Primitive Averages, Directional Expansivity,
and Quantitative Twisted Recurrence for Ergodic \(\mathbb{Z}^d\)-Actions
June 11, 2026
We prove two new results about probability preserving actions \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\). First, for a function \(f\in L^2(\mu)\), we provide an explicit formula for the \(L^2(\mu)\)-limit of the average \[\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v \in Q_N^\mathcal{P}} T_v f\] where \(\mathcal{P}\subset \mathbb{Z}^d\) is the set of primitive vectors, i.e. those for which the greatest common divisor of its components is \(1\), and \(Q_N^\mathcal{P}= [-N,N]^d\cap \mathcal{P}\). Second, for a set \(A\subset X\) with \(\mu(A)>0\), we provide a spectral condition under which the set of \(\varepsilon\)-expansive directions \[\left\{ v\in \mathbb{Z}^d \, : \, \mu\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\] has lower density very close to \(1\). As an application of our techniques we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Björklund, Fish and the first author [1].
Fix an integer \(d\geqslant 2\). Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be a probability preserving system1 and denote \(Q_N=[-N,N]^d\cap \mathbb{Z}^d\). For any \(f\in L^2(\mu)\), von Neumann’s mean ergodic theorem implies that \[\label{eq:32MET} \lim_{N\to \infty} \frac{1}{\left\lvert Q_N\right\rvert}\sum_{v \in Q_N} T_v f = P_{\mathcal{I}}f\tag{1}\] in \(L^2(\mu)\) where \(P_\mathcal{I}\) denotes the \(L^2(\mu)\)-projection onto the subspace of \(T\)-invariant functions \[\mathcal{I}= \{f\in L^2(\mu) \, : \, T_v f = f \text{ for all }v\in \mathbb{Z}^d\}.\] Our first result gives an explicit analogue of the formula in equation 1 when the averages are taken only over the primitive vectors \(\mathcal{P}\subset \mathbb{Z}^d\), where a non-zero vector \(v\in \mathbb{Z}^d\) is called primitive if the greatest common divisor of its non-zero components is equal to \(1\).
Unlike equation 1 , the limiting value for the primitive averages also depends on how \(f\) correlates with eigenfunctions of finite order, and the strength of this dependence is determined by the arithmetic distribution of primitive lattice points in residue classes. We denote \(Q_N^\mathcal{P}:=Q_N\cap \mathcal{P}\).
Theorem 1. Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be a probability preserving action and let \(f\in L^2(\mu)\). Then \[\label{eq:32Formula32for32primitive32MET32lim} \lim_{N\to \infty} \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v \in Q_N^\mathcal{P}} T_v f = P_{\mathcal{I}}f + \sum_{q\geqslant 2} \frac{\boldsymbol{\mu}(q)}{J_d(q)} P_{q}f,\qquad{(1)}\] where \(\boldsymbol{\mu}: \mathbb{Z}_{>0}\to \mathbb{R}\) is the Möbius function2, \(J_d: \mathbb{Z}_{>0}\to \mathbb{R}\) is the Jordan totient function \[J_d(q): = \left\lvert\left\{ a=(a_1,\ldots,a_d)\in (\mathbb{Z}/q\mathbb{Z})^d\, : \, \gcd(a,q) = 1\right\}\right\rvert,\] and \(P_q\) is the orthogonal projection onto the \(L^2(\mu)\) subspace generated by eigenfunctions3 of order \(q\).
In particular, the limit always exists and depends only on the projection of \(f\) to the rational Kronecker factor, i.e. the closed subspace generated by eigenfunctions of finite order. Since \(J_d(q)\to \infty\) as \(q\to \infty\), the contribution of the order-\(q\) eigenfunctions in ?? becomes negligible for large \(q\).
If the system is totally ergodic, i.e. if \(\mathbb{Z}^d\) and all of its finite index subgroups act ergodically, then the formula collapses to the expected constant limit. For \(E\subset \mathbb{Z}^d\), we define the lower densities of \(E\) with respect to \(Q_N\) and \(Q_N^\mathcal{P}\) by \[\underline{d}_{Q_N}\left(E \right):= \liminf_{N\to \infty}\frac{\left\lvert E\cap Q_N\right\rvert}{\left\lvert Q_N\right\rvert} \qquad\text{and}\qquad \underline{d}_{Q_N^\mathcal{P}}\left(E \right):= \liminf_{N\to \infty}\frac{\left\lvert E\cap Q_N^\mathcal{P}\right\rvert}{\left\lvert Q_N^\mathcal{P}\right\rvert}\] respectively.
Corollary 1. Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be a totally ergodic system and let \(f\in L^2(X)\). Then \[\lim_{N\to \infty} \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v \in Q_N^\mathcal{P}} T_v f = \int_X f \,d\mu.\] Moreover, if \(A\subset X\) has \(\mu(A)>0\), then for every \(\varepsilon>0\) we have that \[\underline{d}_{Q_N^\mathcal{P}} \left( \left\{v \in \mathcal{P} \, : \, \mu(A\cap T_v A) > \mu(A)^2 - \varepsilon \right\} \right)>0.\]
Our second result concerns cyclic subgroups of \(\mathbb{Z}^d\) along which the orbit of a positive-measure set almost covers the whole space. Given \(A\subset X\) with \(\mu(A)>0\), we say that a vector \(v\in \mathbb{Z}^d\) is an \(\varepsilon\)-expansive direction for \(A\) if \[\mu\left(\bigcup_{n\in \mathbb{Z}}T_{nv} A\right)>1-\varepsilon.\] Expansive directions were first studied by Björklund and Fish [2] where they played a central role in their study of simplicies in large subsets of \(\mathbb{Z}^d\), and have since found further applications in [3] and [1].
The existence of expansive directions depends on how the spectral measure of \(A\) is distributed on points of finite order. Recall that the spectral measure \(\sigma_A\) of \(A\) with respect to \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) is the unique finite Borel measure on \(\mathbb{T}^d\) satisfying that \[\label{eq:32Def32of32spec32measure} \mu(A\cap T^{v}A)= \int_{\mathbb{T}^d} e(v\cdot\alpha)\, d\sigma_A(\alpha) \quad \text{for every }v\in \mathbb{Z}^d.\tag{2}\] The order \(\mathrm{ord}(\alpha)\) of a point \(\alpha \in \mathbb{T}^d\) is defined to be the smallest positive integer \(n\) for which \(n\alpha =0\) if such an integer exists, and \(\infty\) otherwise. In [2] Björklund and Fish proved in that if \[\sigma_A(\{\alpha \in \mathbb{T}^d \, : \, 1<\mathrm{ord}(\alpha)<\infty\})<\varepsilon \mu(A)^2,\] then the set of \(\varepsilon\)-expansive directions for \(A\) is non-empty4. Our second theorem strengthens this conclusion from non-emptiness to large lower density under a weaker spectral assumption. Indeed, instead of requiring that \(\sigma_A\) gives small mass to the infinite set of all points with order less than \(\infty\), we only need to control the mass on the finitely many points of order at most \(M\) for some constant \(M>0\).
Theorem 2. For every \(\delta,\varepsilon,\eta>0\), there exist \[M=M(\delta,\varepsilon,\eta)>0 \qquad\text{and}\qquad \kappa = \kappa(\delta,\varepsilon,\eta)>0\] such that the following is true. For any probability preserving action \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)\geqslant\delta\), if \[\sigma_A(\{\alpha \in \mathbb{T}^d\, : \, 1< \mathrm{ord}(\alpha) \leqslant M\})<\kappa,\] then \[\underline{d}_{Q_N}\left(\left\{ v\in \mathbb{Z}^d \, : \, \mu\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\right)> 1-\eta.\]
Not every set in a probability preserving system satisfies the spectral assumption in Theorem 2. However, when \(T\) acts ergodically, the ergodic measure increment argument from [3] allows one to pass to a bounded finite-index subaction and an ergodic component for which the assumption is satisfied, and so we have the following corollary to Theorem 2.
Corollary 2. For every \(\delta,\varepsilon,\eta>0\), there exists some positive integer \(k_0 = k_0(\delta,\varepsilon,\eta)\) such that the following is true. For every ergodic action \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) and every \(A\subset X\) with \(\mu(A)\geqslant\delta\), there exists an integer \(1\leqslant k \leqslant k_0\) and an ergodic component \(\nu\) of \(\mu\) with respect to the sub-action of \(k\mathbb{Z}^d\) such that \(\nu(A)\geqslant\mu(A)\) and \[\underline{d}_{Q_N}\left(\left\{ v\in \mathbb{Z}^d \, : \, \nu\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\} \right) > 1-\eta.\]
Using Theorems 15 and 2 we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Björklund, Fish and the first author. We first recall their theorem.
Theorem 1 ([1]). For any ergodic probability preserving action \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)>0\), there exists a positive integer \(k=k(A)\) such that for any \(d\)-linearly independent vectors \(v_1, \ldots, v_d\in \mathbb{Z}^d\), there exists some \(\gamma \in \mathrm{SL}_d(\mathbb{Z})\) with \[\mu\left( A\cap T_{\gamma k v_1}A\cap \ldots \cap T_{\gamma k v_d}A\right)>0.\]
Theorem 1 is non-quantitative in the sense that the integer \(k\) may depend on the set \(A\), rather than only on \(\mu(A)\). One might hope to strengthen it by ensuring that \(k\leqslant k_0\) for some \(k_0\) depending only on \(\mu(A)\), however it turns out that this is not possible as the following example shows.
Example 2. For every positive integer \(k_0\), there exists an ergodic system \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) and a set \(A\subset X\) with \(\mu(A)=2^{-d}\) such that for every \(k=1,\ldots,k_0\), there exists a vector \(v_k\in \mathbb{Z}^d\) for which \[\mu(A\cap T_{\gamma kv_k}A)=0 \quad \text{for every }\gamma \in \mathrm{SL}_{d}(\mathbb{Z}).\]
In light of Example 2, any quantitative variant of Theorem 1 must require some further restriction on the configurations \(v_1,\ldots,v_d\) for which the conclusion is satisfied. Indeed, restricting ourselves to only those configurations for which \(\left\lvert\det(v_1,\ldots,v_d)\right\rvert\) is bounded, we prove the following quantitative variant of Theorem 1.
Theorem 3. For any \(D\in \mathbb{Z}_{>0}\) and any \(\delta>0\), there exists a positive integer \(k_0=k_0(D,\delta)\) such that for every ergodic action \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) and every \(A\subset X\) with \(\mu(A)\geqslant\delta\), there exists an integer \(1\leqslant k\leqslant k_0\) such that the following holds. For any \(v_1,\ldots,v_d \in \mathbb{Z}^d\) with \[0< \left\lvert\det(v_1,\ldots,v_d)\right\rvert\leqslant D,\] there exists some \(\gamma \in \mathrm{SL}_d(\mathbb{Z})\) with \[\mu\left( A\cap T_{\gamma k v_1}A\cap \ldots \cap T_{\gamma k v_d}A\right)>0.\]
Unlike the proof of Theorem 1 in [1], our proof of Theorem 3 does not use random walk theory, and in particular does not rely on the deep equidistribution results of [4]. Our methods could also be used to give a new proof Theorem 1 which would remove the reliance on random walks entirely, but we have decided not to pursue that direction here, leaving the details to the motivated reader.
By a routine application of Furstenberg’s correspondence principle, Theorem 3 has the following combinatorial consequence. Recall that the upper Banach density of \(E\subset \mathbb{Z}^d\) is \[d^*(E):=\limsup_{N\to \infty} \sup_{v \in \mathbb{Z}^d} \frac{\left\lvert E\cap (v+ [0,N-1]^d)\right\rvert}{N^d}.\]
Corollary 3. For any \(D\in \mathbb{Z}_{>0}\) and any \(\delta>0\), there exists a positive integer \(k_0=k_0(D,\delta)\) such that for every \(E\subset \mathbb{Z}^d\) with \(d^*(E)\geqslant\delta\), there exists an integer \(1\leqslant k\leqslant k_0\) such that for any \(v_1,\ldots,v_d \in \mathbb{Z}^d\) with \[0< \left\lvert\det(v_1,\ldots,v_d)\right\rvert\leqslant D,\] there exists some \(v_0\in E\) and \(\gamma \in \mathrm{SL}_d(\mathbb{Z})\) with \[v_0 + k\gamma v_i \in E \qquad \text{for all }i=1,\ldots,d.\]
Asymptotic notation. All asymptotic notation is taken as \(N\to\infty\). We write \(o_N(1)\) for any quantity which tends to \(0\) as \(N\to\infty\), and write \(a_N=O(b_N)\) if there is a constant \(C>0\), independent of \(N\), such that \(|a_N|\leqslant C|b_N|\) for all sufficiently large \(N\).
Acknowledgments. We are grateful to Michael Björklund and Alexander Fish for their guidance and encouragement. S.S. is particularly thankful to Michael Björklund and Chalmers University for their hospitality in June 2025, when this work began. S.S. was supported by the Australian Research Council through grant DP240100472.
We begin by recalling some relevant background material. Let \(d\) be a positive integer and let \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) be a probability preserving action.
Given any \(f\subset L^2(\mu)\) the spectral measure \(\sigma_f\) of \(f\) with respect to \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) is the unique finite Borel measure on \(\mathbb{T}^d\) with \[\label{lhzquxac} \int_X f \cdot T_v \overline{f}\, d\mu= \int_{\mathbb{T}^d} e(v\cdot\alpha)\, d\sigma_f(\alpha) \quad \text{for every }v\in \mathbb{Z}^d,\tag{3}\] where \(e(x):=\exp(2\pi i x)\) and \(\cdot\) is the standard dot product. If \(A\subset X\) then we write \(\sigma_A\) for \(\sigma_{\mathbf{1}_A}\). We always have that \(\sigma_A(\{0\})\geqslant\mu(A)^2\), with equality in the case that \(T\) acts ergodically6.
The order \(\mathrm{ord}(\alpha)\) of a point \(\alpha \in \mathbb{T}^d\) is defined to be the smallest positive integer \(n\) such that \(n\alpha = 0\) in \(\mathbb{T}^d\) if such an integer exists, and \(\infty\) otherwise. For \(M>0\) we denote \[\mathrm{Rat}(M): = \{\alpha \in \mathbb{T}^d\, : \, 1< \mathrm{ord}(\alpha) \leqslant M\}.\]
For any \(\alpha \in \mathbb{T}^d\), a non-zero function \(f\in L^2(\mu)\) is called an \(\alpha\)-eigenfunction or an eigenfunciton with eigenvalue \(\alpha\) if \[T_v f = e(v\cdot \alpha) f \qquad\text{for all }v\in \mathbb{Z}^d\] and we denote the closed \(L^2(\mu)\) subspace spanned by all \(\alpha\)-eigenfunctions by \(\mathrm{Eig}_T(\alpha)\). The order of an \(\alpha\)-eigenfunction is defined to be \(\mathrm{ord}(\alpha)\) and for each positive integer \(q\) we write \[\mathcal{K}_q := \bigoplus_{\substack{\alpha \in \mathbb{T}^d \\ \mathrm{ord}(\alpha)=q}} \mathrm{Eig}_T(\alpha)\] for the subspace generated by eigenfunctions of order \(q\). The rational Kronecker factor is then defined to be \[\mathcal{K}_\text{rat}: = \overline{\bigoplus_{q\geqslant 1} \mathcal{K}_q}.\]
We will also make use of the fact that \(\mathcal{P}\) has positive density in \(\mathbb{Z}^d\) with respect to \(Q_N\), a proof of which can be found in [5]. More precisely, \[\label{eq:32Primitive32density32exists} \frac{\left\lvert Q_N^\mathcal{P}\right\rvert}{\left\lvert Q_N\right\rvert} = \frac{1}{\zeta(d)} + o_N(1)\tag{4}\] where \(\zeta\) is the Riemann zeta function. For more details, see Section 10, where in particular we also provide a proof of equation 4 .
As the notions of primitive vectors and expansive directions are slightly degenerate in dimension \(1\), we have only stated our results in the introduction for \(d\geqslant 2\). We remark however that the analogous version of Theorem 3 for \(d=1\) is still true, and in fact follows easily from Poincaré recurrence. In particular no ergodicity assumption is required. Indeed, the dimension \(d=1\) analogue of Theorem 3 states that for any positive integer \(D\) and any \(\delta>0\), there exists a positive integer \(k_0 = k_0(D,\delta)\) such that for any probability preserving action \(T: \mathbb{Z}\curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)\geqslant\delta\), there exists some \(1\leqslant k\leqslant k_0\) such that \[\mu(A\cap T_{km}A)>0 \quad \text{for all } m=1,\ldots,D.\] Applying Poincaré recurrence to the \(\mathbb{Z}\) action of \(T\times T^2 \times \ldots \times T^D\) on \((X^D,\mu^{\otimes^D})\) and the set \(B = A\times\ldots \times A\subset X^D\) yields the desired result as the first non-trivial return time can always be bounded in terms of \((\mu^{\otimes^D}(B))^{-1}\), which of course is at most \(\delta^{-D}\).
For the remainder of the paper let us fix a dimension \(d\geqslant 2\), with the understanding that all constants, explicit and implied, will in general depend on the dimension \(d\).
In this section we prove Theorem 1 using the following exponential-sum formula as a black box, and we use it to deduce Corollary 1.
Proposition 3. For any \(\alpha \in \mathbb{T}^d\) we have that \[g(\alpha): = \lim_{N\to \infty}\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^{\mathcal{P}}} e(v\cdot \alpha)= \begin{cases} 1 & \text{ if } \alpha=0\\ \frac{1}{J_d(q)}\boldsymbol{\mu}(q) & \text{ if } \mathrm{ord}(\alpha)=q\geqslant 2\\ 0 & \text{ if }\mathrm{ord}(\alpha)=\infty\\ \end{cases}\] where \(\boldsymbol{\mu}: \mathbb{Z}_{>0}\to \mathbb{R}\) is the Möbius function and \(J_d: \mathbb{Z}_{>0}\to \mathbb{R}\) is the Jordan totient function \[J_d(q) = \left\lvert\left\{ a=(a_1,\ldots,a_d)\in (\mathbb{Z}/q\mathbb{Z})^d\, : \, \gcd(a,q) = 1\right\}\right\rvert.\]
The proof of Proposition 3 is purely number-theoretic and is postponed to Section 10, so that the ergodic-theoretic details can be presented without interruption.
Proof of Theorem 1 via Proposition 3. Fix some probability preserving action \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) and for any positive integers \(N\) and \(q\) denote \[A_N:= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^\mathcal{P}} T_v \qquad \text{and} \qquad c_q:=\frac{\boldsymbol{\mu}(q)}{J_d(q)}.\] Denote by \(L: L^2(\mu) \to L^2(\mu)\) the proposed limit operator \[Lf := \sum_{q=1}^\infty c_q P_q f \qquad \text{for any } f\in L^2(\mu)\] where \(P_q\) is orthogonal projection onto \(\mathcal{K}_q\). It is easy to see that \(L\) is a linear operator, and in fact it is an \(L^2(\mu)\) contraction since by pairwise orthogonality of each \(\mathcal{K}_q\) and the fact that each \(\left\lvert c_q\right\rvert\leqslant 1\), for any \(f\in L^2(\mu)\) we can estimate \[\left\lVert Lf\right\rVert_2^2 = \sum_{q=1}^\infty \left\lVert c_q P_q f\right\rVert_2^2 \leqslant\sum_{q=1}^\infty \left\lVert P_q f\right\rVert_2^2 \leqslant\left\lVert f\right\rVert_2^2.\] We must show that \[\label{eq:32AN32f32goes32to32Lf} \lim_{N\to \infty} A_{N}f = L f \qquad \text{for any } f\in L^2(\mu).\tag{5}\] First consider the case that \(h\in L^2(\mu)\) is orthogonal to \(\mathcal{K}_\text{rat}\). Then for each \(\alpha \in \mathbb{T}^d\), equation 2 together with the mean ergodic theorem applied to the unitary action of \(\overline{e(v\cdot \alpha)}T_v\) implies7 that \(\left\lVert P_{\mathrm{Eig}_T(\alpha)}h\right\rVert_2^2 = \sigma_h(\{\alpha\})\) where \(P_{\mathrm{Eig}_T(\alpha)}\) is the orthogonal projection onto \(\mathrm{Eig}_T(\alpha)\). As \(h\perp \mathcal{K}_\text{rat}\) then we must have that \(\sigma_h(\mathbb{Q}^d/\mathbb{Z}^d)=0\). We can then calculate \[\begin{align} \left\lVert A_N h\right\rVert_2^2 = \left\lVert\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^{\mathcal{P}}} T_v h\right\rVert_2^2 &= \left \langle \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^{\mathcal{P}}} T_v h ,\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{w\in Q_N^{\mathcal{P}}} T_w h \right\rangle\\ &=\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert^2} \sum_{v,w\in Q_N^{\mathcal{P}}} \left \langle T_{v-w} h, h\right \rangle \\ &= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert^2} \sum_{v,w\in Q_N^{\mathcal{P}}} \int_{\mathbb{T}^d} e((v-w)\cdot \alpha) \, d\sigma_h(\alpha)\\ &= \int_{\mathbb{T}^d} \left\lvert\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^{\mathcal{P}}} e(v\cdot \alpha)\right\rvert^2 d\sigma_h(\alpha). \end{align}\] By Proposition 3 and the dominated convergence theorem then \[\label{eq:32prim32avg32kills32Krat32perp} \lim_{N\to \infty} \left\lVert A_N h\right\rVert_2^2 = \int_{\mathbb{T}^d} \left\lvert g(\alpha)\right\rvert^2 d\sigma_h(\alpha) = 0\tag{6}\] where in the final equality we use that \(g(\alpha)=0\) for \(\mathbb{T}^d\) with \(\mathrm{ord}(\alpha)=\infty\) and that \(\sigma_h(\mathbb{Q}^d/\mathbb{Z}^d)=0\). Clearly \(Lh = 0\), and so we have that equation 5 holds on \(\mathcal{K}_{\text{rat}}\).
Now let \(\alpha \in \mathbb{T}^d\) have \(\mathrm{ord}(\alpha)=q<\infty\) and let \(f_\alpha \in \mathrm{Eig}_T(\alpha)\). As \(\mathrm{Eig}_T(\alpha)\subset \mathcal{K}_q\) then \(L f_\alpha = c_q f_\alpha\) so we can calculate \[\begin{align} \left\lVert A_N f_{\alpha} - L f_{\alpha}\right\rVert_2^2 &= \left\lVert\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^{\mathcal{P}}} T_v f_\alpha - c_q f_\alpha\right\rVert^2_2\\ &= \left\lVert\left(\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^{\mathcal{P}}} e(v\cdot \alpha)-\frac{\boldsymbol{\mu}(q)}{J_d(q)}\right)f_\alpha\right\rVert_2^2\\ &\leqslant\left\lvert\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^{\mathcal{P}}} e(v\cdot \alpha)-\frac{\boldsymbol{\mu}(q)}{J_d(q)}\right\rvert^2 \left\lVert f_\alpha\right\rVert_2^2 \end{align}\] which goes to \(0\) as \(N\to \infty\) by Proposition 3 and so equation 5 also holds for any eigenfunction of finite order. By linearity of \(L\), the desired formula also holds for finite linear combinations of finite order eigenfunctions. Since finite linear combinations of finite order eigenfunctions are dense in \(\mathcal{K}_\text{rat}\), then by a standard density argument using that \(A_N\) and \(L\) are both \(L^2(\mu)\) contractions we conclude that equation 5 also holds for any \(f\in \mathcal{K}_{\text{rat}}\). Since \(L^2(\mu) = \mathcal{K}_{\text{rat}} \oplus \mathcal{K}_{\text{rat}}^\perp\) then we are done. ◻
Proof of Corollary 1. Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be totally ergodic. It is easy to see that the only finite order eigenfunctions are constant a.e., and so for any \(f\in L^2(\mu)\) we have that \(P_q f = 0\) for all \(q\geqslant 2\). By Theorem 1 and ergodicity of \(T\) we then have that \[\lim_{N\to \infty}\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^\mathcal{P}} T_v f = P_{\mathcal{I}} f = \int_X f \, d\mu.\] Now suppose \(f =\mathbf{1}_A\) for some \(A\subset X\) with \(\mu(A)>0\). Then by continuity of the inner product and the previous equation we have that \[\begin{align} \lim_{N\to \infty}\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^\mathcal{P}} \mu(A\cap T_v A)& = \left\langle \lim_{N\to \infty}\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^\mathcal{P}}T_v \mathbf{1}_A,\mathbf{1}_A\right\rangle \nonumber \\ &= \mu(A)^2. \label{eq:32avg32of32A32cap32Tv32A32for32TE} \end{align}\tag{7}\] For any \(\varepsilon>0\) we must then have that \[\underline{d}_{Q_N^\mathcal{P}} (\{ v\in \mathcal{P} \, : \, \mu(A\cap T_v A)>\mu(A)^2 - \varepsilon\})>0\] since otherwise we have a contradiction to equation 7 . ◻
In this section we prove Theorem 2.
Lemma 4. Let \(\alpha \in \mathbb{T}^d\) and consider the homomorphism \(\phi_\alpha : \mathbb{Z}^d \to S^1\) taking \(v \mapsto e(v\cdot \alpha)\). Then \[\label{eq:32index32of32kernel32is32order} [ \mathbb{Z}^d : \ker \phi_\alpha] = \mathrm{ord}(\alpha)\qquad{(2)}\] and \[\label{eq:32density32of32kernel32is32147order} d_{Q_N}(\ker \phi_\alpha) := \lim_{N\to \infty} \frac{\left\lvert\ker \phi_\alpha \cap Q_N\right\rvert}{\left\lvert Q_N\right\rvert} = \frac{1}{[\mathbb{Z}^d : \ker \phi_\alpha]}\qquad{(3)}\] where \(1/\infty : = 0\).
Proof. We start with equation ?? . First consider the case that \(\mathrm{ord}(\alpha)=\infty\). Suppose that \([\mathbb{Z}^d : \ker \phi_\alpha] < \infty\). By the first isomorphism theorem \(\phi_\alpha(\mathbb{Z}^d) \cong \mathbb{Z}^d / \ker \phi_\alpha\) must be finite so there exists a positive integer \(n\) such that \(\phi_\alpha(v)^n = 1\) for all \(v\in \mathbb{Z}^d\). But this means that \(n\alpha \cdot v = 0\) in \(\mathbb{T}^d\) for all \(v \in \mathbb{Z}^d\), contradicting that \(\mathrm{ord}(\alpha) = \infty\). Now if \(\mathrm{ord}(\alpha)=q\) then there exist integers \(0\leqslant a_1, \ldots,a_d \leqslant q-1\) with \(\gcd(a_1,\ldots,a_d,q)=1\) such that \(\alpha = (a_1,\ldots,a_d)/q\). It follows that \[\phi_\alpha(\mathbb{Z}^d) \subset \{ e(a/q) \, : a \in \mathbb{Z}/ q \mathbb{Z}\}\] and so \(|\phi_\alpha(\mathbb{Z}^d)|\leqslant q\). On the other hand, Bézout’s identity ensures there exists \(v_1,\ldots,v_d,t \in \mathbb{Z}\) such that \((v_1,\ldots,v_d)\cdot(a_1,\ldots,a_d) + tq =1\), i.e. \(\sum_{i=1}^d v_i a_i \equiv 1 \pmod{q}\) and so \(e(1/q) \in \phi_\alpha(\mathbb{Z}^d)\) which implies that \(|\phi_\alpha(\mathbb{Z}^d)|\geqslant q\). By the first isomorphism theorem then \[[\mathbb{Z}^d : \ker \phi_\alpha] = \left\lvert\mathbb{Z}^d / \ker \phi_\alpha\right\rvert = \left\lvert\phi_\alpha(\mathbb{Z}^d)\right\rvert=q.\]
We now prove equation ?? . Clearly if \(H\leqslant\mathbb{Z}^d\) has infinite index then \(d_{Q_N}(H) =0\), since otherwise for \(\{v_i\}_{i=1}^\infty = \mathbb{Z}^d / H\) we can write \[\begin{align} d_{Q_N}(\mathbb{Z}^d) &= d_{Q_N}\left( \bigsqcup_{i=1}^\infty H+v_i\right)\\ &\geqslant d_{Q_N}\left( \bigsqcup_{i=1}^M H+v_i\right)\\ &= M d_{Q_N}\left( H\right) \to \infty \text{ as }M\to \infty. \end{align}\] If \(H\leqslant\mathbb{Z}^d\) has \([\mathbb{Z}^d : H ] = q\), then similarly \[1 = d_{Q_N} (\mathbb{Z}^d) = d_{Q_N} \left( \bigsqcup_{i=1}^q H+v_i \right) = q d_{Q_N} (H)\] which proves the lemma. ◻
For a vector \(v\in \mathbb{Z}^d\) we define its annihilator \(L_v^\perp \subset \mathbb{T}^d\) to be \[L_v^\perp : = \left \{ \alpha \in \mathbb{T}^d \, : \, v\cdot \alpha = 0 \text{ in }\mathbb{T}^d \right \}.\]
Lemma 5. For \(\alpha \in \mathbb{T}^d\) consider \[f(\alpha) :=\lim_{N\to \infty} \frac{1}{|Q_N|} \sum_{v \in Q_N} \mathbf{1}_{L_v^\perp}(\alpha).\] Then for every \(\alpha \in \mathbb{T}^d\), \(f(\alpha)\) exists and equals \(1/\mathrm{ord}(\alpha)\), where \(1/\infty : = 0\).
Proof. Note that for fixed \(\alpha\), \(\mathbf{1}_{L_v^\perp}(\alpha) = \mathbf{1}_{\ker\phi_\alpha}(v)\), so by Lemma 4 we have that \[f(\alpha) = d_{Q_N}(\ker \phi_\alpha) = \frac{1}{\mathrm{ord}(\alpha)}.\] ◻
We will also need the follow essential fact from [2].
Lemma 6 ([2]). Given an probability preserving action \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\), a set \(A\subset X\) with \(\mu(A)>0\) and a vector \(v\in \mathbb{Z}^d\), we have that \[\mu \left(\bigcup_{n \in \mathbb{Z}} T_{nv} A\right) \geqslant\frac{\sigma_A(\{0\})}{\sigma_A(L_v^\perp)}.\]
Proof of Theorem 2. Fix some \(\delta,\varepsilon,\eta>0\). Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be a probability preserving action and suppose \(A\subset X\) has \(\mu(A)\geqslant\delta\). Suppose that \(\sigma_A(\mathrm{Rat}(M))<\kappa\) for \(M\) and \(\kappa\) to be determined later. By Lemma 5 and the dominated convergence theorem we have that \[\begin{align} \lim_{N\to \infty} \frac{1}{\left\lvert Q_N\right\rvert}\sum_{v\in Q_N} \sigma_A(L_{v}^\perp) & = \lim_{N\to \infty}\int_{\mathbb{T}^d} \frac{1}{\left\lvert Q_N\right\rvert}\sum_{v\in Q_N} \mathbf{1}_{L_{v}^\perp}(\alpha)\, d \sigma_A(\alpha) \nonumber \\ &=\sigma_A(\{0\}) + \sum_{q=2}^\infty \frac{1}{q} \sigma_A(\{\alpha\in \mathbb{T}^d \, : \, \mathrm{ord}(\alpha)=q\}). \label{eq:32Lv32perp32average} \nonumber \end{align}\tag{8}\] Set \(Y(v): = \sigma_A(L_v^\perp) - \sigma_A(\{0\})\geqslant 0\). The previous equation is then equivalent to \[\lim_{N\to \infty} \frac{1}{\left\lvert Q_N\right\rvert}\sum_{v\in Q_N} Y(v) = \underbrace{\sum_{q=2}^\infty \frac{1}{q} \sigma_A(\{\alpha\in \mathbb{T}^d \, : \, \mathrm{ord}(\alpha)=q\})}_{:=L(A)}.\] For \(t>0\) consider \[E(t) = \{ v\in \mathbb{Z}^d \, : \, Y(v)\geqslant t\}.\] By Markov’s inequality we have that \(\overline{d}_{Q_N}(E(t))\leqslant L(A)/t\) which is equivalent to \[\label{eq:32Markov32on32E40t4132compliment} \underline{d}_{Q_N}(E(t)^c) \geqslant 1-\frac{L(A)}{t}.\tag{9}\] By definition, any \(v\in E(t)^c\) has that \(\sigma_A(L_v^\perp)<\sigma_A(\{0\}) + t\), and so by Lemma 6 must satisfy that \[\mu\left(\bigcup_{n\in \mathbb{Z}}T_{nv}A\right) > 1-\frac{t}{\sigma_A(\{0\})+t} > 1-\varepsilon,\] where the last inequality will hold provided that we pick \(t\) small enough in terms of \(\sigma_A(\{0\})\) and \(\varepsilon\). In particular, we can take \(t = \sigma_{A}(\{0\})\varepsilon\) so that \[\left\{v \in \mathbb{Z}^d \, : \, \mu\left(\bigcup_{n\in \mathbb{Z}}T_{nv}A\right)>1-\varepsilon\right \} \supset E(t)^c.\] By equation 9 then \[\begin{align} \underline{d}_{Q_N} \left(\left\{v \in \mathbb{Z}^d \, : \, \mu\left(\bigcup_{n\in \mathbb{Z}}T_{nv}A\right)>1-\varepsilon\right \}\right) &\geqslant\underline{d}_{Q_N}(E(t)^c) \nonumber \\ &\geqslant 1-\frac{L(A)}{\sigma_A(\{0\})\varepsilon}.\label{eq:32ineq32with32LA} \end{align}\tag{10}\] Now pick \[M = \frac{2}{\delta^2 \varepsilon \eta} \qquad \text{and} \qquad \kappa = \frac{\delta^2 \varepsilon \eta}{2}.\] Then \[\begin{align} L(A) &= \sum_{q=2}^{\infty} \frac{1}{q}\sigma_A(\{\alpha\in \mathbb{T}^d \, : \, \mathrm{ord}(\alpha)=q\}) \nonumber \\ &\leqslant\sigma_A(\mathrm{Rat}(M)) + 1/M \nonumber \\ &<\delta^2 \varepsilon \eta \leqslant\sigma_{A}(\{0\}) \varepsilon \eta, \label{eq:32LA32bound} \end{align}\tag{11}\] where in the last inequality we use that \(\delta^2\leqslant\mu(A)^2 \leqslant\sigma_A(\{0\})\). The conclusion then follows from equations 10 and 11 . ◻
We now show how Corollary 2 follows from Theorem 2 via the ergodic measure increment argument introduced in [3]. We recall the relevant background.
Proposition 7 (\(T^k\)-ergodic components [6]). Let \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) act ergodically. For any positive integer \(k\) there exist finitely many \(k\mathbb{Z}^d\)-invariant and ergodic probability measures \(\nu_1,\ldots,\nu_n\) with disjoint supports such that \[\mu = \frac{1}{n}\sum_{i=1}^n \nu_i.\] In particular \(\nu_i \ll \mu\) for each \(i=1,\ldots,n\). We call \(\nu_1,\ldots,\nu_n\) the \(T^k\)-ergodic components of \(\mu\).
Lemma 8 ([3]). Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be an ergodic action and let \(A\subset X\) have \(\mu(A)>0\). For any \(M\in \mathbb{Z}_{>0}\) there exists some \(T^{M!}\)-ergodic component \(\nu\) of \(\mu\) such that \[\nu(A)\geqslant\sqrt{\mu(A)^2 + \sigma_A(\mathrm{Rat}(M))}.\]
Proof of Corollary 2. Fix \(\delta,\varepsilon,\eta>0\) and let \(M= M(\delta,\varepsilon,\eta)\) and \(\kappa=\kappa(\delta,\varepsilon,\eta)\) be as in Theorem 2. Let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) act ergodically and suppose \(A\subset X\) has \(\mu(A)\geqslant\delta\). If the conclusion holds with \(k=1\) and \(\nu=\mu\) we are done. Otherwise by Theorem 2 we must have that \(\sigma_A(\mathrm{Rat}(M))\geqslant\kappa\), and so by Lemma 8 we can find some \(T^{M!}\)-ergodic component \(\nu_1\) of \(\mu\) with \[\nu_1(A)\geqslant\sqrt{\mu(A)^2 + \kappa} \geqslant\mu(A) + \frac{\kappa}{3}.\] Let \(k_1 = M!\) and set \(S_v:= T_{k_1 v}\) for all \(v\in \mathbb{Z}^d\). If our conclusion holds with \(k=k_1\) and \(\nu=\nu_1\) then we are done. Otherwise, since \(S:\mathbb{Z}^d \curvearrowright(X,\nu_1)\) is an ergodic system and \(\nu_1(A)> \mu(A)\geqslant\delta\) then by Theorem 2 and Lemma 8 again we can find some \(T^{k_1^2}\)-ergodic component \(\nu_2\) of \(\mu\) with \[\nu_2(A)\geqslant\nu_1(A)+\frac{\kappa}{3}\geqslant\mu(A) + \frac{2\kappa}{3}.\] Repeating in this way, we must reach our conclusion with \(k = k_1^j = (M!)^j\) for some \(j\leqslant\ceil{3/\kappa}\), since otherwise we will find a probability measure \(\nu\) with \(\nu(A) > 1\), which is clearly a contradiction. Hence the theorem holds with \(k_0 = (M!)^{\ceil{3/\kappa}}\). ◻
Theorem 3 follows from the exact same measure increment argument combined with the following Theorem, which states, in complete analogy with Theorem 2, that the conclusion of Theorem 3 can always be reached with \(k=1\) provided that \(\sigma_A(\mathrm{Rat}(M))\) is sufficiently small.
Theorem 9. For any \(D\in\mathbb{Z}_{>0}\) and \(\delta>0\) there exist some \(\kappa=\kappa(D,\delta)>0\) and \(M=M(D,\delta)>0\) such that the following holds. For any probability preserving action \(T:\mathbb{Z}^d\curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)\geqslant\delta\), if \(\sigma_A(\mathrm{Rat}(M))<\kappa\) then for any \(v_1,\ldots,v_d\in \mathbb{Z}^d\) with \(0< \left\lvert\det(v_1,\ldots,v_d)\right\rvert\leqslant D\) there exists some \(\gamma \in \mathrm{SL}_d(\mathbb{Z})\) with \[\label{eq:32Recurrence32conclusion32with32k611} \mu\left( A\cap T_{\gamma v_1}A\cap \ldots \cap T_{\gamma v_d}A\right)>0.\qquad{(4)}\]
Remark 10. Implicit in their proof of Theorem 1, the authors of [1] show that under the stronger spectral assumption that \[\sigma_A\left(\{\alpha \in \mathbb{T}^d \, : \, 1<\mathrm{ord}(\alpha) < \infty\}\right)\] is sufficiently small in terms of \(\mu(A)\), then for any linearly independent \(v_1,\ldots,v_d\in \mathbb{Z}^d\) there exists \(\gamma \in \mathrm{SL}_d(\mathbb{Z})\) for which ?? holds. Here we see the fundamental difference between Theorem 1 and Theorem 3; the conclusion of Theorem 1 depends on the entire infinite rational spectrum, and indeed Example 2 shows that this really is unavoidable, whereas the conclusion of Theorem 3 depends only on the finite piece of the rational spectrum corresponding to \(\mathrm{Rat}(M)\).
Proof of Theorem 3 via Theorem 9 and Lemma 8.. Repeat the measure increment argument used in the proof of Corollary 2 with Theorem 9 replacing the role of Theorem 2. ◻
In this section we show how the conclusion of Theorem 9 follows from the existence of \(\varepsilon\)-expansive vectors \(v\) which also satisfy that \(\mu(A\cap T_v A)\) is large. Modulo some additional bookkeeping of quantitative constants, the argument is essentially identical to the one presented in the proof of Theorem 1 in [1].
Proposition 11. For any \(D\in\mathbb{Z}_{>0}\) and \(\delta>0\) there exist some \(\kappa=\kappa(D,\delta)>0\) and \(M=M(D,\delta)>0\) such that for any probability preserving action \(T:\mathbb{Z}^d\curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)\geqslant\delta\) and \(\sigma_A(\mathrm{Rat}(M))<\kappa\) the following holds. For all \(m=1,\ldots, D\) there exists some \(v\in m \mathcal{P}\) with \[\label{eq:32big32return32time} \mu(A\cap T_{v}A) > \frac{\mu(A)^2}{2}\qquad{(5)}\] and \[\label{eq:32first32D32multiples32are32expansive} \mu\left(\bigcup_{n\in \mathbb{Z}} T_{nlv}A\right) > 1-\frac{\delta^2}{2(d-1)} \quad \text{for all } l=1,\ldots,D.\qquad{(6)}\]
Proof of Theorem 9 via Proposition 11. Fix some positive integer \(D\) and \(\delta>0\). Let \(\kappa=\kappa(D,\delta)\) and \(M=M(D,\delta)\) be as in the statement of Proposition 11. We will show that this \(\kappa\) and \(M\) satisfy the conclusion of Theorem 9. So let \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) be a probability preserving system and suppose \(A\subset X\) has \(\mu(A)\geqslant\delta\) and \(\sigma_A(\mathrm{Rat}(M))<\kappa\). Let \(v_1,\ldots,v_d \in \mathbb{Z}^d\) have \(0 < \left\lvert\det(v_1,\ldots,v_d)\right\rvert\leqslant D\). Denote \(L:=\left\lvert\det(v_1,\ldots,v_d)\right\rvert\). Since \(\gcd(v_1) \leqslant L \leqslant D\), then by Proposition 11 there exists \(v\in \gcd(v_1) \mathcal{P}\) satisfying equations ?? and ?? . It is easy to see that \(\mathrm{SL}_d(\mathbb{Z})\) acts transitively on \(\mathcal{P}\) so there exists some \(\gamma_0 \in \mathrm{SL}_d(\mathbb{Z})\) with \(\gamma_0 v_1 = v\). Since \(\mu(A)\geqslant\delta\) then equation ?? then implies that \[\mu(A\cap T_{\gamma_0 v_1}A)> \frac{\delta^2}{2},\] and since \(T\) preserves \(\mu\), the \(l=L\) case of equation ?? implies that \[B_j:= \bigcup_{n\in \mathbb{Z}} T_{nL \gamma_0 v_1 + \gamma_0 v_j}A\qquad \text{has} \qquad \mu(B_j) > 1-\frac{\delta^2}{2(d-1)}\] for all \(j=2,\ldots, d\). Then \[\begin{align} \mu\left( (A\cap T_{\gamma_0 v_1}A) \cap \bigcap_{j=2}^d B_j \right)&=1 - \mu\left((A\cap T_{\gamma_0 v_1}A)^c \cup \bigcup_{j=2}^d B_j^c \right)\\ &>\mu(A\cap T_{\gamma_0 v_1}A) - \sum_{j=2}^d \mu(B_j^c)\\ &> \frac{\delta^2}{2} - (d-1) \frac{\delta^2}{2(d-1)} = 0. \end{align}\] It then follows from the definition of each \(B_j\) that there exist \(n_2,\ldots,n_d \in \mathbb{Z}\) such that \[\mu\left(A\cap T_{\gamma_0 v_1}A\cap T_{n_2L \gamma_0 v_1 + \gamma_0v_2}A\cap \ldots \cap T_{n_d L \gamma_0 v_1 + \gamma_0 v_d} A\right)>0.\] All that remains is to check that there exists some \(\gamma \in \mathrm{SL}_d(\mathbb{Z})\) with \[\gamma v_1 = \gamma_0 v_1 \qquad\text{and} \qquad \gamma v_j = \gamma_0 v_j + n_j L \gamma_0 v_1 \quad \text{for all } j =2 ,\ldots, d.\] Set \(u_i:= \gamma_0 v_i\) for all \(i=1,\ldots,d\) and for each \(j=2,\ldots, d\) define a \(\mathbb{Q}\)-linear map on \(\mathbb{Q}^d\) by setting \[S_j u_i = u_i + \delta_{ij}L u_1 \qquad \text{for all } i=1,\ldots,d\] where \(\delta_{ij}\) is the Kronecker delta, and extending by linearity. We claim that that each \(S_j \in \mathrm{SL}_d(\mathbb{Z})\) and so \[\gamma: = S_2^{n_2}S_3^{n_3}\ldots S_d^{n_d} \gamma_0 \in \mathrm{SL}_d(\mathbb{Z})\] is as required. Each \(S_j\) is a shear and so clearly has \(\det(S_j)=1\), so it suffices to prove that \(S_j\) has integer entries. Let \(U\in M_{d\times d}(\mathbb{Z})\) have columns \(u_1,\ldots,u_d\) and notice that \(\det(U) = L\). Each \(S_j\) is then of the form \[S_j = U (I + L E_{1j}) U^{-1}\] where \(E_{1j}\) is the elementary matrix with a \(1\) in position \((1,j)\) and zeros elsewhere. Now \(U^{-1} = \frac{1}{L} \mathrm{adj}(U)\) where \(\mathrm{adj}(U)\in M_{d\times d}(\mathbb{Z})\) so \[S_j = U(I+L E_{1j})U^{-1} = I + UL E_{1j}U^{-1}= I +U E_{1j} \mathrm{adj}(U)\] which implies that \(S_j\) has integer entries as required. ◻
We prove Proposition 11 by first estimating the size of the set of vectors satisfying equations ?? and ?? separately. Non-emptiness of the intersection of these two sets will then follow from size considerations alone.
Proposition 12. For any \(D\in \mathbb{Z}_{>0}\) and \(\delta>0\) there exist some \(\kappa=\kappa(D,\delta)>0\) and \(M=M(D,\delta)>0\) such that for any probability preserving action \(T:\mathbb{Z}^d\curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)\geqslant\delta\) and \(\sigma_A(\mathrm{Rat}(M))<\kappa\) the following is true. For all \(m=1,\ldots,D\) the set \[R(m):= \left\{v \in \mathbb{Z}^d \, : \, \mu(A\cap T_{mv} A) > \frac{\mu(A)^2}{2}\right\}\] satisfies that \[\underline{d}_{Q_N^\mathcal{P}}(R(m))>\frac{\mu(A)^2}{3}.\]
Proof. Let \(\delta>0\) and \(D\in \mathbb{Z}_{>0}\). Let \(M\) and \(\kappa\) be positive constants to be later determined. Let \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) be probability preserving and suppose \(A\subset X\) satisfies that \(\mu(A)\geqslant\delta\) and \(\sigma_A(\mathrm{Rat}(M))<\kappa\). Fix some \(m\in \{1,\ldots,D\}\) and denote by \(\sigma_A^m\) the pushforward of \(\sigma_A\) under the map \(\alpha \mapsto m \alpha\). Define the average \[S_{N}(A):= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^\mathcal{P}} \mu(A\cap T_{mv}A).\] Using the definition of \(\sigma_A\) we have that \[S_{N}(A) = \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^\mathcal{P}} \int_{\mathbb{T}^d} e(mv \cdot \alpha) \, d \sigma_A(\alpha) = \int_{\mathbb{T}^d} \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^\mathcal{P}} e(v\cdot \alpha) \, d \sigma_A^m(\alpha).\] By Proposition 3 and the dominated convergence theorem it then follows that \[\lim_{N\to \infty} S_{N}(A) = \sigma_A^m(\{0\}) + \sum_{q \geqslant 2} \frac{\boldsymbol{\mu}(q)}{J_d(q)} \sigma_A^m(\{ \alpha \in \mathbb{T}^d \, : \, \mathrm{ord}(\alpha)=q\}).\] Let \(\varepsilon>0\) be later determined. Clearly8 we have that \(J_d(q) \to \infty\) as \(q\to \infty\), and since \(\sigma_A^m\) is a finite measure then there exists \(Q=Q(\varepsilon)>0\) such that \[\label{eq:32Tail32bound} \left\lvert\sum_{q >Q} \frac{\boldsymbol{\mu}(q)}{J_d(q)} \sigma_A^m(\{ \alpha \in \mathbb{T}^d \, : \, \mathrm{ord}(\alpha)=q\})\right\rvert<\frac{\varepsilon}{2}.\tag{12}\] For any \(\alpha \in \mathbb{T}^d\) with finite order, it is not hard to show that \(\mathrm{ord}(m\alpha) = \mathrm{ord}(\alpha)/\gcd(\mathrm{ord}(\alpha),m)\), and since \(\gcd(\mathrm{ord}(\alpha),m)\leqslant m \leqslant D\) it follows that \(\sigma_A^m(\mathrm{Rat}(Q))\leqslant\sigma_A(\mathrm{Rat}(DQ))\). We can then calculate \[\begin{align} \left\lvert\sum_{q=2}^Q \frac{\boldsymbol{\mu}(q)}{J_d(q)} \sigma_A^m(\{ \alpha \in \mathbb{T}^d \, : \, \mathrm{ord}(\alpha)=q\})\right\rvert&\leqslant\sigma^m_A(\mathrm{Rat}(Q)) \nonumber\\ &\leqslant\sigma_A(\mathrm{Rat}(DQ)) < \frac{\varepsilon}{2}\label{eq:32Middle32bound} \end{align}\tag{13}\] where the last inequality follows provided we chose \(M = DQ\) and \(\kappa = \varepsilon/2\). Now we also have that \[\label{eq:32Atom32bound} \sigma_A^m(\{0\})\geqslant\sigma_A(\{0\}) = \mu(A)^2\tag{14}\] so together with the reverse triangle inequality, equations 12 , 13 , and 14 imply that \[\label{eq:32return32time32average32lower32bound} \lim_{N\to \infty} S_{N}(A) \geqslant\mu(A)^2 - \varepsilon.\tag{15}\] Write \[\delta_N:= \frac{\left\lvert R(m)\cap Q_N^\mathcal{P}\right\rvert }{\left\lvert Q_N^\mathcal{P}\right\rvert}\] and estimate \[S_N(A) \leqslant\delta_N \mu(A) + \frac{\mu(A)^2}{2}(1-\delta_N) = \delta_N\left(\mu(A) - \frac{\mu(A)^2}{2}\right) + \frac{\mu(A)^2}{2}.\] By taking the limit infimum of both sides and applying equation 15 we then have that \[\frac{\mu(A)^2}{2} - \varepsilon \leqslant\underline{d}_{Q_N^\mathcal{P}}(R(m)) \left(\mu(A) - \frac{\mu(A)^2}{2}\right)< \underline{d}_{Q_N^\mathcal{P}}(R(m)).\] Taking \(\varepsilon = \delta^2/6\) then ensures that \(\underline{d}_{Q_N^\mathcal{P}}(R(m))>\mu(A)^2/3\), and since \(m=1,\ldots,D\) was arbitrary we are done. ◻
Proposition 13. For any \(D\in \mathbb{Z}_{>0}\) and any \(\delta, \varepsilon,\eta>0\) there exist some \(M=M(D,\delta,\varepsilon,\eta)>0\) and \(\kappa=\kappa(D,\delta,\varepsilon,\eta)>0\) such that for any probability preserving action \(T:\mathbb{Z}^d\curvearrowright(X,\mu)\) and any \(A\subset X\) with \(\mu(A)\geqslant\delta\) the following is true. If \(\sigma_A(\mathrm{Rat}(M))<\kappa\) then \[\underline{d}_{Q_N}\left( \left\{v \in \mathbb{Z}^d \, : \, \mu\left( \bigcup_{n\in \mathbb{Z}}T_{nlv}A\right) >1- \varepsilon\quad \text{ for each }l=1,\ldots,D\right\} \right)>1-\eta.\]
Proof. Fix \(D\in \mathbb{Z}_{>0}\) and \(\delta, \varepsilon,\eta>0\). Let \(M_1 = M_1(\delta,\varepsilon,\eta/D)\) and \(\kappa_1 = \kappa_1(\delta,\varepsilon,\eta/D)\) be as in Theorem 2. We claim that \[M: = DM_1 \qquad \text{and }\qquad \kappa:= \kappa_1\] are the desired constants.
So let \(T:\mathbb{Z}^d\curvearrowright(X,\mu)\) be probability preserving and suppose \(A\subset X\) satisfies that \(\mu(A)\geqslant\delta\) and \(\sigma_A(\mathrm{Rat}(M))<\kappa\). Fix \(l\in \{1,\ldots,D\}\) and consider the sub-action \(S\) defined by \(S_v:= T_{lv}\) for each \(v\in \mathbb{Z}^d\). Then the spectral measure of \(A\) with respect to \(S:\mathbb{Z}^d \curvearrowright(X,\mu)\) is precisely \(\sigma^l_A\), the pushforward of \(\sigma_A\) (where \(\sigma_A\) is the spectral measure of \(A\) with respect to \(T:\mathbb{Z}^d \curvearrowright(X ,\mu)\)) under the map \(\alpha \mapsto l\alpha\). Just as in the proof of Proposition 12 we have that \[\sigma^l_A(\mathrm{Rat}(M_1)) \leqslant\sigma_A(\mathrm{Rat}(DM_1)) = \sigma_A(\mathrm{Rat}(M)<\kappa\] and so by Theorem 2 applied to the action \(S:\mathbb{Z}^d \curvearrowright(X,\mu)\) we have that \[\underline{d}_{Q_N}\left(\underbrace{ \left\{v \in \mathbb{Z}^d \, : \, \mu\left( \bigcup_{n\in \mathbb{Z}}T_{nlv}A\right) >1- \varepsilon\right\}}_{:=E(l,\varepsilon)} \right)>1-\frac{\eta}{D}.\] As \(l\in \{1,\ldots,D\}\) was arbitrary then \[\underline{d}_{Q_N}\left(\bigcap_{l=1}^D E(l,\varepsilon)\right) > 1-\eta\] as required. ◻
Proof of Proposition 11 using Propositions 12 and 13. Let \(\delta>0\) and \(D\in \mathbb{Z}_{>0}\). Let \(M_1 = M_1(D,\delta)\) and \(\kappa_1=\kappa_1(D,\delta)\) be as in Proposition 12, and let \(M_2=M_2(D,\delta,\varepsilon,\eta)\) and \(\kappa_2=\kappa_2(D,\delta,\varepsilon,\eta)\) be as in Proposition 13 with the choices \(\varepsilon=\delta^2/(2(d-1))\) and \(\eta = \delta^2/(3\zeta(d))\).
We claim that \[M := \max(M_1,M_2)\qquad \text{and} \qquad \kappa := \min(\kappa_1,\kappa_2)\] will be the constants required in the statement of Proposition 11.
So let \(T: \mathbb{Z}^d \curvearrowright(X,\mu)\) be probability preserving and suppose \(A\subset X\) has \(\mu(A)\geqslant\delta\) and satisfies that \(\sigma_A(\mathrm{Rat}(M))<\kappa\). Then by Proposition 12, we have that \[\label{eq:32Size32of32R40M41} \underline{d}_{Q_N^\mathcal{P}}\left(\underbrace{\left\{ v \in \mathbb{Z}^d \, : \, \mu(A\cap T_{mv}A)>\frac{\mu(A)^2}{2}\right\}}_{=R(m)}\right) > \frac{\delta^2}{3}\tag{16}\] for each \(m=1,\ldots,D\). By Proposition 13 we have that \[E:= \left\{v \in \mathbb{Z}^d \, : \, \mu\left( \bigcup_{n\in \mathbb{Z}}T_{nlv}A\right) >1- \frac{\delta^2}{2(d-1)}\quad \text{ for each }l=1,\ldots,D\right\}\] satisfies that \[\label{eq:32Size32of32E} \underline{d}_{Q_N}(E) >1- \frac{\delta^2}{3\zeta(d)}.\tag{17}\]
By inclusion-exclusion we have that \[\left\lvert E\cap \mathcal{P}\cap Q_N\right\rvert \geqslant\left\lvert E\cap Q_N\right\rvert +\left\lvert\mathcal{P}\cap Q_N\right\rvert -\left\lvert Q_N\right\rvert.\] Dividing by \(\left\lvert Q_N\right\rvert\) and taking the limit infimum of both sides then shows that \[\label{eq:32Using32inclusion32exclusion32for32density} \underline{d}_{Q_N}(E\cap \mathcal{P}) \geqslant\underline{d}_{Q_N}(E) + \underline{d}_{Q_N}(\mathcal{P}) -1> \frac{1}{\zeta(d)}\left(1-\frac{\delta^2}{3}\right)\tag{18}\] where in the last inequality we use equations 4 and 17 . On the other hand, equation 4 also implies that \[\lim_{N\to \infty} \frac{\left\lvert\mathcal{P}\cap Q_N\right\rvert}{\left\lvert Q_N\right\rvert} = \frac{1}{\zeta(d)},\] which together with equation 18 implies that \[\label{eq:32Relative32size32of32E} \underline{d}_{Q_N^\mathcal{P}}(E) > 1-\frac{\delta^2}{3}.\tag{19}\] Together then equations 16 and 19 ensure that \[\underline{d}_{Q_N^\mathcal{P}}(E) + \underline{d}_{Q_N^\mathcal{P}}(R(m))>1 \qquad \text{ for all }m=1,\ldots,D,\] and so \(E\cap R(m)\cap \mathcal{P}\neq \emptyset\) for each \(m=1\ldots,D\) as required. ◻
We show that for every positive integer \(k_0\) there exists an ergodic system \(T:\mathbb{Z}^d \curvearrowright(X,\mu)\) and a set \(A\subset X\) with \(\mu(A)=2^{-d}\) such that for each \(k=1,\ldots,k_0\), there exists some \(v_k \in \mathbb{Z}^d\) for which \[\mu(A\cap T_{\gamma kv_k}A)=0 \quad \text{for every }\gamma \in \mathrm{SL}_d(\mathbb{Z}).\]
So fix \(k_0\) and set \(L:= k_0!\). Let \(X=(\mathbb{Z}/(2L\mathbb{Z}))^d\) with counting probability measure \(\mu\), and let \(T\) be the \(\mathbb{Z}^d\) action on \(X\) by translations. Clearly \(T\) acts ergodically. Consider the set \(A:=\{0,1,\ldots,L-1\}^d\subset X\). Clearly \(A\) has \(\mu(A)=2^{-d}\). For any \(v\in \mathcal{P}\) we have that \(v \not \equiv (0,\ldots,0) \pmod{2}\) and so at least one component of \(Lv\) is congruent to \(L\pmod{2L}\). It follows that \[A\cap T_{Lv}A = \emptyset \qquad \text{for all }v\in \mathcal{P}.\] Since the \(SL_{d}(\mathbb{Z})\) action on \(\mathbb{Z}^d\) preserves the \(\gcd\) of any vector, for each \(k=1,\ldots,k_0\), and any \(w\in \mathcal{P}\) the vector \(v = (L/k)w\) then has \[\mu(A\cap T_{\gamma kv}A) = 0\quad \text{for every }\gamma \in \mathrm{SL}_d(\mathbb{Z}).\]
In this section we prove Proposition 3, restated here for convenience.
Proposition 14. For any \(\alpha \in \mathbb{T}^d\) we have that \[g(\alpha): = \lim_{N\to \infty}\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^{\mathcal{P}}} e(v\cdot \alpha)= \begin{cases} 1 & \text{ if } \alpha=0\\ \frac{1}{J_d(q)}\boldsymbol{\mu}(q) & \text{ if } \mathrm{ord}(\alpha)=q\geqslant 2\\ 0 & \text{ if }\mathrm{ord}(\alpha)=\infty. \end{cases}\]
We first recall some facts from analytic number theory, all of which can be found in [7].
The Möbius function \(\boldsymbol{\mu}: \mathbb{Z}_{>0}\to \mathbb{R}\) is defined by \[\boldsymbol{\mu}(n) : = \begin{cases} 1 & \text{if }n=1\\ (-1)^k &\text{if }n \text{ is a product of }k\text{ distinct primes}\\ 0 &\text{otherwise} \end{cases}\] and satisfies that for any positive integer \(m\), \[\label{eq:32Mobius32inversion} \sum_{n | m} \boldsymbol{\mu}(n) = \begin{cases} 1 & \text{if }m=1\\ 0 &\text{otherwise.} \end{cases}\tag{20}\] The Riemann zeta function is defined by \[\zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{for any }s\in \mathbb{C}\text{ with }\mathrm{Re}(s)>1\] and satisfies the Euler product formula \[\label{eq:32Euler32prod32for32zeta} \zeta(s) = \prod_{p\text{ prime}}\frac{1}{1-p^{-s}}\qquad \text{for any }s\in \mathbb{C}\text{ with }\mathrm{Re}(s)>1,\tag{21}\] and that \[\label{eq:32Dirichlet32series32formula} \frac{1}{\zeta(s)} = \sum_{n=1}^\infty\frac{\boldsymbol{\mu}(n)}{n^s}\qquad \text{for any }s\in \mathbb{C}\text{ with }\mathrm{Re}(s)>1.\tag{22}\]
The Jordan totient function \(J_q: \mathbb{Z}_{>0}\to \mathbb{R}\) is defined by \[J_d(q): = \left\lvert\left\{ a=(a_1,\ldots,a_d)\in [q]^d\, : \, \gcd(a,q) = 1\right\}\right\rvert\] where \([q]:=\{1,\ldots,q\}\).
Lemma 15. Let \(N\) and \(M\) be positive integers and let \(r \in (\mathbb{Z}/M\mathbb{Z})^d\) have \(\gcd(r,M)=1\). Then we have that \[\label{eq:32Density32of32primitives} \left\lvert Q_N^\mathcal{P}\right\rvert = \frac{\left\lvert Q_N\right\rvert}{\zeta(d)} + O\left(\log(N)N^{d-1}\right)\qquad{(7)}\] and \[\label{eq:32Counting32primtives32in32congruence32class} \left\lvert\left\{v\in Q_N^\mathcal{P} \, : \, v \equiv r \:(\mathrm{mod }\,M) \right\}\right\rvert = \frac{\left\lvert Q_N\right\rvert}{\zeta(d)J_d(M)} + O\left( \frac{\log(N)N^{d-1}}{M^{d-1}}\right).\qquad{(8)}\]
Proof. Let \(M\) be a positive integer. For any \(r_0 \in \mathbb{Z}/M\mathbb{Z}\) clearly we have that \[\left\lvert \left\{ n\in [-N,N]\cap \mathbb{Z}\, : \, n \equiv r_0 \:(\text{mod }M) \right\}\right\rvert = \frac{\left\lvert 2N+1\right\rvert}{M} + O\left( 1 \right),\] and so for any \(r\in (\mathbb{Z}/M\mathbb{Z})^d\) \[\label{eq:3240141} \left\lvert\left\{ v\in Q_N \, : \, v \equiv r \:(\text{mod }M) \right\}\right\rvert = \frac{\left\lvert Q_N\right\rvert}{M^d} + O\left(\frac{N^{d-1}}{M^{d-1}}\right).\tag{23}\] Taking \(r\equiv 0\) in 23 yields \[\label{eq:3240241} \left\lvert\left\{ v\in Q_N \, : \, M \, |\, v \right\}\right\rvert = \frac{\left\lvert Q_N\right\rvert}{M^d} + O\left(\frac{N^{d-1}}{M^{d-1}}\right).\tag{24}\] Use equation 20 to write for any \(v\in \mathbb{Z}^d\), \[\label{eq:3240441} \mathbf{1}_{\gcd(v)=1} = \sum_{n | \gcd(v)} \boldsymbol{\mu}(n) = \sum_{n\geqslant 1} \boldsymbol{\mu}(n) \mathbf{1}_{n | v}.\tag{25}\] We can then use 24 and 25 to calculate \[\begin{align} \left\lvert Q_N^\mathcal{P}\right\rvert&= \sum_{v \in Q_N} \mathbf{1}_{\gcd(v)=1} = \sum_{v \in Q_N}\sum_{n\geqslant 1} \boldsymbol{\mu}(n) \mathbf{1}_{n | v}\\ &=\sum_{v\in Q_N} \sum_{n=1}^N \boldsymbol{\mu}(n) \mathbf{1}_{n | v}\\ &= \sum_{n=1}^N \boldsymbol{\mu}(n) \left\lvert\left\{ v\in Q_N \, : \, n \,|\, v \right\}\right\rvert\\ &=\sum_{n=1}^N \boldsymbol{\mu}(n)\left(\frac{\left\lvert Q_N\right\rvert}{n^d} + O\left(\frac{N^{d-1}}{n^{d-1}}\right).\right)\\ &= \left\lvert Q_N\right\rvert \sum_{n=1}^N \frac{\boldsymbol{\mu}(n)}{n^d} + \sum_{n=1}^N \boldsymbol{\mu}(n) O\left(\frac{N^{d-1}}{n^{d-1}}\right)\\ &=\left\lvert Q_N\right\rvert\left( \sum_{n=1}^\infty\frac{\boldsymbol{\mu}(n)}{n^d} + O\left(\frac{1}{N^{d-1}}\right)\right) + O(\log(N)N^{d-1}) \end{align}\] where in the final line we bound the absolute value of the tail \(\sum_{n>N}\frac{\boldsymbol{\mu}(n)}{n^d}\) by an integral and use the crude bound \[\left\lvert\sum_{n=1}^N \frac{\boldsymbol{\mu}(n)}{n^{d-1}}\right\rvert \leqslant\sum_{n=1}^N \frac{1}{n} = O(\log(N))\] for any \(d\geqslant 2\). Applying the formula from 22 we have that \[\left\lvert Q_N^\mathcal{P}\right\rvert=\left\lvert Q_N\right\rvert\left( \frac{1}{\zeta(d)} + O\left(\frac{1}{N^{d-1}}\right)\right) + O(\log(N)N^{d-1})\] and collecting error terms yields equation ?? .
We now turn to proving equation ?? . So let \(r\in (\mathbb{Z}/M\mathbb{Z})^d\) have \(\gcd(r,M)=1\) and use equation 20 again to write \[\begin{align} \left\lvert\left\{v\in Q_N^\mathcal{P} \, : \, v \equiv r \:(\text{mod }M) \right\}\right\rvert &= \sum_{v\in Q_N} \mathbf{1}_{v \equiv r (\text{mod }M)} \mathbf{1}_{\gcd(v)=1} \\ &=\sum_{v\in Q_N} \mathbf{1}_{v \equiv r (\text{mod }M)} \sum_{n=1}^N \boldsymbol{\mu}(n) \mathbf{1}_{n | v}. \end{align}\] Notice that since \(\gcd(r,M)=1\) then \(n \, | \, v\) implies that \(\gcd(n,M)=1\). Indeed writing \(v = r + Ml\) for some \(l\in \mathbb{Z}^d\) then \[n \, | \, v \implies \gcd(n,M) \, | \, v \implies \gcd(n,M) \, | \, r\implies \gcd(n,M)=1.\] Continuing with our earlier calculation then \[\begin{align} \left\lvert\left\{v\in Q_N^\mathcal{P} \, : \, v \equiv r \:(\text{mod }M) \right\}\right\rvert &= \sum_{v\in Q_N} \mathbf{1}_{v \equiv r (\text{mod }M)} \sum_{n=1}^N \boldsymbol{\mu}(n) \mathbf{1}_{\gcd(n,M)=1}\mathbf{1}_{n | v} \nonumber \\ &= \sum_{n=1}^N \mathbf{1}_{\gcd(n,M)=1} \boldsymbol{\mu}(n) \sum_{v\in Q_N} \mathbf{1}_{v \equiv 0 (\text{mod }n)} \mathbf{1}_{v\equiv r (\text{mod }M)}.\label{eq:3240541} \end{align}\tag{26}\] Now for each \(n\) and \(M\) with \(\gcd(n,M)=1\) the Chinese remainder theorem implies there exists some \(r_n \in (\mathbb{Z}/ (nM)\mathbb{Z})^d\) so that \[\label{eq:3240641} \sum_{v\in Q_N} \mathbf{1}_{v \equiv 0 (\text{mod }n)} \mathbf{1}_{v\equiv r (\text{mod }M)}=\Big|\left\{v \in Q_N \, : \, v\equiv r_n \pmod{nM}\right\}\Big|.\tag{27}\] Equation 23 provides us with a bound for the set in the right hand side of 27 , which combined with equation 26 allows us to calculate \[\begin{align} \Big|\{v\in Q_N^\mathcal{P} &\, : \, v \equiv r \:(\text{mod }M) \}\Big| \nonumber\\ &=\sum_{n=1}^N \mathbf{1}_{\gcd(n,M)=1} \boldsymbol{\mu}(n) \left(\frac{\left\lvert Q_N\right\rvert}{n^d M^d} + O\left(\frac{N^{d-1}}{n^{d-1}M^{d-1}}\right) \right)\nonumber\\ &= \frac{\left\lvert Q_N\right\rvert}{M^d}\sum_{n=1}^N \mathbf{1}_{\gcd(n,M)=1} \frac{\boldsymbol{\mu}(n)}{n^d}+O\left(\frac{\log(N)N^{d-1}}{M^{d-1}}\right)\nonumber\\ &=\frac{\left\lvert Q_N\right\rvert}{M^d}\left(\sum_{n=1}^\infty \mathbf{1}_{\gcd(n,M)=1} \frac{\boldsymbol{\mu}(n)}{n^d} + O\left(\frac{1}{N^{d-1}}\right)\right)+O\left(\frac{\log(N)N^{d-1}}{M^{d-1}}\right)\nonumber\\ &= \frac{\left\lvert Q_N\right\rvert}{M^d}\sum_{n=1}^\infty \mathbf{1}_{\gcd(n,M)=1} \frac{\boldsymbol{\mu}(n)}{n^d} + O\left(\frac{\log(N)N^{d-1}}{M^{d-1}}\right). \label{eq:3240741} \end{align}\tag{28}\] We claim that \[\frac{1}{M^d}\sum_{n=1}^\infty \mathbf{1}_{\gcd(n,M)=1} \frac{\boldsymbol{\mu}(n)}{n^d} = \frac{1}{\zeta(d) J_d(M)},\] which, once combined with equation 28 , will yield equation ?? as required. Indeed, by equation 20 again we have that \[\begin{align} J_d(M)= \sum_{a \in [M]^d} \mathbf{1}_{\gcd(a,M)=1} &= \sum_{a\in [M]^d} \sum_{n \,| \,M} \boldsymbol{\mu}(n) \mathbf{1}_{n \, | \, \gcd(a,M)}\\ &=\sum_{n \,| \,M} \boldsymbol{\mu}(n)\left\lvert\left\{ a\in [M]^d \, : \, n \, | \, a \right\} \right\rvert\\ &=\sum_{n \,| \,M} \boldsymbol{\mu}(n) \frac{M^d}{n^d}\\ &=M^d \sum_{n\geqslant 1} \mathbf{1}_{n\, | \, M} \frac{\boldsymbol{\mu}(n)}{n^d}. \end{align}\] The function \(f(n)= \mathbf{1}_{n\, | \, M} \boldsymbol{\mu}(n)/n^d\) is multiplicative with an absolutely convergent sum \(\sum_{n\geqslant 1}f(n)\) so we can use an Euler product9 to write \[\begin{align} \sum_{n\geqslant 1} \mathbf{1}_{n\, | \, M} \frac{\boldsymbol{\mu}(n)}{n^d}&= \prod_{p}\left( 1+ \sum_{n\geqslant 1}\frac{\mathbf{1}_{p^n \, | \, M} \boldsymbol{\mu}(p^n)}{p^{nd}}\right)\\ &=\prod_{p} \left(1-\frac{\mathbf{1}_{p \, | \, M}}{p^d}\right)\\ &=\prod_{p \, | \, M} \left(1-\frac{1}{p^d}\right), \end{align}\] and so \[\label{eq:32euler32prod32for32Jordan32totient} J_d(M) = M^d\prod_{p \, | \, M} \left(1-\frac{1}{p^d}\right).\tag{29}\] Similarly we have that \[\begin{align} \frac{1}{M^d}\sum_{n\geqslant 1} \mathbf{1}_{\gcd(n,M)=1} \frac{\boldsymbol{\mu}(n)}{n^d} &= \frac{1}{M^d} \prod_{p}\left( 1+ \sum_{n\geqslant 1}\frac{\mathbf{1}_{\gcd(p^n,M)=1} \boldsymbol{\mu}(p^n)}{p^{nd}}\right)\\ &=\frac{1}{M^d} \prod_{p}\left( 1-\frac{\mathbf{1}_{p \,\nmid \,M}}{p^d}\right)\\ &=\frac{1}{M^d} \prod_{p \, \nmid \, M}\left(1-p^{-d}\right). \end{align}\] Combining equations 21 and 29 we then have that \[\frac{1}{M^d} \prod_{p \, \nmid \, M}\left(1-p^{-d}\right) =\frac{1}{M^d} \frac{\prod_{p}\left(1-p^{-d}\right)}{\prod_{p \, | \, M}\left(1-p^{-d}\right)}\\ =\frac{1}{\zeta(d)J_d(M)}\] as required. ◻
Proof of Proposition 3. For \(\alpha \in \mathbb{T}^d\) define \[\label{eq:32Def32of32gN} g_N(\alpha):=\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v\in Q_N^{\mathcal{P}}} e(v\cdot \alpha).\tag{30}\] First consider the case that \(\mathrm{ord}(\alpha)=q\) for some positive integer \(q\). Clearly if \(\alpha = 0\) the conclusion is obvious, so take \(q\geqslant 2\). We can then write \(\alpha = a/q\) for some \(a\in [q]^d\) with \(\gcd(a,q)=1\). Then \(v \mapsto e((v\cdot \alpha)/q)\) is constant on residue classes modulo \(q\mathbb{Z}^d\), so we can use ?? from Lemma 15 to write \[\begin{align} g_N(\alpha) &= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert}\sum_{v\in Q_N^\mathcal{P}} e \left( \frac{v\cdot a}{q}\right) \nonumber \\ &= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{r \in [q]^d \: \gcd(r,q)=1}\left\lvert\left\{v\in Q_N^\mathcal{P} \, : \, v \equiv r \:(\mathrm{mod }\,q) \right\}\right\rvert e \left( \frac{r\cdot a}{q}\right) \nonumber \\ &= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \left(\frac{\left\lvert Q_N\right\rvert}{\zeta(d)J_d(q)} + O\left( \frac{\log(N)N^{d-1}}{q^{d-1}}\right) \right)\sum_{\substack{r \in [q]^d \\ \gcd(r,q)=1}} e \left( \frac{r\cdot a}{q}\right). \label{eq:32Reducing32to32Ramanujan32sum} \end{align}\tag{31}\] By equation ?? we have that \[\label{eq:32little32o32equation} \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \left(\frac{\left\lvert Q_N\right\rvert}{\zeta(d)J_d(q)} + O\left( \frac{\log(N)N^{d-1}}{q^{d-1}}\right) \right) =\frac{1}{J_d(q)}\frac{1}{1+o_{N}(1)} + o_{N}(1).\tag{32}\] The sum in equation 31 is a classical \(d\)-dimensional Ramanujan sum which we can calculate using equation 20 again as follows. \[\begin{align} \sum_{\substack{r \in [q]^d \\ \gcd(r,q)=1}} e \left( \frac{r\cdot a}{q}\right)&=\sum_{r \in [q]^d } \mathbf{1}_{\gcd(r,q)=1} e \left( \frac{r\cdot a}{q}\right) \nonumber \\ &=\sum_{r \in [q]^d } \sum_{t \, | \, r,q} \boldsymbol{\mu}(t) e \left( \frac{r\cdot a}{q}\right) \nonumber\\ &=\sum_{r \in [q]^d } \sum_{t \, | \, q} \mathbf{1}_{t \, | \,r} \boldsymbol{\mu}(t) e \left( \frac{r\cdot a}{q}\right) \nonumber\\ &=\sum_{t \, | \, q} \boldsymbol{\mu}(t) \sum_{\substack{r \in [q]^d \\ t \, | \, r} } e \left( \frac{r\cdot a}{q}\right) \nonumber \\ &=\sum_{t \, | \, q}\boldsymbol{\mu}(t)\sum_{r\in [q/t]^d} e \left( \frac{r\cdot a}{q/t}\right) \nonumber \\ &=\sum_{t \, | \, q}\boldsymbol{\mu}(t) \mathbf{1}_{q/t \, | \, a} \left(\frac{q}{t}\right)^d = \boldsymbol{\mu}(q) \label{eq:32formula32for32ramanujan32sum} \end{align}\tag{33}\] In the second last equality we used orthogonality of the characters on \(\mathbb{Z}/(q/t)\mathbb{Z}\) and in the final equality we used that \(\gcd(a,q)=1\). Substituting equations 32 and 33 into equation 31 and taking \(N\to \infty\) then gives the desired formula.
It remains to show that \(g(\alpha)=0\) when \(\mathrm{ord}(\alpha)=\infty\). Use equation 25 again to write \[\begin{align} g_N(\alpha) = \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{v \in Q_N^\mathcal{P}} e(v\cdot \alpha)&=\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{n=1}^N \boldsymbol{\mu}(n) \sum_{\substack{v \in Q_N \\ n \, | \, v}} e(v\cdot \alpha) \nonumber\\ &=\frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \sum_{n=1}^N \boldsymbol{\mu}(n) \sum_{v \in Q_{\floor{N/n}}} e(v\cdot n\alpha). \label{eq:32dealing32with32irrational} \end{align}\tag{34}\] Now for any positive integer \(M\) we have that \[\sum_{v\in Q_M} e(v\cdot n\alpha) = \sum_{v_1,\ldots,v_d \in [-M,M]} \prod_{i=1}^d e(v_i\cdot n\alpha_i) = \prod_{i=1}^d \sum_{m=-M}^M e(mn \alpha_i).\] Since \(\mathrm{ord}(\alpha)=\infty\) then at least one \(\alpha_i\) is irrational, and so by summing a finite geometric series in the above formula we have that \[\begin{align} \left\lvert\sum_{v\in Q_M} e(v\cdot n\alpha)\right\rvert &\leqslant\left\lvert 2M+1\right\rvert^{d-1} \left\lvert\sum_{m=-M}^M e(n\alpha_i)^m\right\rvert\\ &\leqslant\left\lvert 2M+1\right\rvert^{d-1} \left\lvert\frac{2}{1-e(n\alpha_i)}\right\rvert = O_\alpha(M^{d-1}), \end{align}\] and applying this to equation 34 shows that \[\begin{align} \left\lvert g_N(\alpha)\right\rvert &= \frac{1}{\left\lvert Q_N^\mathcal{P}\right\rvert} \left\lvert\sum_{n=1}^N \boldsymbol{\mu}(n) O_{\alpha}\left(\frac{N^{d-1}}{n^{d-1}} \right)\right\rvert\\ &\leqslant\frac{N^{d-1}}{\left\lvert Q_N^\mathcal{P}\right\rvert} C_{\alpha} \sum_{n=1}^N \frac{1}{n^{d-1}} = \frac{N^{d-1}}{\left\lvert Q_N^\mathcal{P}\right\rvert} O_{\alpha}(\log(N)) \to 0 \text{ as } N \to \infty \end{align}\] since \(\left\lvert Q_N^\mathcal{P}\right\rvert\) is on the order of \(\left\lvert Q_N\right\rvert = (2N+1)^d\) by equation ?? . ◻
We choose not to include the underlying \(\sigma\)-algebra in our notation and moving forward all subsets of a measurable space will be assumed to be measurable. We will also always assume that \(L^2(\mu)\) is separable.↩︎
We will always use the bold symbol \(\boldsymbol{\mu}\) for the Möbius function, reserving the un-bolded \(\mu\) for a probability measure.↩︎
For the precise definition of eigenfunctions and their order see Section 2.↩︎
In fact, they showed that under the same assumption, \(\varepsilon\)-expansive directions for \(A\) exist in any haystack \(H\subset \mathbb{Z}^d\), where a haystack is an infinite subset \(H\subset \mathbb{Z}^d\) such that every distinct \(v_1,\ldots, v_d \in H\) are linearly independent. See [2].↩︎
Actually the proof we present does not use Theorem 1 per se, as it turns out to be more direct to use the main technical input to Theorem 1 instead, namely Proposition 3.↩︎
In fact it follows immediately from the Euler product formulae in equations 21 and 29 in Section 10 that \(J_d(q)\zeta(d) \geqslant q^d\).↩︎
Recall that a function \(f: \mathbb{Z}_{>0} \to \mathbb{C}\) is multiplicative if \(f(nm)= f(n)f(m)\) for all \(n\) and \(m\) with \(\gcd(n,m)=1\). If a multiplicative function \(f\) has that the series \(\sum_{n \geqslant 1}f(n)\) is absolutely convergent, then the series is equal to its Euler product \[\sum_{n \geqslant 1}f(n) = \prod_{p\text{ prime}}(1+f(p)+f(p^2)+\ldots).\] See [7] for more details.↩︎