hat{\delta} q_{r_n}\log q_{r_n})^{-1}\). Then either \[x<z\implies x_{\min,q_{r_n}}^-<(\hat{\delta} q_{r_n}\log q_{r_n})^{-1},\] or \[\begin{align} x_{\min,q_{r_n}}^-&=y_{\min,q_{r_n}}^-+(y-x)=z_{\min,q_{r_n}}^-+(z-x) \\&\leqslant(\hat{\delta} q_{r_n}\log q_{r_n})^{-1} +{\gamma}(C_4 \,q_{r_n}\log q_{r_n})^{-1}, \end{align}\] where we have used \[(z-x)C_4 q_{r_n}\log q_{r_n}\leqslant-\int_x^z S_{q_{r_n}}(\varphi')(u)\,du\leqslant S_{q_{r_n}}(\varphi)(x)-S_{q_{r_n}}(\varphi)(y).\] The existence of \(C_7\) then follows from 34 : \[S_{q_{r_n}}(\varphi'')(x)\geqslant C_3 x_{\min,q_{r_n}}^{-2}-C_3^{-1}q_{r_n}\log q_{r_n}=O((q_{r_n}\log q_{r_n})^2).\]

Finally, recall that, by convexity, each interval of \(H_n\) is nested on a different interval of \(\mathbb{T}\setminus\cup_{m=0}^{q_{r_n}-1}\{R_\alpha^{-m}0\}\), which cover \((a,b)\cap R_\alpha^{-q_{r_n}}(a,b)\) up to a measure zero set and have length bounded by \(2/q_{r_n}\). Therefore the number of intervals of \(H_n\) fully contained in \((a,b)\cap R_\alpha^{-q_{r_n}}(a,b)\) is lower bounded by \[\frac{q_{r_n}}{2}(b-a-\left| \left| \alpha q_{r_n} \right| \right|)-2.\] ◻

Proof of Proposition 9. Without loss of generality, let \(\hat{t}_1,\hat{t}_2\in\{c,c+\varepsilon,\dots,d-\varepsilon\}\), not necessarily different, and consider \(A_1\), \(A_2\) such that \[A_i=(a,b)\times[\hat{t}_i,\hat{t}_i+\varepsilon],\quad i=1,2.\]

Notice that \(A_2\) and \(\phi_\varepsilon A_2\) are stacked one on top of each other.

Let \(N_*\leqslant l\leqslant q_{r_n}.\) First, we claim that for any but (at most) the leftmost and rightmost maximal intervals \(I\subset F_n( \hat{t}_1,l)\) it holds \[\mu(\phi_{t_n}(I\times (\hat{t}_1,\hat{t}_1+\varepsilon))\cap\phi_\varepsilon A_2)\geqslant\mu(\phi_{t_n}(I\times (\hat{t}_1,\hat{t}_1+\varepsilon))\cap A_2).\] It follows from Lemma 17 that the leftmost and rightmost intervals of \(F_n(\hat{t}_1,l)\) may be cut off by the boundary of \((a,b)\), but for the rest of the intervals Lemma 11 applies with \(f(x)=t_n-S_{q_{r_n}}(\varphi)(x)+\hat{t}_1-\hat{t}_2\).

Secondly, note that Lemmas 17 and 16 imply that the portion of the area of \(A_1\) lying on the extreme intervals of all \(F_n(\hat{t}_1,l)\), together with the portion on \((a,b)\setminus F_n(\hat{t}_1)\), is bounded above by \[\frac{4}{q_{r_n}}(q_{r_n}-N_*+1)\varepsilon+ \frac{\delta C_6}{\log q_{r_n}}\varepsilon= \frac{4}{q_{r_n}}\left(\frac{2C_1 }{\delta}\log q_{r_n}+2\right)+ \frac{\delta C_6}{\log q_{r_n}}\varepsilon.\]

Therefore it is enough to show that there exists enough different subintervals \(I\) of \((a,b)\) satisfying \[\hat{t}_2+2\varepsilon=\sup\Pi_s( \phi_{t_n}(I\times \{\hat{t}_1\}) ),\qquad\hat{t}_2-\varepsilon=\inf\Pi_s( \phi_{t_n}(I\times \{\hat{t}_1\}) ),\] \[\mu(\phi_{t_n}(I\times (\hat{t}_1,\hat{t}_1+\varepsilon))\cap\phi_\varepsilon A_2)-\mu(\phi_{t_n}(I\times (\hat{t}_1,\hat{t}_1+\varepsilon))\cap A_2)>\frac{\varepsilon^4}{q_{r_n}\log q_{r_n}}. \label{eq:positive95diff}\tag{40}\] We claim that this holds for a subinterval of every interval of \(H_n\) fully contained in \((a,b)\cap R_\alpha^{-q_{r_n}}(a,b)\). From Lemma 18, this will imply there are at least \((b-a)q_{r_n}/2-3\) of such intervals, and the result will follow from considering \(\delta\) small enough (with respect to \(\varepsilon^3\)) and \(n\) large enough.

Now, let \(J\) be a maximal interval of \(H_n\) fully contained in \((a,b)\cap R_{\alpha}^{-q_{r_n}}(a,b)\), and let \(I\subset J\) be the interval for which \(-\varepsilon\leqslant t_n-S_{q_{r_n}}(\varphi)(x)+\hat{t}_1-\hat{t}_2\leqslant 2\varepsilon.\) Let \(f(x)=t_n-S_{q_{r_n}}(\varphi)(x)+\hat{t}_1-\hat{t}_2\) on \(I\). By Lemmas 12 and 18, \(f\) satisfy the hypotesis of Lemma 9, with \[k=1,\qquad M=q_{r_n}\log q_{r_n},\qquad \text{and}\qquad C= \max\{\hat{\delta}^{-1}C_1,C_7^{-1}\}\] so that Lemma 10 applies. Hence 40 holds for \(I\) and summing over all the intervals gives \[\begin{align} \mu(\phi_{t_n}A_1\cap \phi_\varepsilon A_2)-\mu(\phi_{t_n}A_1\cap A_2)>\left((b-a)\frac{q_{r_n}}{2}-3\right)\frac{\varepsilon^4}{q_{r_n}\log q_{r_n}}- \frac{4}{q_{r_n}}N_*\varepsilon- \frac{\delta C_6}{\log q_{r_n}}\varepsilon \\=\varepsilon^4 O(({\log q_{r_n}})^{-1})-O(\log q_{r_n} q_{r_n}^{-1})-\delta \varepsilon O(({\log q_{r_n}})^{-1}). \end{align}\] Taking \(\delta\) small enough and \(n\) large enough completes the proof. ◻

Proof. The partition \(\mathcal{P}'_{n,k}\) is obtained by placing \(k\) successive \(q_{n}\)-orbits on \(\mathbb{T}\): \[\mathbb{T} \setminus \bigcup_{m=0}^{k-1} \left( \{-l\alpha\}_{l=0}^{q_{n}-1} - m q_{n} \alpha \right).\]

By the three-distance theorem applied to the \(q_{n}\) points \(\{-j\alpha\}_{j=0}^{q_{n}-1}\), the torus \(\mathbb{T}\) is partitioned into intervals of exactly two lengths: \[\|q_{n-1}\alpha\| + \|q_{n}\alpha\| \quad \text{(type A)} \qquad \text{and} \qquad \|q_{n-1}\alpha\| \quad \text{(type B)}.\] We refer to both types collectively as long intervals.

Recall \(q_{2n}\alpha>p_{2n}\) and \(q_{2n+1}\alpha<p_{2n+1}\) for any \(n\). Consequently, when the \(m\)-th orbit \(\{-l\alpha\}_{l=0}^{q_{n}-1} - mq_{n}\alpha\) is added for \(m \geqslant 1\), each of its \(q_{n}\) points lands exactly \(\|q_{n}\alpha\|\) to the left (if \(n\) is even) or the right (if \(n\) is odd) of the corresponding point of the \((m-1)\)-th orbit, thereby cutting a strip of width \(\|q_{n}\alpha\|\) (short interval) from the left end (\(n\) odd) or the right end (\(n\) even) of each long interval as left by the \((m-1)\)-th orbit.

Since \(k \leqslant a_{n+1}\), the total displacement after \(k\) steps satisfies \[k\|q_{n}\alpha\| \leqslant a_{n+1}\|q_{n}\alpha\| < \|q_{n-1}\alpha\|,\] where the last inequality uses the continuant relation \(\|q_{n-1}\alpha\| = a_{n+1}\|q_{n}\alpha\| + \|q_{n+1}\alpha\|\). Hence every cut lands strictly inside a long interval.

Each new orbit (\(m = 1, \ldots, k-1\)) cuts every long interval once, producing \((k-1)q_{n}\) short intervals in total. The \(q_{n}\) remaining intervals are the left-over pieces of the original long intervals after \(k-1\) cuts each.

A type A interval after \(k-1\) cuts of width \(\|q_{n}\alpha\|\), has remaining length \(\|q_{n-1}\alpha\| + \|q_{n}\alpha\| - (k-1)\|q_{n}\alpha\| = \|q_{n-1}\alpha\| - (k-2)\|q_{n}\alpha\|.\) A type B interval after \(k-1\) cuts, has remaining length \(\|q_{n-1}\alpha\| - (k-1)\|q_{n}\alpha\|.\) This completes the proof. ◻

Proof. Note that \[|(R_\alpha^{mq_{n}}x)_{\min,q_{r_n}}^\pm-x_{\min,q_{r_n}}^\pm|=m\left| \left| q_{n}\alpha \right| \right|\leqslant e^{-\log^2q_{n}}\tfrac{1}{q_{n+1}}\leqslant\tfrac{1}{2} x_{\min,q_{r_n}}^\pm,\] for \(n\geqslant 2\), where we have used \(q_{n+1}\geqslant e^{3\log^2q_{n}}q_n\) for every \(n\). ◻

Proof. From Lemma 20, for \(\text{dist}(x,\partial I)\geqslant{\left| \left| q_{n}\alpha \right| \right|}^{1/2}\), it holds \[\tfrac{1}{2}x_{\min,q_{r_n}}^\pm\leqslant(R_\alpha^{mq_{n}}x)_{\min,q_{r_n}}^\pm \leqslant\tfrac{3}{2}x_{\min,q_{r_n}}^\pm,\] for any \(1\leqslant m\leqslant k\). Hence we can apply Lemma 12 to \(\{R_\alpha^{k_0 q_n+m} x\}_{m=0}^{l_0-1}\) and to each \(q_{n}\)-sub-orbit of \(\{R_\alpha^{m} x\}_{m=0}^{k_0q_n-1}\), and 42 –43 follow.

Note that 44 and Lemma 20 imply 45 after renaming the constant \(C_3'\). To prove 44 , consider \(l\geqslant q_{n}\). Then \[S_{l}(\varphi'')(x)\geqslant S_{l_0}(\varphi'')(R_\alpha^{k_0q_{n}}x)+ \sum_{m=0}^{k_0-1} S_{q_n}(\varphi'')(R_\alpha^{mq_{n}}x).\] We can bound each term using Lemma 12. Since the minimum distance to the singularity of each sub-orbit is upper bounded by \(q_{n}^{-1}\), such \(C_3'\) exists for \(k_0\geqslant 1\). ◻

Proof. Fix \(I\in\mathcal{P}_{n,k}\). The function \(S_{kq_n}(\varphi)(x)\) is \(C^2\) on \(I\) and, by 44 , \(S_{kq_n}(\varphi')\) is strictly monotone and has exactly one zero. Note that then 46 follows from 47 .

The argument is the same as Lemma 13 with two differences. First, the Denjoy–Koksma inequality is applied to \(\varphi'\) restricted away from singularities at a slightly smaller scale rather to include each of the \(k\) sub-orbits. Second, the bounds have an extra factor \(k\).

Fix \(1>\hat{\delta}>0\) and let \(\tilde{\varphi}'\) denote the restriction of \(\varphi'\) to \([\frac{1}{2\hat{\delta} q_{n}\log q_{n}},1-\frac{1}{2\hat{\delta} q_{n}\log q_{n}}]\). Applying the Denjoy–Koksma inequality to \(\tilde{\varphi}'\), we get that there exists \(C_4'=C_4'(\varphi)>{4}/{c_0}\) such that \[\begin{cases} \left|S_{q_{n}}(\tilde{\varphi}')(x)\right|\leqslant(2c_0+C_4')q_{n}\log q_{n},\\ -S_{q_{n}}(\tilde{\varphi}')(x)\geqslant(c_0/2-C_4'\hat{\delta})q_{n}\log q_{n}, \end{cases}\] for \(n\) large enough.

Let \(n\) be large enough such that \(\frac{1}{\hat{\delta} q_{n}\log q_{n}}\leqslant\frac{1}{2q_{n}}.\)

If \(x_{\min,q_{n}}\geqslant({\hat{\delta} q_{n}\log q_{n}})^{-1}\), from Lemma 20 we have, for \(n\) large enough, \(x_{\min,kq_n}>(2{\hat{\delta} q_{n}\log q_{n}})^{-1}\). Therefore, \[-S_{kq_n}(\varphi')(x)=-S_{kq_n}(\tilde{\varphi}')(x)\geqslant k(c_0/2-C_4'\hat{\delta})q_{n}\log q_{n}.\]

For \(x_{\min,q_{n}}^-<{\hat \delta}({ q_{n}\log q_{n}})^{-1}\), \[S_{{kq_n}}(\varphi')(x)\geqslant\sum_{l=0}^{k-1}\left[\tfrac{c_0}{2}(R_\alpha^{lq_{n}}x)_{\min,q_{n}}^{-1} -4q_nc_0-O(|\log (R_\alpha^{lq_{n}}x)_{\min,q_{n}}|)\right]-k(2c_0+C_4')q_{n}\log q_{n},\] and, for \(x_{\min,q_{n}}^+<{\hat \delta}({ q_{n}\log q_{n}})^{-1}\), \[\begin{align} S_{{kq_n}}(\varphi')(x)&\leqslant\sum_{l=0}^{k-1}\left[-{2c_0}(R_\alpha^{lq_{n}}x)_{\min,q_{n}}^{-1} +4q_nc_0+O(|\log (R_\alpha^{lq_{n}}x)_{\min,q_{n}}|)\right]\\&+k({c_0}/{2}-C_4'\hat{\delta})q_{n}\log q_{n}. \end{align}\]

If \(e^{-2\log^2q_{n}}<x_{\min,q_{n}}^\pm<{\hat \delta}({ q_{n}\log q_{n}})^{-1}\), Lemma 20 again implies \[\max_{l\in\{0,\dots,k-1\}}(R_\alpha^{lq_{n}}x)_{\min,q_{n}}^\pm<2x_{\min,q_{n}}^\pm,\] and if \(e^{-2\log^2q_{n}}\geqslant x_{\min,q_{n}}^\pm,\)\[\max_{l\in\{0,\dots,k-1\}}(R_\alpha^{lq_{n}}x)_{\min,q_{n}}^\pm\leqslant x_{\min,q_{n}}^\pm+l\left| \left| \alpha q_n \right| \right|\leqslant e^{-2\log^2q_{n}}+q_n^{-1}e^{-2\log^2q_{n}}\leqslant(2\hat{\delta} q_{n}\log q_{n})^{-1},\]for \(n\) large enough. Applying this to the previous inequalities and taking \(\hat{\delta}\) small enough gives 47 . ◻

Proof. We will use the same ideas as in the proof of Lemma 14.

Let \(I\in\mathcal{P}_{n,k}\) be the rightmost interval of \(\mathcal{P}_{n,k}\) when ordered in \((0,1)\), and let \(\{y\}=I\cap\Gamma_{n,k}\) Every point in the orbit \(\{y-l\alpha\}_{l=0}^{q_n-1}\) lies in an interval of \(\mathcal{P}_{n,k}\). This is clear by looking at the proof of Lemma 19 and noting that the left and the right boundary points of the elements of \(\mathcal{P}_{n,k}\) form two \(q_n\)-orbits. By the same reason, for \(0\leqslant l \leqslant{q_n-1}\), \[(y-l\alpha)_{\min,q_n}^-=y_{\min,q_n}^-=\left| \left| y \right| \right|,\quad (y-l\alpha)_{\min,kq_n}^-=y_{\min,kq_n-l}^-=y_{\min,kq_n}^-.\]

Since \(S_{kq_n}(\varphi')(y)=0\), we write \[S_{kq_n}(\varphi')(y-l\alpha)= S_{kq_n}(\varphi')(y-l\alpha)-S_{kq_n}(\varphi')(y).\]

Let \(n\) be large enough such that \(\left| \left| kq_n\alpha \right| \right|\leqslant\frac{k}{q_{n+1}}\leqslant({q_{n}e^{2\log^2q_{n}}})^{-1}\leqslant\frac{\hat{\delta}}{4q_{n}\log q_{n}}.\) We can combine the bound 43 with 41 by applying the mean value theorem. We obtain \[|S_{kq_n}(\varphi')(y-l\alpha)|\leqslant|S_{l}(\varphi'')(\theta)|\left| \left| kq_n\alpha \right| \right| \leqslant\frac{C_2'k\,\theta_{\min,kq_n}^{-2}}{q_{n+1}}\leqslant\frac{C_2'\theta_{\min,kq_n}^{-2}}{q_{n}e^{2\log^2q_{n}}},\] for some \(\theta\in(y-l\alpha,y-l\alpha+kq_n\alpha)\) or \((y-l\alpha+kq_n\alpha,y-l\alpha)\).

Therefore, using 46 and the bound on \(\left| \left| kq_n\alpha \right| \right|\), we have \[e^{- 2\log ^2 q_n}<\frac{\hat{\delta} q_{n}\log q_{r_n}}{4}\leqslant q_{n}\log q_{n}\left(\frac{2}{\hat{\delta}} +\frac{\hat{\delta}}{4}\right)^{-1}\leqslant\theta_{\min,kq_n}^{-1}\leqslant\frac{4q_{n}\log q_{n}}{\hat{\delta}}.\]

Again, following the steps of Lemma 14, by the convexity result 45 , if \(y'\in\Gamma_{n,k}\) lies in the same interval of \(\mathcal{P}_{n,k}\) as \(y-l\alpha\), then \[\begin{align} |y'-y+l\alpha|&\leqslant\frac{C_2'}{q_{n}e^{2\log^2q_{n}}}\left(\frac{4q_{n}\log q_{n}}{\hat{\delta}}\right)^2 \frac{1}{C_3'}\left(\frac{2}{\hat{\delta} q_{n}\log q_{n}}\right)^2\leqslant\tilde{C} \left(q_{n}e^{2\log^2q_{n}}\right)^{-1}, \end{align}\] for some \(\tilde{C}(\varphi,\hat{\delta})>0\). Thus, by 42 and 41 , the main value gives \[\begin{align} |S_{q_{r_n}}(\varphi)(y')-S_{q_{r_n}}(\varphi)(y-l\alpha)|&\leqslant\tilde{C} \left(q_{n}e^{2\log^2q_{n}}\right)^{-1}C_1'k\frac{2q_{n}\log q_{n}}{\hat{\delta}}=O(\log q_{n}e^{-\log^2q_{n}}), \end{align}\] and \[|S_{kq_n}(\varphi)(y)-S_{{kq_n}}(\varphi)(y-l\alpha)|\leqslant\left| \left| kq_n\alpha \right| \right|C_1'\frac{2q_{n}\log q_{n}}{\hat{\delta}}= O(\log q_{n}e^{-\log^2q_{n}}).\]

Altogether, \(|S_{q_{r_n}}(\varphi)(y)-S_{q_{r_n}}(\varphi)(y')|\) decays asymptotically with \(n\), for any \(y,y'\in\Gamma_{r_n}\). So, by definition of \(t_n\), for \(n\) large enough we obtain \[t_n < S_{q_{r_n}}(\varphi)(y)+3\varepsilon\quad \text{for all } y\in\Gamma_{r_n}.\] ◻

Proof. The factor \(\sqrt{k}\) in the denominator of the lower bound reflects the fact that, at scale \(k\), the convexity of \(S_{kq_n}(\varphi)\) is of order \(k q_n^2 \log^2 q_n\) while the first derivative bound is of order \(\sqrt{k} q_n \log q_n\). Applying Lemmas 9 and 10 then gives the \(1/\sqrt{k}\) decay.

We first claim that there exists \(C_5'=C_5'(\varphi)>0\) such that \(\forall x \in H_{n,k}\) \[|S_{kq_n}(\varphi')(x)|\leqslant C_5' \sqrt{k}\, q_{n}\log q_{n},\] for \(n\) large enough, for every \(1 \leqslant k \leqslant e^{\log^2q_{n}}\).

Indeed, fix \(I\in\mathcal{P}_{n,k}\) and let \(x\in I\cap H_{n,k}\), \(\{y\}=I\cap \Gamma_{n,k}\). First, notice that \(x\) is to the left of \(y\), and by the same argument as in the proof of Lemma 18, the estimates 46 and 47 imply \[e^{-\log^2 q_n}<\hat{\delta}\,(2{\, q_{n}\log q_{n}})^{-1}\leqslant x_{\min,kq_n}^- \leqslant 2\,({\hat{\delta}\, q_{n}\log q_{n}})^{-1}+{\gamma}\,(C_4' \,k \,q_{n}\log q_{n})^{-1}.\] Hence \(x_{\min,kq_n}=x_{\min,kq_n}^-\) and it follows from 43 and 45 that \[\tilde{C}^{-1}\, k\, q_{n}^2\log ^2q_{n} \leqslant S_{kq_n}(\varphi'')\leqslant\tilde{C}\, k\, q_{n}^2\log ^2q_{n}\] on \((x,y)\), for some \(\tilde{C}(\varphi)>0\). Now, the Taylor expansion about \(y\) of \(f=S_{kq_n}(\varphi)\) yields \[\gamma\geqslant f(x)-f(y)=f''(\theta)\tfrac{(y-x)^2}{2}\geqslant\tfrac{1}{2}\tilde{C}^{-1}\, k\, q_{n}^2\log ^2q_{n}\,(y-x)^2,\] for some \(\theta\in(x,y)\). Therefore \[f'(x)=f''(\theta)(x-y)\geqslant-\tilde{C}\, k\, q_{n}^2\log ^2q_{n}\,\sqrt{\frac{2\tilde{C}\gamma}{kq_{n}^2\log ^2q_{n}}}=:-C_5' \sqrt{k}\, q_{n}\log q_{n}.\]

Let \(I\in\mathcal{P}_{n,k}\) and let \(J(t)= I\cap\{t_{n,k}+t-S_{kq_n(\varphi)}(x)\in(-\varepsilon,2\varepsilon)\}\cap H_{n,k}\). From the definition of \(t_{n,k}\) and \(H_{n,k}\), \(J(t)\) is an interval satisfying \[\inf_{J(t)}[ t_{n,k}+t-S_{kq_n(\varphi)}(x)]=-\varepsilon,\quad\sup_{J(t)}[ t_{n,k}+t-S_{kq_n(\varphi)}(x)]=2\varepsilon.\]

By the estimate we just showed we can apply Lemmas 9 and 10 to \(f(x)=t_{n,k}+t-S_{kq_n}(\varphi)(x)\) with \(k\), \(M=q_{n}\log q_{n}\), and \(C=\max\{C_5',\tilde{C}\}\). We get \[\mu\left[\phi_{t_{n,k}+t}(J(t)\times[0,\varepsilon])\cap \phi_\varepsilon A\right]-\mu\left[\phi_{t_{n,k}+t}(J(t)\times[0,\varepsilon])\cap A\right]>\frac{\varepsilon^4}{\sqrt{k}\,q_{n}\log q_{n}},\] for every \(t\in[0,\varepsilon]\). Summing over the \(q_{n}\) intervals of \(\mathcal{P}_{n,k}\) gives the result. ◻

Proof. Note that \(N(y,t_{n,k})=kq_n\) \(\forall y\in\Gamma_{n,k}\) and, by 42 and the MVT, \[\begin{align} N(x,t_{n,k})&\geqslant N(y,t_{n,k})-\frac{1}{\gamma}(S_{N(x,t_{n,k})}(\varphi)(x)-S_{N(x,t_{n,k})}(\varphi)(y))-1\\&\geqslant kq_n-\frac{2C_1' k}{\delta}\log q_{n}-2\geqslant(k-1)q_n, \end{align}\] for any \(x\) in the same component of \(F_{n,k}'\) as \(y\) and for \(n\) large enough, where we have used \(\frac{\delta}{kq_{n}\log q_{n}}\geqslant\frac{\delta}{q_{n}\log q_{n}}e^{-\log^2 q_n}\geqslant e^{-2\log^2 q_n}\). ◻

Proof. From the definition of \(F_{n,k}\), 44 applies. Therefore, similarly to Lemma 17, on any \(I\in \mathcal{P}_{n,k}\) we have that, for any \(t>0\), if \(\min_{I} S_l(\varphi)<t-\gamma,\) the set \(\{x\in I:-\varepsilon<t-S_l(\varphi)<2\varepsilon\},\) is either empty or the union of at most two disjoint intervals, one in which \(S_l(\varphi')<0\) and one in which \(S_l(\varphi')>0\). Moreover, on both intervals \(\inf S_l(\varphi)=t-2\varepsilon\).

From our choice of \(\varepsilon\), \(t\), \(H_{n,k}\) contains exactly the interval with \(l=kq_n\) and \(S_l(\varphi')<0\). Therefore for \(l<kq_n\) all these intervals are contained in \(H_{n,k}'\), as well as the interval for which \(l=kq_n\) and \(S_l(\varphi')>0\). The convexity of \(S_l(\varphi)\) in \(F_{n,k}\) allows us to apply Lemma 11 to each of these intervals. This gives the first result.

To prove 50 , recall from Lemma 25 that \(F_{n,k}'\subset F_{n,k}\), which implies \[\lambda (\mathbb{T}\setminus F_{n,k})=q_n\frac{2\delta}{k\,q_n\log q_n}+(k-1)q_n\left| \left| q_n\alpha \right| \right|\leqslant\frac{2\delta}{k\,\log q_n}+\frac{k q_n}{q_{n+1}}\leqslant\frac{\varepsilon^3}{2\sqrt{k}\log q_{n}},\] for \(0<\delta<\delta^*\), \(\delta^*(\varepsilon)>0\) small enough, and \(n\) large enough, where we have used 41 to bound the second term of the sum: \(\frac{kq_n}{q_{n+1}} \leqslant e^{-2\log^2 q_n}\).

Therefore, putting together these results and Lemma 24, and noting \[A=[H_{n,k}'\cup H_{n,k}\cup (\mathbb{T}\setminus F_{n,k})]\times[0,\varepsilon],\] the result 50 holds. ◻

Proof of Theorem 3. Let \(\alpha\) satisfy 41 and consider \(N\) given by 26. Then, for \(n\geqslant N\), \(2 \leqslant k\leqslant e^{\log^2q_{n}}\), and \(t\in[0,\varepsilon]\), 50 holds. Noting \(\mu(A)=\mu(\phi_\varepsilon A)\) and writing 50 as a difference of correlations, we get that at least one of the following hold \[|\text{Cor}_{t_{n,k}+t-\varepsilon}(\chi_A,\chi_{A})|\geqslant\frac{\varepsilon^4}{4\sqrt{k}\log q_{n}},\quad \text{or}\quad |\text{Cor}_{t_{n,k}+t}(\chi_A,\chi_{A})|\geqslant\frac{\varepsilon^4}{4\sqrt{k}\log q_{n}}.\] Squaring and integrating over \(t \in [0,\varepsilon]\) gives \[\int_{t_{n,k}-\varepsilon}^{t_{n,k}+\varepsilon} |\mathrm{Cor}_t(\chi_A,\chi_A)|^2\,dt \geqslant\frac{\varepsilon^9}{16k\log^2 q_n}.\]

To see that the intervals \([t_{n,k} - \varepsilon, t_{n,k} + \varepsilon]\) are pairwise disjoint, note that by definition we can bound the difference \(t_{n,k+1} - t_{n,k}\) below by the minimum of \(S_{q_n}(\phi)\). Therefore \[t_{n,k+1} - t_{n,k} \geqslant\gamma q_n - 2\varepsilon \geqslant 2\varepsilon,\] for all \(1 \leqslant k \leqslant e^{\log^2 q_n}\). Since \(k \leqslant e^{\log^2 q_n} \ll a_{n+1}\), the same applies to \(t_{n+1,1} - t_{n,k}\) for any such \(k\).

Since the intervals \([t_{n,k}-\varepsilon, t_{n,k}+\varepsilon]\) are pairwise disjoint over all \((n,k)\) with \(n \geqslant N\) and \(2 \leqslant k \leqslant e^{\log^2 q_n}\), summing gives \[\int_0^\infty |\mathrm{Cor}_t(\chi_A,\chi_A)|^2\,dt \geqslant\sum_{n \geqslant N} \sum_{k=2}^{e^{\log^2 q_n}} \frac{\varepsilon^9}{16k\log^2 q_n} \geqslant\sum_{n \geqslant N} \frac{\varepsilon^9}{32} = \infty,\] where we used \(\sum_{k=2}^{K} k^{-1} \geqslant(\log K)/2\). This completes the proof. ◻

Proof. Recall from Section 2.2 that \(\Phi : X \to \mathcal{M}'\) is a measure-preserving isomorphism intertwining \((\phi_t)\) with \((\phi_t^{\mathcal{M}})\). It is defined by \(\Phi(x,0) \in \Sigma\) (the base transversal, parametrized by the local Hamiltonian \(H\)) and \(\Phi(x,s) = \phi_s^{\mathcal{M}}(\Phi(x,0))\) for \(x\in\mathbb{T}\) and \(0 \leqslant s\). At a smooth point \((x,s)\) with \(x \notin D_0\), the differential of \(\Phi\) satisfies \[d\Phi_{(x,s)}(\partial_s) = W(\Phi(x,s)), \qquad d\Phi_{(x,s)}(\partial_x) = J(x,s),\] where \(W\) is the Hamiltonian vector field generating \((\phi_t^{\mathcal{M}})\) and \[J(x,s) :=\partial_\epsilon\big|_{\epsilon=0}\phi_s^{\mathcal{M}}(\Phi(x+\epsilon,0))\] is the transverse variation field.

At each \(p \in \mathcal{M}'\) with \(W(p)\neq 0\), define the orthonormal frame \[e_1(p) :=\frac{W(p)}{\|W(p)\|}, \qquad e_2(p) :=e_1(p)^\perp,\] where \(e_2\) is the unit vector orthogonal to \(e_1\) with \(\omega(e_1,e_2)=1\). We decompose the variation field in this frame (see Figure 3) \[J(x,s) = \alpha(x,s)\,W(\Phi(x,s)) + \beta(x,s)\,e_2(\Phi(x,s))= \alpha(x,s)\|W\|\,e_1 + \beta(x,s)\,e_2.\] Since \(\Phi\) is area-preserving and \(\Sigma\) is parametrized by \(H\) with \(dH = \omega(W,\cdot)\), we have at \(s = 0\), \[\omega(W(\Phi(x,0)),J(x,0)) = \omega\bigl( W(\Phi(x,0)),\partial_x\Phi(x,0)\bigr) = dH\bigl(\partial_x\Phi(x,0)\bigr) = 1,\] and, by area preservation \((\phi_s^{\mathcal{M}})^*\omega = \omega\), for \(s > 0\) it holds \[\omega(W(\Phi(x,s)),J(x,s)) = \omega(W(\Phi(x,0)),J(x,0)) = 1.\] Substituting the decomposition of \(J\) we get \(\beta(x,s)= 1/\|W\|\): \[1 = \omega\left(\|W\|\,e_1,\alpha\|W\|\,e_1 + \beta\,e_2\right) = \beta\|W\|\,\omega(e_1, e_2) = \beta\|W\|.\] The variation field is therefore \[J(x,s) = \alpha(x,s)\|W\|\,e_1 + \frac{1}{\|W\|}\,e_2.\]

Applying the the chain rule to \(f = \tilde{f} \circ \Phi\) yields \[\begin{align} \partial_s f &= d\tilde{f}(W) = \|W\|\,d\tilde{f}(e_1), \\ \partial_x f &= d\tilde{f}(J) = \alpha\|W\|\,d\tilde{f}(e_1) + \frac{1}{\|W\|}\,d\tilde{f}(e_2) = \alpha\,\partial_s f + \frac{1}{\|W\|}\,d\tilde{f}(e_2). \end{align}\] Inverting, we get \[d\tilde{f}(e_1) = \frac{\partial_s f}{\|W\|}, \qquad d\tilde{f}(e_2) = \|W\|\bigl(\partial_x f -\alpha(x,s)\,\partial_s f\bigr).\] Since \(\{e_1,e_2\}\) is orthonormal, we obtain the pointwise bound \[\|d\tilde{f}(p)\| \leqslant|d\tilde{f}(e_1)| + |d\tilde{f}(e_2)| \leqslant\frac{|\partial_s f|}{\|W\|} + \|W\|\,|\partial_x f| + |\alpha(x,s)|\,\|W\|\,|\partial_s f|.\] ◻

Proof. Recall that \(\alpha(x,s)\) is the longitudinal component of the variation field \(J(x,s) = \partial_\epsilon|_{\epsilon=0}\, \phi_s^{\mathcal{M}}(\Phi(x+\epsilon,0))\), measuring the drift in the flow direction between the orbits of \(x\) and \(x + \epsilon\) at time \(s\).

Let \(N=N(x,s)\) be the number of complete returns to \(\Sigma\) before time \(s\) and define \[\alpha_{\mathrm{exc}}(x,s) := \alpha(x,s) - \alpha(x, S_{N}(\varphi)(x)),\] to be the increment during the incomplete excursion starting from \(\Phi(T^Nx,0)\).

Differentiating in \(\epsilon\) and applying the chain rule, we get \[\begin{align} J(x,s) &= \partial_\epsilon|_{\epsilon=0}\,\phi_{s}^{\mathcal{M}}(\Phi(x+\epsilon,0))= \partial_\epsilon|_{\epsilon=0}\,\phi_{s - S_N(\varphi)(x+\epsilon)}^{\mathcal{M}}(\Phi(T^N(x+\epsilon),0)) \\ &= d\phi_{s-S_N(\varphi)(x)}^{\mathcal{M}}(\Phi(T^Nx,0))\left[\partial_\epsilon|_{\epsilon=0}\,\Phi(T^N(x+\epsilon),0)\right]- S_N(\varphi')(x)\cdot W(\Phi(x,s)), \end{align}\] where the first term is the variation field of \(\phi_{s-S_N(\varphi)(x)}^{\mathcal{M}}\) started from the point \(\Phi(T^N(x),0) \in \Sigma\), and the second term accounts for the shift in elapsed time, which displaces the orbit by \(-S_N(\varphi')(x)\) units along the flow direction \(W\). Projecting along the flow direction, the first term on the right contributes \(\alpha_{\mathrm{exc}}(x,s)\) (the variation coefficient of the incomplete excursion starting from \(\Phi(T^Nx,0)\)), while the second term contributes \(-S_N(\varphi')(x)\), giving \[\label{eq:alpha95decomp95explicit} \alpha(x,s) = \alpha\left(T^Nx,s-S_N(\varphi)(x)\right)-S_N(\varphi')(x).\tag{53}\]

Since the iterates \(T^k x\) for \(k = 0, \ldots, N-1\) all satisfy \(\mathrm{dist}(T^k x, D_0) \geqslant\delta\) by 11 , and since \(T\) acts by translation on each continuity interval, the iterates falling within distance \(r \geqslant\delta\) of any fixed \(y \in D_0\) are mutually separated by at least \(\delta\). This well-separation (cf.Lemma 3) gives that each singularity of \(\varphi'\) contributes at most \(O(\delta^{-1} \log N)\) to the Birkhoff sum. Summing over the finitely many points in \(D_0\) yields \[|S_{N(x,s)}(\varphi')(x)| \lesssim \frac{\log N(x,s)}{\delta}.\]

For the incomplete excursion, first note that \(\Phi[0,1]=\Sigma\) is a closed curve, so \(\alpha(\cdot,0)\) is uniformly bounded by a constant depending only on the choice of \(\Sigma\) and the geometry of \(\mathcal{M}'\). More generally, outside fixed neighborhoods of the saddles \(p_i\), the vector field \(W\) is bounded away from zeros, and trajectories can spend only a uniformly bounded amount of time outside these neighborhoods before returning to \(\Sigma\), generating only bounded contributions to the variation field. The dominant contributions to \(\alpha(x,s)\) arise from passages near the saddle points.

For any fixed point \(p_i\in \mathcal{M}'\), let \(\tau_i(h) \sim -C_i \log h\) be the passage time through a fixed ball around \(p_i\) at energy level \(h\) (cf.[16]). Since \(\alpha_{\mathrm{exc}}\) measures the cumulative variation of the passage times of the orbit segment \(\{\Phi(x,\sigma)\}_{\sigma\in[S_N(\varphi)(x),\,s]}\) with respect to the initial condition on \(T^N x \in \Sigma\), we have \[|\alpha_{\mathrm{exc}}| \lesssim \max_i\left|\frac{d}{dx}\tau_i(h(x))\right| = \max_i \left|\tau_i'(h)\frac{dh}{dx}\right| \lesssim \max_i |\tau_i'(h)| \sim \max_i\frac{C_i}{h} \lesssim \frac{1}{\delta},\] where we used that \(dh/dx\) is uniformly bounded, that the piece of orbit intersects each neighborhood at energy level \(h \gtrsim \mathrm{dist}(T^N x, D_0) \geqslant\delta\) by 11 , and that each excursion passes near each singularity at most once (cf. [5]).

Combining 53 with the two bounds above, and using \((\log N)/\delta \geqslant 1/\delta\) for \(N\) large enough, we get \[|\alpha(x,s)| \lesssim \frac{\log N(x,s)}{\delta}.\] Substituting \(N(x,s) \leqslant(\log t)^{1+\nu}\) and \(\delta = (\log t)^{-1-\nu}\) from 11 : \[|\alpha(x,s)| \lesssim \frac{(1+\nu)\log\log t}{(\log t)^{-(1+\nu)}} = (\log t)^{1+\nu} \log\log t.\] ◻

Proof. Since \(\mathcal{M}'\) is compact and \(W\) is a smooth vector field, \(\|W\|\) is bounded above by some constant \(C_W > 0\) on all of \(\mathcal{M}'\).

For the lower bound, near a non-degenerate saddle \(p_i\) there exist local coordinates \((u,v)\) centered at \(p_i\) in which \(H(u,v) = uv\) and \(W \asymp u\,\partial_u - v\,\partial_v\) (see [16]), so on a level set \(H = h\) one has by the AM-GM inequality \(\|W(p)\| \gtrsim \sqrt{h}\) for \(p\) on the level set \(H = h.\)

The transversal \(\Sigma\) is parametrized by \(H\), so a point \(x \in \mathbb{T}\) at distance \(\rho = \mathrm{dist}(x, D_0)\) from the nearest singularity corresponds to a level set with \(h \gtrsim \rho\). Since level sets are preserved by the flow, for all \(s \in [0, \varphi(x))\) \[\|W(\Phi(x,s))\| \gtrsim \sqrt{\mathrm{dist}(x, D_0)} .\] By 11 , \(\mathrm{dist}(T^k x, D_0) \geqslant\delta := (\log t)^{-1-\nu}\) for all \(x \in [a,b]\) and \(0\leqslant k \leqslant N(x, s)\), giving \(\|W(\Phi(x,s))\| \gtrsim \delta^{1/2} = (\log t)^{-(1+\nu)/2}\) on the support. ◻

Proof of Proposition 2. We combine Lemmas 27, 28 and 29 by substituting the bounds from 10 , 52 , and 54 into 51 .

From \(\|\partial_s f\|_\infty = O((\log t)^{-1+\nu})\) and \(\left| \left| W \right| \right|\gtrsim(\log t)^{-(1+\nu)/2}\) we get \[\frac{|\partial_s f|}{\|W\|} \lesssim \frac{(\log t)^{-1+\nu}}{(\log t)^{-(1+\nu)/2}} = (\log t)^{(1+\nu)/2 - 1} = (\log t)^{(3\nu-1)/2}.\]

Since \(\|W\| \leqslant C_W\) uniformly on \(\mathcal{M}'\) and \(\|\partial_x f\|_\infty = O((\log t)^{1+\nu})\), we have \[\|W\|\,|\partial_x f| = O((\log t)^{1+\nu}).\]

Lastly, 52 , \(\|W\| \leqslant C_W\), and \(\|\partial_s f\|_\infty = O((\log t)^{-1+\nu})\) give \[|\alpha|\,\|W\|\,|\partial_s f| \lesssim (\log t)^{1+\nu}\log\log t \cdot C_W \cdot (\log t)^{-1+\nu} = O((\log t)^{2\nu} \log\log t).\] For \(\nu<1\), the three terms are \(O((\log t)^{1+\nu}).\) Together with \(\|\tilde{f}\|_\infty = \|f\|_\infty=1\), we get the desired control on the \(C^1\) norm \(\|\tilde{f}\|_{C^1(\mathcal{M}')} = \|\tilde{f}\|_\infty + \|d\tilde{f}\|_\infty = O((\log t)^{1+\nu}).\) ◻

Proof of Proposition 3 (Condition 16 ). Fix \(\pi \in \mathfrak{R}\) and pick \(\underline{\tau_*}\in \Theta_\pi\) such that the associated geometric height vector \(\hat{\underline{h_*}}= -\Omega_\pi \underline{\tau_*}\) is strictly positive and set \(\kappa=2\max_{i,j} \hat{h_*}_i/\hat{h_*}_j\). Let \(\{\delta_m\}\subset\mathbb{R}^+\) be a decreasing sequence, \(\delta_m\to0\), and let \(\underline{\lambda_m}=(1-\tfrac{\delta_m}2,\tfrac{\delta_m}{2(d-1)},\cdots,\tfrac{\delta_m}{2(d-1)})\).

For each \(m\), define the normalized \(\underline{\tau_m}={\underline{\tau_*}}{\text{Area}(\underline{\lambda_m}, \pi, \underline{\tau_*})}^{-1}\) so that the triple \((\underline{\lambda_m}, \pi, \underline{\tau_m})\) lies in \(\widehat{\mathcal{M}}_d(\mathfrak{R})\); and let \(K_m \subset \widehat{\mathcal{M}}_d(\mathfrak{R})\) be a compact neighborhood of \((\underline{\lambda_m}, \pi, \underline{\tau_m})\) on which both \(\kappa\)-balance of \(\hat{\underline{h}}\) and \(\lambda_1 > (1-\delta_m)\) hold. Since \(\widehat{\nu}_{\mathfrak{R}}\) is equivalent to Lebesgue on each fiber, \(K_m\) has positive \(\widehat{\nu}_{\mathfrak{R}}\)-measure.

By ergodicity of \(\widehat{\nu}_{\mathfrak{R}}\), for a.e. \((\underline\lambda, \pi, \underline\tau)\) there exists an increasing sequence \((r_m)\) such that \(\hat{\mathcal{Z}}^{r_m}(\underline\lambda, \pi, \underline\tau)\in K_m\), for \(m\geqslant 1\). The \(\kappa\)-balance of \(\hat{\underline{h}}\) implies \(\eta\)-balance of the Rokhlin tower heights \((h_1^{(r_m)},\dots,h_d^{(r_m)})\), for some constant \(\eta(\kappa)\) independent of \(m\). Moreover, knowing \[\lambda^{(r_m)}-\sum_{j\neq 1}\lambda_j^{(r_m)}=\lambda_1^{(r_m)}>(1-\delta_m) \lambda^{(r_m)},\] and the partition identity \(\sum_j \lambda_j^{(r_m)} h_j^{(r_m)} = 1\), we get \[\sum_{j\neq 1}\lambda_j^{(r_m)} <\delta_m\lambda^{(r_m)},\quad\text{ and }\quad h_1^{(r_m)} <((1-\delta_m)\lambda^{(r_m)})^{-1}.\] Therefore \[1 - \lambda\bigl(\mathcal{T}_1^{(r_m)}\bigr) = \sum_{j \neq 1} \lambda_j^{(r_m)} h_j^{(r_m)} <\delta_m\lambda^{(r_m)} \cdot \frac{\eta}{(1-\delta_m)\lambda^{(r_m)}} = \frac{\delta_m\eta}{1-\delta_m}.\]

Projecting into \(\mathcal{M}_d(\mathfrak{R})\), we get that the former holds for almost every \((\underline{\lambda},\pi)\). Given any \(\varepsilon>0\), we restrict the subsequence to \(m\) large enough so that \(0<\delta_m<1/4\) and \(\delta_m\eta/(1-\delta_m) < \varepsilon\), yielding \({\lambda_{j^*}^{(r_m)}} > \tfrac{3}{4}{\lambda^{(r_m)}}\) and \(\lambda(\mathcal{T}_1^{(r_m)}) > 1-\varepsilon\). ◻