Proof of Theorem 7. We separately address the cases where \(\mathcal{L}_1\) is nonempty or empty.

1) If \(\mathcal{L}_1 \ne \varnothing\) , then Proposition 16 shows that for each \(v \in \mathcal{V}\), \(A_v(W)\) satisfies the hypotheses of Proposition 14 with \(\gamma_j=\gamma_v\) and \(\beta_j=\beta_v\). Therefore for all \(v \in \mathcal{V}\), there exist \(C_v, \lambda_v >0\), such that for \(\lambda \geq \lambda_v\), and for all \(E \in \mathbb{R}\), \[\left| \left| (-\partial_s^2 + i \lambda A_v(W)(s) - E)^{-1} \right| \right|_{\mathcal{L}(L^2)} \leq C_v \lambda^{\frac{1}{\beta_v+2}} \leq C_v \lambda^{\frac{1}{\beta'+2}},\] where the final inequality holds because \(\min \beta_v = \beta'\). Then, by Proposition 13, 13 holds with \(\alpha = 1 - \frac{1}{\beta'+3}\), and this completes the proof.

2) If \(\mathcal{L}_1 = \varnothing\), then Proposition 16 shows that for all \(v \in \mathcal{V}\), \(A_v(W)\) satisfies the hypotheses of the second part of Proposition 14 with \(\gamma_j=\gamma_v\). Therefore for all \(v \in \mathcal{V}\), there exist \(C_v, \lambda_v >0\), such that for \(\lambda \geq \lambda_v\), and for all \(E \in \mathbb{R}\), \[\left| \left| (-\partial_s^2 + i \lambda A_v(W)(s) - E)^{-1} \right| \right|_{\mathcal{L}(L^2)} \leq C_v \lambda^{-\frac{2}{\gamma_v+2}} \leq C_v \lambda^{-\frac{2}{\gamma'+2}},\] where the final inequality holds because \(\max \gamma_v=\gamma'\). Then, by Proposition 13, 13 holds with \(\alpha = 1+\frac{2}{\gamma'}\), and this completes the proof. ◻

Proof. The first part follows by multiplying \((-\partial_s^2+i\lambda V(s)-E)u=f\) by \(\bar{u}\). Then integrate by parts and take the imaginary part.

To prove the second part we consider the left hand side, integrate by parts twice, and use the equation to obtain \[\begin{align} \int \psi|u'|^2 ds &= - \text{Re }\int \psi' u' \bar{u} ds - \text{Re }\int \psi u'' \bar{u} ds \\ &=\frac{1}{2} \int \psi''|u|^2 + E \int \psi |u|^2 ds + \text{Re }\int \psi f \bar{u} ds. \end{align}\] Now use that \(\psi \leq CV\), \(|\psi''| \le CV\), and Hölder’s inequality to write \[\int \psi|u|^2 ds \leq C (1+\max(0,E)) \int V |u|^2 ds + C \left(\int |f|^2 ds \right)^{1/2} \left( \int V |u|^2 ds \right)^{1/2} \left| \left| \psi \right| \right|_{L^\infty}^{1/2} .\] Then apply Young’s inequality for products to the second term, then apply part 1 and take square roots to conclude. ◻

Proof of Proposition 14. 1) Order the finite intervals \([a_j,b_j]\) and zeroes \(s_j\) in a single list of length \(N\). Then let \(\chi_j\) be a partition of unity on \(\mathbb{S}^1\), such that \(\chi_k\) is identically 1 on the \(k\)th element of the list \(\{ [a_j,b_j], s_j\}\), and for some \(\varepsilon>0\), \(\text{supp }\chi_{N+1} \subset \{V \geq \varepsilon\}\).

Let \(u,f\) solve \((-\partial_s^2+i\lambda V(s)-E)u=f\). Then, by Lemma 2, \[\left| \left| \chi_{N+1} u \right| \right|_{L^{2}} \leq \frac{1}{\varepsilon^{1/2}} \left| \left| V^{1/2} u \right| \right|_{L^{2}} \leq \frac{C}{{\sqrt{\lambda}}} |\left\langle f,u\right\rangle|^{1/2}.\] Applying Young’s inequality for products, we have for any \(\delta>0\) \[\left| \left| \chi_{N+1}u \right| \right|_{L^{2}} \leq \frac{C}{{\lambda} \delta} \left| \left| f \right| \right|_{L^{2}} + C\delta \left| \left| u \right| \right|_{L^{2}}.\]

We now separately consider those \(j\) associated to intervals \([a_j, b_j]\) or single points \(s_j\). Each of these cases will be further split into sub-cases based on the size of \(E\). The technique is the same for intervals and points, but the constants involved are different.

Before doing so we make a common definition. \[V_j(s) = \begin{cases} V(s), \quad & \text{ for } s\in \text{supp }(\chi_j),\\ \varepsilon, \quad &\text{ otherwise}. \end{cases}\] Note that \(V \chi_j = V_j \chi_j\), and if \(s_j \in \text{supp }\chi_j\) then \(V_j\) satisfies the hypotheses of [13]. Additionally if \([a_j,b_j] \subset \text{supp }\chi_j\), then \(V_j\) satisfies the hypotheses of [2]. Furthermore \[\begin{align} (-\partial_s^2 + i \lambda V_j(s) -E) \chi_j u &= \chi_j(-\partial_s^2 + i \lambda V (s) -E) u + [-\partial_s^2, \chi_j]u \\ &= \chi_j f - \chi_j'' u - 2\chi_j' \partial_s u.\label{eq:1dchijSDWE} \end{align}\tag{18}\] 2) For intervals \([a_j, b_j]\) we consider separately the cases \(E \leq \lambda^{\frac{\beta'+1}{\beta'+2}}\) and \(E \geq \lambda^{\frac{\beta'+1}{\beta'+2}}\).

2a) Assume \(E \leq \lambda^{\frac{\beta'+1}{\beta'+2}}\). Then by [2], there exists \(C>0\) such that \[\left| \left| \chi_j u \right| \right|_{L^{2}} \leq C \lambda^{\frac{1}{\beta_j+2}} \left| \left| \chi_j f {-} \chi_j'' u {-} 2\chi_j' \partial_s u \right| \right|_{L^{2}}.\] Now since \(\beta'=\min \beta_j\), note that \(\lambda^{\frac{1}{\beta_j+2}} \leq \lambda^{\frac{1}{\beta'+2}}\). Therefore applying Lemma 2, since \(|\chi_j''| \leq CV\) and \(\chi_j'=0\) on a neighborhood of \(V=0\), we have \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} &\leq C \lambda^{\frac{1}{\beta'+2}} \left( \left| \left| \chi_j f \right| \right|_{L^{2}} + \left| \left| \chi_j'' u \right| \right|_{L^{2}} + 2 \left| \left| \chi_j' \partial_s u \right| \right|_{L^{2}} \right)\\ &\leq C \lambda^{\frac{1}{\beta'+2}} \left| \left| f \right| \right|_{L^{2}} + C (\lambda^{\frac{1}{\beta'+2}-\frac{1}{2}} + \lambda^{\frac{1}{\beta'+2}-\frac{1}{2}} \max(0,E)^{1/2})|\left\langle f,u\right\rangle|^{1/2}\\ &\leq C \lambda^{\frac{1}{\beta'+2}} \left| \left| f \right| \right|_{L^{2}} + C \lambda^{\frac{1}{2(\beta'+2)}} |\left\langle f,u\right\rangle|^{1/2} , \end{align}\] where the third inequality uses that \(E \leq \lambda^{\frac{\beta'+1}{\beta'+2}}\) and that \(\lambda\) is large. Then using Young’s inequality for products we have, for any \(\delta>0\) \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} &\leq \frac{C}{\delta} \lambda^{\frac{1}{\beta'+2}} \left| \left| f \right| \right|_{L^{2}} + \delta \left| \left| u \right| \right|_{L^{2}}.\label{eq:intervalcombineEsmall} \end{align}\tag{19}\] 2b) Assume \(E \geq \lambda^{\frac{\beta'+1}{\beta'+2}}\). Rewrite equation 18 as \[(-\partial_s^2-E) \chi_j u= \chi_j f + \chi_j'' u - 2 \partial_s(\chi_j' u) - i\lambda V \chi_j u.\] Then by a standard 1-d propagation estimate, see [11] or [19], and using that \(|\chi_j'| + |\chi_j''| \leq CV\), there exists \(C>0\) such that \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} &\leq C E^{-1/2} \left| \left| f+\chi_j'' u - i\lambda V \chi_j u \right| \right|_{L^{2}} + \left| \left| \partial_s (\chi_j' u) \right| \right|_{H^{-1}} + \left| \left| Vu \right| \right|_{L^{2}}\\ &\leq C E^{-1/2} \left| \left| f \right| \right|_{L^{2}} + C(\lambda E^{-1/2} +1) \left| \left| V^{1/2} u \right| \right|_{L^{2}}.\label{eq:highE1dest} \end{align}\tag{20}\] Note that to obtain 20 we did not use the form of \(V\) near \([a_j, b_j]\) nor the exact size of \(E\).

Now since \(E \geq \lambda^{\frac{\beta'+1}{\beta'+2}}\), \(E^{-1/2} \leq \lambda^{-\frac{\beta'+1}{2(\beta'+2)}}\), and applying Lemma 2 we obtain \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} & \leq C \left| \left| f \right| \right|_{L^{2}} + C \lambda^{1-\frac{\beta'+1}{2(\beta'+2)}} \left| \left| V^{1/2}u \right| \right|_{L^{2}}\\ &\leq C \left| \left| f \right| \right|_{L^{2}} + C \lambda^{\frac{1}{2(\beta'+2)}} |\left\langle f,u\right\rangle|^{1/2}. \end{align}\] Then applying Young’s inequality for products, for any \(\delta>0\) we have \[\label{eq:intervalcombineElarge} \left| \left| \chi_j u \right| \right|_{L^{2}} \leq \frac{C}{\delta} \lambda^{\frac{1}{\beta'+2}} \left| \left| f \right| \right|_{L^{2}} + \delta \left| \left| u \right| \right|_{L^{2}}.\tag{21}\] 2c) Now by 19 and 21 for all \(E \in \mathbb{R}\), if \(V\) vanishes on \([a_j, b_j]\), then, for any \(\delta>0\), \[\left| \left| \chi_j u \right| \right|_{L^{2}} \leq \frac{C}{\delta} \lambda^{\frac{1}{\beta'+2}} \left| \left| f \right| \right|_{L^{2}} + \delta \left| \left| u \right| \right|_{L^{2}}. \label{eq:intervalchijest}\tag{22}\] 3) For single points \(s_j\) we consider separately the cases \(E \leq \lambda^{\frac{\gamma'+4}{\gamma'+2}}\), and \(E \geq \lambda^{\frac{\gamma'+4}{\gamma'+2}}\).

3a) Assume \(E \leq \lambda^{\frac{\gamma'+4}{\gamma'+2}}\). Applying [13] to 18 there exists \(C>0\) \[\left| \left| \chi_j u \right| \right|_{L^{2}} \leq C \lambda^{-\frac{2}{\gamma_j+2}} \left| \left| \chi_j f {-} \chi_j'' u {-} 2\chi_j' \partial_s u \right| \right|_{L^{2}}\] Now because \(\gamma'=\max \gamma_j\), we have \(\lambda^{-\frac{2}{\gamma_j+2}} \leq \lambda^{-\frac{2}{\gamma'+2}}\). Therefore applying Lemma 2, since \(|\chi_j''| \leq CV\) and \(\chi_j'=0\) on a neighborhood of \(\{V=0\}\), we have \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} &\leq C \lambda^{-\frac{2}{\gamma'+2}} \left( \left| \left| \chi_j f \right| \right|_{L^{2}} + \left| \left| \chi_j'' u \right| \right|_{L^{2}} + 2 \left| \left| \chi_j' \partial_s u \right| \right|_{L^{2}} \right)\\ &\leq C \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + C (\lambda^{-\frac{2}{\gamma'+2}-\frac{1}{2}} + \lambda^{-\frac{2}{\gamma'+2}-\frac{1}{2}} \max(0,E)^{1/2})|\left\langle f,u\right\rangle|^{1/2}\\ &\leq C \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + C \lambda^{-\frac{1}{\gamma'+2}} |\left\langle f,u\right\rangle|^{1/2}. \end{align}\] Where the third inequality uses that \(E \leq \lambda^{\frac{\gamma'+4}{\gamma'+2}}\). Then using Young’s inequality for products we have, for any \(\delta>0\) \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} &\leq \frac{C}{\delta} \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + \delta \left| \left| u \right| \right|_{L^{2}}.\label{eq:pointcombineEsmall} \end{align}\tag{23}\] 3b) Assume \(E \geq \lambda^{\frac{\gamma'+4}{\gamma'+2}}\). Following the same argument as in case 2b) we obtain 20 \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} &\leq C E^{-1/2} \left| \left| f \right| \right|_{L^{2}} + C(\lambda E^{-1/2} +1) \left| \left| V^{1/2} u \right| \right|_{L^{2}}. \end{align}\] Now since \(E \geq \lambda^{\frac{\gamma'+4}{\gamma'+2}}\), \(E^{-1/2} \leq \lambda^{-\frac{\gamma'+4}{2(\gamma'+2)}}\leq \lambda^{-\frac{2}{\gamma'+2}}\), and applying Lemma 2 we obtain \[\begin{align} \left| \left| \chi_j u \right| \right|_{L^{2}} & \leq \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + C \lambda^{1-\frac{\gamma'+4}{2(\gamma'+2)}} \left| \left| V^{1/2}u \right| \right|_{L^{2}}\\ &\leq C \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + C \lambda^{-\frac{1}{\gamma'+2}} |\left\langle f,u\right\rangle|^{1/2}. \end{align}\] Then applying Young’s inequality for products, for any \(\delta>0\) we have \[\label{eq:pointcombineElarge} \left| \left| \chi_j u \right| \right|_{L^{2}} \leq \frac{C}{\delta} \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + \delta \left| \left| u \right| \right|_{L^{2}}.\tag{24}\] 3c) Combining 23 and 24 , for all \(E \in \mathbb{R}\) if \(V\) vanishes at \(s_j\) then for any \(\delta>0\) we have \[\left| \left| \chi_j u \right| \right|_{L^{2}} \leq \frac{C}{\delta} \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + \delta \left| \left| u \right| \right|_{L^{2}}. \label{eq:pointchijest}\tag{25}\] 4) We now combine the cases together to obtain the two resolvent estimates. First, we assume that \(V\) vanishes on at least one interval \([a,b]\). Combining 22 and 25 , and noting that \(\lambda^{-\frac{2}{\gamma'+2}}\leq \lambda^{\frac{1}{\beta'+2}}\), then we have for any \(\delta>0\) \[\begin{align} \left| \left| u \right| \right|_{L^{2}} &\leq \sum_{j=1}^N \left| \left| \chi_j u \right| \right|_{L^{2}} + \left| \left| \chi_{N+1} u \right| \right|_{L^{2}} \\ &\leq \frac{C}{\delta} (\lambda^{\frac{1}{\beta'+2}} + \lambda^{-\frac{2}{\gamma'+2}} ) \left| \left| f \right| \right|_{L^{2}} + C \delta \left| \left| u \right| \right|_{L^{2}}\\ &\leq \frac{C}{\delta} \lambda^{\frac{1}{\beta'+2}} \left| \left| f \right| \right|_{L^{2}} + C \delta \left| \left| u \right| \right|_{L^{2}}. \end{align}\] Choosing \(\delta>0\) small enough, we can absorb the final term on the right-hand side back to obtain the first resolvent estimate.

Now assume that \(V\) vanishes only at points \(s_j\). Then applying 25 , for any \(\delta>0\), \[\begin{align} \left| \left| u \right| \right|_{L^{2}} &\leq \sum_{j=1}^N \left| \left| \chi_j u \right| \right|_{L^{2}} + \left| \left| \chi_{N+1} u \right| \right|_{L^{2}} \\ &\leq \frac{C}{\delta} \lambda^{-\frac{2}{\gamma'+2}} \left| \left| f \right| \right|_{L^{2}} + C \delta \left| \left| u \right| \right|_{L^{2}}. \end{align}\] Choosing \(\delta>0\) small enough, we can absorb the final term on the right-hand side back to obtain the second resolvent estimate. ◻

Proof of Proposition 16(1). 1) Recall the definition of \(A_v(W)\) \[A_v(W)(s) = \frac{1}{T_v} \int_0^{T_v} W(s,t) dt.\] Thus if \(s \in \partial\{A_v(W)>0\}\), then for some \(t\), we have \(sv^{\bot}+tv \in {\mathcal{G}}(v)\). By Remark 4, \({\mathcal{G}}\) is finite, and so there are finitely many points in \(\partial\{A_v(W)>0\}\). Therefore \(\{A_v(W)=0\}\) has finitely many connected components. Furthermore \(\{A_v(W)=0\}\) is closed as the level set of a continuous function. The only closed connected sets in \(\mathbb{S}^1\) are closed intervals or points. Thus \(\{A_v(W)=0\}\) is a finite union of closed intervals and points.

2) Now we show that if \({\mathcal{L}}_1(v) = \varnothing\), then \(A_v(W)=0\) only on a finite set. To see this, consider \(s_0 \in \partial\{A_v(W)>0\}\), so for some \(t_0\), we have \(t_0 v +s_0 v^{\bot}\in {\mathcal{G}}(v)\). If \[A_v(W)(s)=0 \text{ for } s \in (s_0, s_0+\varepsilon_0)\quad (\text{or } s\in (s_0-\varepsilon_0,s_0)),\] for some \(\varepsilon_0>0\), then for all \(t\) \[W(s,t) = 0 \text{ for } s \in (s_0, s_0+\varepsilon_0) \quad (\text{or } s\in (s_0-\varepsilon_0,s_0)).\] That is, for all \(\varepsilon\in (0,\varepsilon_0)\) \[L_{t_0v+(s_0+\varepsilon) v^{\bot}, v} \cap \omega =\varnothing, \quad (\text{or }L_{t_0v+(s_0-\varepsilon) v^{\bot}, v} \cap \omega =\varnothing).\] Therefore \(L_{t_0v+s_0 v^{\bot},v} \in {\mathcal{L}}_1(v) =\varnothing\), which is a contradiction. Thus \(A_v(W)>0\) on a punctured neighborhood of \(s_0\), and \(A_v(W)=0\) only on a discrete, and hence finite, set. ◻

Proof. We will consider the case where \(L_s \cap \omega \cap U \neq \varnothing\) for all \(s \in [s_0,s_0+\delta)\) as the proof for \(s \in (s_0-\delta, s_0]\) is analogous. Without loss of generality we may also assume that \(a>0\) in 26 so that \(s>s_0\) is mapped to \(x>0\) by \(\psi\).

When \(\varepsilon>0\) is taken small enough, for \((s,t) \in \omega \cap U\) we have \[C^{-1} \text{dist}\left( (s,t), \omega^c \cap U\right)^{\beta} \le W(s,t) \le C \text{dist}\left( (s,t), \omega^c \cap U\right)^{\beta }.\] By its definition \(\psi\) is bi-Lipschitz. Using this, 26 , and writing \(\widetilde{t}=b(s-s_0)+c(t-t_0)\), \[\begin{align} \text{dist}((s,t),\omega^c \cap U) \simeq \text{dist}(\psi(s,t), \psi(\omega^c \cap U)) =\text{dist}\left(\begin{pmatrix}a(s-s_0) \\ \widetilde{t} \end{pmatrix}, \psi(\omega^c \cap U) \right) . \label{eq:JbiLipschitz} \end{align}\tag{27}\] Now let \[\begin{align} \Gamma_{out} &=\{(x,y) \in \psi(U) : C_{out} |y|^{\eta} \leq x\}^c , \\ \Gamma_{in} &= \{(x,y) \in \psi(U) : y \geq 0, C_{in}^{-1} |y|^{\eta} \leq x \leq C_{in}|y|^{\eta}\}^c. \end{align}\] By Definition 3, \[\Gamma_{in} \supset \psi(\omega^c \cap U) \supset \Gamma_{out}.\] Therefore \[\text{dist}(\psi(z), \Gamma_{in}) \leq \text{dist}(\psi(z), \psi(\omega^c\cap U)) \leq \text{dist}(\psi(z), \Gamma_{out}),\] and so \[\label{e:sandwich-in-out} \text{dist}(\psi(z), \Gamma_{in}) \lesssim \text{dist}(z, \omega^c \cap U) \lesssim \text{dist}(\psi(z), \Gamma_{out}).\tag{28}\] We now focus on the upper bound. By Lemma 5, we have for any \(z\) such that \(\psi(z) \not \in \Gamma_{out}\): \[\label{e:dist-Gamma-out} \begin{align} \text{dist}(\psi(z),\Gamma_{out})=\text{dist}\left(\begin{pmatrix} a(s-s_0) \\ \widetilde{t} \end{pmatrix}, \Gamma_{out}\right) \lesssim \left\{ \begin{aligned} \left| |\tilde{t}|^{\eta} - C_{out}^{{-1}} a (s - s_0) \right|, & \;\; \eta \ge 1 , \\ \left| |\tilde{t}| - (C_{out}^{{-1}} a (s - s_0))^{1/\eta} \right|, & \;\; \eta \le 1 . \end{aligned} \right. \end{align}\tag{29}\] When \(\eta \geq 1\) we combine this with 28 to obtain \[\begin{align} \int_{L_s \cap U} W(s,t) dt~ &{\lesssim} \int_{L_s \cap U} \text{dist}\left( (s,t), \omega^c \cap U \right)^{\beta} d t \\ &\lesssim \int_{|\tilde{t}|^{\eta} \le C_{out}^{{-1}} a (s - s_0)} \bigg(C_{out}^{{-1}} a(s-s_0) - |\widetilde{t}|^{\eta}\bigg)^{\beta} d\widetilde{t} \\ &\lesssim |s - s_0|^{\beta + 1/\eta} , \end{align}\] where we used Lemma 4 in the last step, with \((\eta, k, \epsilon) = (\eta, C_{out}^{{-1}} a (s - s_0), 1)\). This is exactly the upper bound in part 1. When \(\eta \le 1\), we combine 29 with 28 to obtain \[\begin{align} \int_{L_s \cap U} W(s,t) dt ~&{\lesssim} \int_{L_s \cap U} \text{dist}\left((s,t), \omega^c \cap U \right)^{\beta} dt\\ &\lesssim \int_{|\tilde{t}|^{\eta} \le C_{out}^{{-1}} a (s - s_0)} \bigg((C_{out}^{{-1}} a(s-s_0))^{1/\eta} - |\widetilde{t}|\bigg)^{\beta} d\widetilde{t} \\ &\lesssim |s - s_0|^{(\beta + 1)/\eta} , \end{align}\] where we used Lemma 4 in the last step, with \((\eta, k, \epsilon) = ( 1, (C_{out}^{{-1}} a (s - s_0))^{1/\eta}, 1)\). This is exactly the upper bound in part 2.

Now we turn to the lower bound. By definition, \(\psi(z) = (x, y) \not\in \Gamma_{in}\) if and only if \[C_{in}^{-1} |y|^{\eta}\le x \le C_{in} |y|^{\eta} .\] Therefore, by Lemma 5, we have for any \(z\) such that \(\psi(z) \not\in \Gamma_{in}\): \[\begin{align} \text{dist}(\psi(z), \Gamma_{in})&=\text{dist}\left(\begin{pmatrix} a(s-s_0) \\ \widetilde{t} \end{pmatrix}, \Gamma_{in} \right) \\ &\gtrsim \left\{ \begin{aligned} \min\left\{ \left| |\tilde{t}|^{\eta} - C_{in} a (s - s_0) \right|, \left| |\tilde{t}|^{\eta} - C_{in}^{-1} a (s - s_0) \right| \right\} &\quad \textrm{if} \;\; \eta \ge 1 , \\ \min\left\{ \left| |\tilde{t}| - (C_{in} a (s - s_0))^{1/\eta} \right|, \left| |\tilde{t}| - (C_{in}^{-1} a (s - s_0))^{1/\eta} \right| \right\} &\quad \textrm{if} \;\; \eta \le 1 . \end{aligned} \right. \end{align}\] Splitting into cases based on whether \(|\widetilde{t}|^{\eta}\) is closer to \(C_{in} a(s-s_0)\) or \(C_{in}^{-1} a(s-s_0)\), with \(\tilde{C}_{in} := \frac{C_{in} + C_{in}^{-1}}{2}\), we can rephrase this as follows: if \(\eta \ge 1\), we have \[\begin{align} \label{e:l-bound-g} \text{dist}(\psi(z), \Gamma_{in}) &\gtrsim \left( C_{in} a (s - s_0) - |\tilde{t}|^{\eta} \right) \mathbf{1}_{C_{in} a (s - s_0) \ge |\tilde{t}|^{\eta} \ge \tilde{C}_{in} a (s - s_0)} \nonumber\\ &\qquad\qquad + \left( |\tilde{t}|^{\eta} - C_{in}^{-1} a (s - s_0) \right) \mathbf{1}_{C_{in}^{-1} a (s - s_0) \le |\tilde{t}|^{\eta} \le \tilde{C}_{in} a (s - s_0)} \nonumber\\ &\ge \left( C_{in} a (s - s_0) - |\tilde{t}|^{\eta} \right) \mathbf{1}_{C_{in} a (s - s_0) \ge |\tilde{t}|^{\eta} \ge \tilde{C}_{in} a (s - s_0)} . \end{align}\tag{30}\] For \(\eta \le 1\), we have similarly \[\label{e:l-bound-l} \text{dist}(\psi(z), \Gamma_{in}) \gtrsim \left( (C_{in} a (s - s_0))^{1/\eta} - |\tilde{t}| \right) \mathbf{1}_{C_{in} a (s - s_0) \ge |\tilde{t}|^{\eta} \ge \tilde{C}_{in} a (s - s_0)} .\tag{31}\] In the case where \(\eta \ge 1\), we raise both sides of  30 to the power \(\beta\) and integrate in \(\tilde{t}\), then we use the left-hand side of 28 to obtain \[\int_{L_s \cap U} \text{dist}\left( (s,t), \omega^c \cap U \right)^\beta dt \gtrsim \int_{C_{in} a (s - s_0) \ge |\tilde{t}|^\eta \ge \tilde{C}_{in} a (s - s_0)} \left( C_{in} a (s - s_0) - |\tilde{t}|^\eta \right)^\beta \, d \tilde{t} .\] Next, we decompose the right-hand side into two terms and apply Lemma 4 to each of them with \((\eta ,k) = (\eta, C_{in} a(s-s_0))\) and \(\epsilon = 1\) or \(\epsilon=\tilde{C}_{in}/C_{in} < 1\) \[\begin{align} \int_{L_s \cap U} W(s,t) dt~ &{\gtrsim} \int_{L_s \cap U} \text{dist}\left( (s,t), \omega^c \cap U \right)^{\beta} d t \\ & \gtrsim \int_{|\tilde{t}|^{\eta} \le C_{in} a (s - s_0)} \left( C_{in} a (s - s_0) - |\tilde{t}|^{\eta} \right)^{\beta} \, d \tilde{t} \\ & - \int_{|\tilde{t}|^{\eta} \le \tilde{C}_{in} a (s - s_0)} \left( C_{in} a (s - s_0) - |\tilde{t}|^{\eta} \right)^{\beta} \, d \tilde{t} \\ &= c_0(1) \left(C_{in} a (s - s_0)\right)^{\beta + 1/\eta} - c_0\left(\tfrac{\tilde{C}_{in}}{C_{in}} \right) \left(C_{in} a (s - s_0)\right)^{\beta + 1/\eta} \\ &\gtrsim (s - s_0)^{\beta + 1/\eta} \end{align}\] (recall that the function \(c_0\) from Lemma 4 is increasing). Following the same process in the case \(\eta \le 1\), this time choosing \(\epsilon = 1\) or \(\epsilon=\tilde{C}_{in}/C_{in} < 1\) and \((\eta ,k)\) \(= ( 1, (C_{in} a(s-s_0))^{1/\eta})\) in Lemma 4, we arrive at \[\int_{L_s \cap U} W(s,t) dt ~{\gtrsim} \int_{L_s \cap U} \text{dist}\left( (s,t), \omega^c \cap U \right)^{\beta} d t \gtrsim (s - s_0)^{(\beta + 1)/\eta} ,\] which yields the desired lower bound. ◻

Proof of 16 . Let \([\alpha, \rho]\) be one of the intervals from Proposition 16(1). Then \(A_v(W)(s)=0\) for \(s \in [\alpha,\rho]\), and for some \(\varepsilon>0\), \(A_v(W)(s)>0\) for all \(s \in (\alpha-\varepsilon,\alpha)\cup(\rho,\rho+\varepsilon)\). We will only estimate \(A_v(W)\) for \(s \in [\rho, \rho+\varepsilon)\) as an analogous argument applies for \(s\) near \(\alpha\).

In view of Proposition 16(1) and the definition of \(A_v(W)\), for all \(t\) such that \((\rho,t) \in {\mathcal{G}}(v)\), we have \((\rho,t) \in {\mathcal{G}}_1(v)\). By Remark 4 we know \({\mathcal{G}}_1(v)\) is finite. Thus there are finitely many points \((\rho, t_k) \in {\mathcal{G}}_1(v)\). Each point \((\rho, t_k)\) is a point of order \(\eta_k\), we let \(U_k\) be the associated coordinate neighborhood from Definition 3. We can also assume that they are pairwise disjoint. We also have for some \(\beta_k\) that \(W(z) \simeq d(z)^{\beta_k}\) near \((\rho, t_k)\).

Then, letting \(z_0=(\rho,t_1)\) and \(L_s=L_{z_0+sv^{\bot},v}\), we have \[\bigcup_{k=1}^n U_k \supset L_{\rho} \cap \partial\omega = \bigcup_{k=1}^n \{(\rho, t_k)\}.\] Furthermore, there exists \(\varepsilon>0\), such that for all \(s \in [\rho,\rho+\varepsilon)\), \[\bigcup_{k=1}^n U_k \supset L_{s} \cap \omega.\]

Therefore, for \(s \in [\rho, \rho+\varepsilon)\), we have \[A(W)(s) = \frac{1}{T_v} \int_{0}^{T_v} W(s,t) dt = \frac{1}{T_v} \sum_{k=1}^n \int_{L_s \cap U_k} W(s,t) dt.\] We apply Lemma 3 to each term of the sum to deduce that there exists \(C \geq 1\), such that \[\begin{align} C^{-1} (s-\rho)^{\beta_v} \leq \int_{L_s \cap U_k} W(s,t) dt \leq C (s-\rho)^{\beta_v}, \qquad \beta_v = \frac{\beta_k}{\min(\eta_k , 1)} + \frac{1}{\eta_k }, \end{align}\] which yields 16 . ◻

Proof of 17 . After some additional care to describe the geometry, the proof is similar to that of 16 . Consider one of the points \(s_0\) from Proposition 16(1). Thus \(A_v(W)(s_0)=0\), and for some \(\varepsilon>0\), \[\label{e:avwsg0} A_v(W)(s)>0 \text{ for all }s \in (s_0-\varepsilon,s_0)\cup(s_0, s_0+\varepsilon).\tag{32}\] The glancing points on \(L_{s_0}\) are the points \((s_0, t)\) for all \(t\) such that \((s_0, t) \in \partial\omega\). By Remark 4, \(\mathcal{G}\) is finite. Thus, there are finitely many points \((s_0, t_k) \in \mathcal{G}(v) \cap L_{s_0}\), which may be one-sided or two-sided. Each \((s_0,t_k)\) is a point of some order \(\eta_k\), and we let \(U_k\) be the associated coordinate neighborhood from Definition 3. We also have \(W(z) \simeq d(z)^{\gamma_k}\) for some \(\gamma_k\) on a neighborhood of \((s_0,t_k)\), either on one side or on both sides of \(L_{s_0}\).

As in the proof of 16 , there exists \(\varepsilon>0\), such that for all \(s \in (s_0-\varepsilon, s_0+\varepsilon)\), we have \(\bigcup_{k=1}^n U_k \supset L_{s} \cap \omega\). Therefore, for \(s \in (s_0-\varepsilon, s_0+\varepsilon)\) we have \[\label{eq:splitUj} A_v(W)(s) = \frac{1}{T_v} \int_0^{T_v} W(s,t) dt = \frac{1}{T_v} \sum_{k=1}^n \int_{L_s \cap U_k} W(s,t) dt ,\tag{33}\] where \(T_v\) is the length of \(L_s\).

By Lemma 3, for each \(s\) and \(k\) we have \[\label{e:wstless} \int_{L_s \cap U_k} W(s,t) dt \le C |s-s_0|^{\gamma_v}, \qquad \gamma_v = \frac{\gamma_k}{\min(\eta_k,1)} + \frac{1}{\eta_k}.\tag{34}\] Furthermore, for each \(s\), \[\label{e:wstgtr} \int_{L_s \cap U_k} W(s,t) dt \ge C^{-1} |s-s_0|^{\gamma_v},\tag{35}\] for all \(k\) such that the left side is not zero, and such a \(k\) exists for each \(s \ne s_0\) by 32 . Hence 17 follows from inserting 34 and 35 into 33 . ◻

Proof of Proposition 10. First, according to Lemma 1, for any \(r > 0\), there is a finite set \(\Upsilon_r \subset \mathbb{S}^1\) of rational directions on \(\mathbb{T}^2\) such that any geodesic with direction \(v \not\in \Upsilon_r\) enters every open ball of radius \(r\) in \(\mathbb{T}^2\). Note that if \(r_1 \le r_2\), then \(\Upsilon_{r_2} \subset \Upsilon_{r_1}\).

Let \(\mathcal{Q}'\) be the set of polygons \(P \in \mathcal{P}_n\) whose directions of edges do not belong to \(\Upsilon_{1/k_P}\), where \(k_P := \lfloor \frac{1}{r(\omega_P)} \rfloor + 1\), and \[\label{e:inradius} r = r(\omega_P) := \sup \left\{ \epsilon > 0 \;:\; \exists z \in \mathbb{T}^2 : B_{\varepsilon}(z) \subset \omega_P \right\} .\tag{36}\] Let \(\mathcal{Q}\) be the interior of \(\mathcal{Q}'\).

Introduce for any rational direction \(v \in \mathbb{S}^1\) and any \(k \in \{1, 2, \dots, n\}\) the set: \[\label{e:cal-E} \mathcal{E}_{v, k} := \mathbb{R}^{2(k-1)} \times \mathop{\mathrm{span}}(v) \times \mathbb{R}^{2(n-k)} \subset \mathbb{R}^{2n},\tag{37}\] which is a subspace of dimension \(2n - 1\). Note that for any \(v\) and \(k\), \(\mathcal{H}_n \not \subset \mathcal{E}_{v,k}\), as \(\mathcal{E}_{v,k}\) does not contain any polygon whose \(k\)th side is an irrational vector. Thus, \(\mathcal{H}_n \cap \mathcal{E}_{v, k}\) is a proper subspace of \(\mathcal{H}_n\), of dimension at most \(\dim \mathcal{H}_n - 1 = 2n - 3\).

We will prove that \(\mathcal{Q}\) is dense by showing that for any \(P \in \mathcal{P}_n\) there is \(\epsilon>0\) such that \[\label{e:calQ39} \mathcal{Q}' \cap B_{\epsilon}(P) \supset \Big(\mathcal{P}_n \backslash \bigcup \{\mathcal{E}_{v, k} \cap \mathcal{H}_n \colon 1 \le k \le n , v \in \Upsilon_{1/k_P}\}\Big) \cap B_{\epsilon}(P);\tag{38}\] this suffices because the right hand side of 38 is open and dense in \(B_{\epsilon}(P)\).

To prove 38 , choose a representation of \(P\), with vertices \(x_1, x_2, \ldots, x_n\), and a point \(x\) in the polygon at distance \(r := r(\omega_P)\) from \(\partial \omega_P\). For any \(\epsilon > 0\) and any other polygon \(\tilde{P}\) defined by a family of vertices \(\tilde{x}_1, \tilde{x}_2, \ldots, \tilde{x}_n\) such that \[|\tilde{x}_j - x_j| \le \epsilon , \qquad \forall j \in \{1, 2, \ldots, n\} ,\] we have for all \(j \in \{1, 2, \ldots, n\}\) and \(t \in [0, 1]\): \[\begin{align} |(1 - t) \tilde{x}_j + t \tilde{x}_{j+1} - x| &= |(1 - t) (\tilde{x}_j - x_j) + t (\tilde{x}_{j+1} - x_{j+1}) + (1 - t) x_j + t x_{j+1} - x| \\ &\ge |(1 - t) x_j + t x_{j+1} - x| - (1 - t) |\tilde{x}_j - x_j| - t |\tilde{x}_{j+1} - x_{j+1}| \\ &\ge r - \epsilon . \end{align}\] We deduce that any perturbed polygon \(\tilde{P}\) in \(B_\epsilon(P)\) is such that \(r(\omega_{\tilde{P}}) \ge r(\omega_P) - \epsilon\). Taking \(\epsilon\) sufficiently small ensures that \[k_{\tilde{P}} := \left\lfloor \dfrac{1}{r(\omega_{\tilde{P}})} \right\rfloor + 1 \le \left\lfloor \dfrac{1}{r(\omega_P) - \epsilon} \right\rfloor + 1 = \left\lfloor \dfrac{1}{r(\omega_P)} \right\rfloor + 1 =: k_P .\] Therefore, we have \(\Upsilon_{1/k_{\tilde{P}}} \subset \Upsilon_{1/k_P}\) for any \(\tilde{P} \in B_\epsilon(P)\). By definition of \(\mathcal{Q}'\), we deduce that \[\mathcal{Q}'^c \cap B_\epsilon(P) \subset \bigcup_{\substack{1 \le k \le n \\ v \in \Upsilon_{1/k_P}}} \mathcal{E}_{v, k} \cap \mathcal{H}_n \cap B_\epsilon(P) , \qquad \dim \mathcal{E}_{v, k} \cap \mathcal{H}_n \le 2n - 3 ,\] which implies 38 .

We now assume that \(P \in \mathcal{Q}\) and prove that the glancing set of \(\omega_P\) consists of vertices of order \(1\). Let \(L\) be a glancing line and \(z \in L \cap \partial \omega_P\) and denote by \(v\) the direction of \(L\). We claim that \(z\) is a vertex of the polygon. If not, since the polygon has no self-intersections, then either \(v\) is tangent to the edge containing \(z\), or \(L\) enters the interior of \(P\). In the first case \(L \cap \partial \omega_P\) would contain an edge, which is impossible in view of the definition of \(\mathcal{Q}' \supset \mathcal{Q}\) and Lemma 1. In the second case this contradicts \(L\) being glancing at \(z\). Thus our claim is true.

Now since the polygon has no self-intersections and projects properly onto the torus, we deduce that there are exactly two edges \(v_j, v_{j+1}\) connected to this vertex, lying on the same side of \(L\). We can construct an affine chart in a neighborhood of \(z\) by mapping \((\delta (v_{j+1} - v_j), \delta v), \delta \ll 1\), to the standard basis vectors \((e_1, e_2)\) in \(\mathbb{R}^2\).

Finally, we need to justify that for any \(P \in \mathcal{P}_n\), only finitely many admissible angles \(\theta \in \Theta(\omega_P)\) are such that \(R_\theta P \not\in \mathcal{Q}\). This is true because by Lemma 1 there are only finitely many directions \(v \in \Upsilon_{1/k_P}\) which can have glancing lines. For each \(j \in \{1, 2, \ldots, n\}\), there are finitely many angles for which \(R_\theta v_j\) coincides with one of these pathological directions, hence the result. ◻

Proof of Corollary 1. Let \(P \in \mathcal{Q}\). The glancing set consists of a finite number of points of order \(1\) by Proposition 10. In the worst-case scenario, we have \({\mathcal{L}}_1 \neq \varnothing\), hence by Theorem 7 decay at rate \(\alpha = 1 - \frac{1}{\beta + 1 + 3}\) (otherwise the decay is faster). For \(P \in \mathcal{P}_n\), this applies to all but a finite number of rotations of \(P\) by Proposition 10. ◻

Proof of Proposition 11. We first construct the set \(\mathcal{Y}\). On the one hand, for any \(\gamma \in \mathcal{U}\), we have a number \(r(\omega_\gamma) > 0\) (see 36 ), corresponding to the radius of a ball contained in \(\omega_\gamma\), the open set enclosed by \(\gamma\). On the other hand, according to Lemma 1, for any \(r > 0\), there is a finite set \(\Upsilon_r \subset \mathbb{S}^1\) of rational directions on \(\mathbb{T}^2\) such that any geodesic with direction \(v \not\in \Upsilon_r\) enters any open ball of radius \(r\) in \(\mathbb{T}^2\). We can also assume that whenever \(r_1 \le r_2\), then \(\Upsilon_{r_2} \subset \Upsilon_{r_1}\).

For any \(\gamma \in \mathcal{U}\), with the curvature \(\kappa\) as in 12 , we define \(f_\gamma \in C(\mathbb{S}^1; [0, + \infty))\) by \[f_\gamma(t) := \kappa\left(\gamma(t)\right) + \prod_{\pm} \prod_{v \in \Upsilon_{1/n}} \left| \dfrac{\dot{\gamma}(t)}{|\dot{\gamma}(t)|} \pm \dfrac{v}{|v|} \right| , \quad n = \left\lfloor \frac{1}{r(\omega_\gamma)} \right\rfloor + 1 , \qquad t \in \mathbb{S}^1 .\] Let \(\mathcal{Y}\) be the subset of \(\mathcal{U}\) of curves \(\gamma\) such that \(f_\gamma\) does not vanish. For such a curve \(\gamma\), the product in the definition of \(f_\gamma\) vanishes at any glancing point, since glancing directions necessarily belong to \(\Upsilon_{1/n}\). Hence we have \(\kappa > 0\) at every glancing point.

Let us prove that \(\mathcal{Y}\) is open. Consider \(\gamma \in \mathcal{Y}\) and \(\tilde{\gamma} \in \mathcal{U}\) such that \(\| \tilde{\gamma} - \gamma \|_{C^2} \le \epsilon\). Then \[r(\omega_{\tilde{\gamma}}) \ge r(\omega_\gamma) - \epsilon.\] This implies for \(\epsilon\) small enough \[n_{\tilde{\gamma}} := \left\lfloor \dfrac{1}{r(\omega_{\tilde{\gamma}})} \right\rfloor + 1 \le \left\lfloor \dfrac{1}{r(\omega_\gamma) - \epsilon} \right\rfloor + 1 = \left\lfloor \dfrac{1}{r(\omega_\gamma)} \right\rfloor + 1 =: n_\gamma.\] Therefore, we have \[\label{e:ups-n} \Upsilon_{1/n_{\tilde{\gamma}}} \subset \Upsilon_{1/n_\gamma} .\tag{39}\] Using the continuity of \[C^2(\mathbb{S}^1; \mathbb{R}^2) \ni \tilde{\gamma} \longmapsto \left| \kappa\left(\tilde{\gamma}(t)\right) \right| + \prod_{\pm} \prod_{v \in \Upsilon_{1/n_\gamma}} \left| \dfrac{\dot{\tilde{\gamma}}(t)}{|\dot{\tilde{\gamma}}(t)|} \pm \dfrac{v}{|v|} \right|,\] where we note that here we put \(n_\gamma\) in place of \(n_{\tilde{\gamma}}\), we deduce that the above function does not vanish for \(\tilde{\gamma}\) sufficiently close to \(\gamma\) in the \(C^2\) topology. In view of 39 , we conclude that \(f_{\tilde{\gamma}}\) does not vanish either, therefore \(\mathcal{Y}\) is open.

Now, let \(\gamma \in \mathcal{U}\) and let us show that “most rotations" are in \(\mathcal{Y}\). Consider the \(C^1\) map \[\begin{align} \mathbb{S}^1 \ni t &\longmapsto \dfrac{\dot{\gamma}(t)}{|\dot{\gamma}(t)|} \in \mathbb{S}^1. \end{align}\] Its critical points are exactly the times at which \(\kappa(\gamma(t)) = 0\), since the curvature is independent of the parametrization. By Sard’s theorem, the set of critical values, that is to say the set \(\mathcal{E} \subset \mathbb{S}^1\) of directions \[\frac{\dot{\gamma}(t)}{|\dot{\gamma}(t)|} \in \mathbb{S}^1 \qquad \textrm{with} \qquad \kappa\left(\gamma(t)\right) = 0\] has Lebesgue measure \(0\), and it is compact. Now note that \(f_\gamma\) does not vanish for \(\gamma\) rotated by some angle \(\theta \in \Theta(\omega_\gamma)\), when \(R_\theta \mathcal{E} \cap \Upsilon_{1/n_\gamma} = \varnothing\). Let us identify the directions in \(\Upsilon_{1/n_\gamma}\) with angles in \(\mathbb{S}^1\): \[\Upsilon_{1/n_\gamma} = \left\{ \theta_1, \theta_2, \ldots, \theta_k \right\} \subset \mathbb{S}^1 .\] Then we have \(R_\theta \mathcal{E} \cap \Upsilon_{1/n_\gamma} \neq \varnothing\) if and only if there exists \(j \in \{1, 2, \ldots, k\}\) such that \[\theta_j \in R_\theta \mathcal{E} \quad \Longleftrightarrow \quad \theta_j - \theta \in \mathcal{E} \quad \Longleftrightarrow \quad - \theta \in R_{- \theta_j} \mathcal{E} .\] We conclude that \[R_\theta \mathcal{E} \cap \Upsilon_{1/n_\gamma} = \varnothing , \qquad \forall \theta \in \mathbb{S}^1 \setminus \bigcup_{j = 1}^k R_{-\theta_j}\mathcal{E} ,\] and the above set of angles \(\theta\) is the complement of a compact set of measure \(0\).

We finally show that the set \(\mathcal{Y}\) is dense in \(\mathcal{U}\). For any \(\gamma \in \mathcal{U}\), the set \(\Theta(\omega_\gamma)\) is open, so it contains an open neighborhood of \(0\), in which we can pick \(\theta\), arbitrarily close to \(0\), such that \(R_\theta \gamma\) belongs to \(\mathcal{Y}\). ◻

Proof of Corollary 2. Let \(\gamma \in \mathcal{Y}\). By Proposition 11, we know that the curvature is non-zero at every glancing point \(z \in {\mathcal{G}}\). We can construct an affine chart in a neighborhood of \(z\) by mapping \(\delta (\ddot \gamma_1(0), \dot{\gamma}_1(0))\), \(\delta \ll 1\), to \((e_1, e_2)\) the standard basis vectors in \(\mathbb{R}^2\), where \(\gamma_1\) is the arc-length reparameterization of \(\gamma\) with \(\gamma_1(0) = z\). In this chart, we have \(\gamma_1(t) = (t^2/2 \delta^2, t/\delta) + o(t^2)\), namely the point \(z\) is of order \(\eta = 2\). Theorem 7 yields decay at rate at least \(1 - \frac{1}{\beta + 1/2 + 3}\).

Given \(\gamma \in \mathcal{U}\), the above applies to rotations outside the compact negligible set \(K\) from Proposition 11, which concludes the proof. ◻

Proof. By symmetry, it is sufficient to study the integral on \(t \ge 0\). We introduce the change of variables \(t' = k^{-1} t^\eta\), \(dt = \eta^{-1} k^{1/\eta} {t'}^{1/\eta-1}\,dt'\), so that \[\int_0^{(\epsilon k)^{1/\eta}} \left(k - t^\eta\right)^\beta \, dt =\int_0^{\epsilon} (k - kt')^\beta \eta^{-1} k^{1/\eta} {t'}^{1/\eta-1} \, dt' = k^{\beta + 1/\eta} \underbrace{\eta^{-1} \int_0^{\epsilon} (1 - t')^\beta t'^{1/\eta - 1} \, dt'}_{=: c_0(\epsilon)/2} ,\] hence the result. ◻

Proof. Case \(\eta < 1\). Let \(z_0 = (x_0, y_0) \in [0, + \infty) \times [0, + \infty)\) (by symmetry we can consider \(y_0 \geq 0\)). Let \(z_1 = (x_1, y_1)\) be the point on \(x=c|y|^\eta\) closest to \((x_0, y_0)\) and denote by \(z_0' = (x_0, (c^{-1} x_0)^{1/\eta})\) the vertical projection of \(z_0\) onto \(\Gamma_c\). In the triangle with vertices \(z_0, z_0', z_1\), let us introduce the sidelengths \[d := |z_1 - z_0|, \qquad d' := |z_0' - z_0|, \qquad \textrm{and} \qquad \ell := |z_1 - z_0'| .\] Our goal is to prove that \(d' \sim d\) for \(\varepsilon\) small enough. The slope of the tangent to \(\Gamma_c\) at a point \(z = (x, (c^{-1}x)^{\frac{1}{\eta}})\) is equal to \(\frac{c^{-1/\eta}}{\eta} x^{\frac{1}{\eta}-1}\). Note that as \(z_0 \rightarrow 0\), we have \(z_1 \rightarrow 0\) and \(z_0' \rightarrow 0\). Therefore, since \(\eta<1\), the slope at \(z_1\) and \(z_0'\) tends to zero as \(z_0 \to 0\), and so does the slope of the line spanned by \(z_1 - z_0'\). Hence, \[\label{e:slope1} (z_1 - z_0') \cdot e_2 = o(|z_1 - z_0'|) = o(\ell) .\tag{40}\] Since \(z_1\) minimizes the distance from \(z_0\) to \(\Gamma_c\), \(z_1-z_0\) is perpendicular to the tangent line of \(\Gamma_c\) at \(z_1\). Thus the slope of the line spanned by \(z_1 - z_0\) tends to infinity as \(z_0 \to 0\). Therefore we have \[\label{e:slope2} \left|(z_1 - z_0) \cdot e_2\right| = |z_1 - z_0| + o(|z_1 - z_0|) = d + o(d) .\tag{41}\] Multiplying 40 and 41 by \(d' = |z_0 - z_0'|\), since \(z_0-z_0'\) is parallel to \(e_2\), we deduce that \[\left\{ \begin{align} \left|(z_1 - z_0') \cdot (z_0' - z_0) \right| &= o(\ell d') , \\ \left|(z_1 - z_0) \cdot (z_0 - z_0')\right| &= d d' + o(d d') . \end{align} \right.\] Notice that, in the second equation, we have \((z_1 - z_0) \cdot (z_0 - z_0') \le 0\), since \(y_1 - y_0\) and \(y_0 - y_0'\) always have opposite signs, regardless of whether \(z_0\) is to the right or to the left of \(\Gamma_c\). Computing directly and applying the above equations we have \[\left\{ \begin{align} d^2 &= |z_1 - z_0|^2 = |z_1 - z_0'|^2 + |z_0' - z_0|^2 + 2 (z_1 - z_0') \cdot (z_0' - z_0) = \ell^2 + d'^2 + o(\ell d') , \\ \ell^2 &= |z_1 - z_0'|^2 = |z_1 - z_0|^2 + |z_0 - z_0'|^2 + 2 (z_1 - z_0) \cdot (z_0 - z_0') = d^2 + d'^2 - 2 d d' + o(d d') . \end{align} \right.\] Rearranging the first equation and then combining it with the second, we obtain \[d^2 - d'^2 = \ell^2 + o(\ell d') = d^2 + d'^2 - 2dd' + o\left( \ell d' + d d' \right).\] Rearranging again \[o\left( (\ell + d) d' \right) = d^2 - d'^2 - d^2 -d'^2 +2dd' = (d - d') 2 d' .\] If \(d' = 0\), the situation is trivial since then \(z_0 = z_0' = z_1 \in \Gamma_c\). Otherwise, we have \[d - d' = o(\ell + d) .\] Since \(\ell \le d + d' \le 2 \max\{d, d'\}\), we conclude that \(d \sim d'\) as \(z_0 \rightarrow 0\), which is the desired result for \(\eta \in (0, 1)\).

Case \(\eta > 1\). For \(\eta \in (1, + \infty)\), once again, we can assume without loss of generality that \(y_0 \ge 0\) and that the distance \(\text{dist}(z_0, \Gamma_c)\) is achieved at a point \(z_1 \in \Gamma_c \cap \{y \ge 0\}\), by symmetry with respect to \(\{y = 0\}\). We reduce to the previous case by applying the isometry \(T : (x, y) \mapsto (y, x)\) on \(\mathbb{R}^2\): we have \(T(\Gamma_c \cap \{y \ge 0\}) = \{x = c' |y|^{\eta'}\} \cap \{y \ge 0\}\) with \(\eta' := 1/\eta < 1\) and \(c'=c^{-1/\eta}\). Applying the previous case to \(T(z_0) = (y_0, x_0)\) in place of \((x_0, y_0)\), we obtain \[\text{dist}(z_0, \Gamma_c) = \text{dist}\left(T(z_0), T(\Gamma_c)\right) \asymp \left| x_0 - \left(\frac{1}{c'} y_0\right)^{1/\eta'} \right| = \left| x_0 - c y_0^\eta \right| ,\] dividing by \(c\) proves the sought result for \(\eta \in (1, + \infty)\).

Case \(\eta = 1\). For \(\eta = 1\), we can compute the distance explicitly, for any \(c > 0\): assuming \(y_0 \ge 0\) again, the distance is achieved at \(z_1 = (x_1, y_1) = (c y_1, y_1)\) such that \[0 = \begin{pmatrix} x_0 - c y_1 \\ y_0 - y_1 \end{pmatrix} \cdot \begin{pmatrix} c \\ 1 \end{pmatrix} = y_1 \left( c x_0 + y_0 - (1 + c^2) y_1 \right) .\] Therefore, by direct calculation, \[\text{dist}(z_0, \Gamma_c) = \dfrac{|c y_0 - x_0|}{\sqrt{1 + c^2}} , \qquad \forall z_0 \in \{x \ge 0\} ,\] which finishes the proof. ◻

Proof of Lemma 1. Geodesics with irrational direction are dense, so they intersect \(\omega\). Let \(v \in \mathbb{S}^1\) be a rational direction. Then \(v = (p, q)/T_v\) for some coprime integers \(p\) and \(q\), where \(T_v = (p^2 + q^2)^{1/2}\) is the period of any geodesic \(L_{z, v}\), \(z \in \mathbb{T}^2\). Let \(a, b\) be Bézout coefficients, such that \(ap + bq = 1\). Then we have \[\begin{align} \dfrac{1}{T_v^2} \begin{pmatrix} -q \\ p \end{pmatrix} - \dfrac{bp - aq}{T_v^2} \begin{pmatrix} p \\ q \end{pmatrix} &= \begin{pmatrix} -b \\ a \end{pmatrix} \in \mathbb{Z}^2 . \end{align}\] We deduce that for any \(z \in \mathbb{T}^2\), we have \[\label{e:transl-geod} z + \dfrac{1}{T_v^2} \begin{pmatrix} -q \\ p \end{pmatrix} \equiv z + \dfrac{bp - aq}{T_v} v \pmod{\mathbb{Z}^2} \in L_{z, v} .\tag{42}\] The vector \((-q, p)/T_v^2\) is orthogonal to \(v\) and has norm \(1/T_v\). If \(1/T_v < 2 \varepsilon\), 42 implies that any geodesic of the form \(L_{z, v}\) must intersect any ball of radius \(\varepsilon\). Therefore, geodesics not entering \(\omega\) must have length \(T_v \le 1/2 \varepsilon\), hence \(|p|, |q| \le 1/2 \varepsilon\), and the result follows. ◻