1 Introduction and Background↩︎

Rectifiability plays a fundamental role in geometric measure theory. Roughly speaking, rectifiable sets behave like subsets of Lipschitz graphs, while purely unrectifiable sets have no such structure.

Investigating geometric characterizations of rectifiability is a central problem in the field. We focus on two results in the plane that relate rectifiability to projection theory: the Besicovitch projection theorem and the two-projection theorem. Before stating each of these results, we give some relevant definitions.

The Besicovitch projection theorem asserts the following:

The Besicovitch theorem can be formulated in terms of Favard length. The Favard length of a Borel set \(E\subseteq\mathbb{R}^2\) is defined as \[\mathop{\mathrm{Fav}}(E):=\int_{S^1} \mathcal{L}^1(\mathop{\mathrm{proj}}_\omega(E))\,d\sigma(\omega).\] So, with \(E\) as in Theorem 3, if \(E\) is purely \(1\)-unrectifiable, then \(\mathop{\mathrm{Fav}}(E)=0\).

The Besicovitch projection theorem is foundational and has inspired several major lines of further research, including (i) the Favard length problem, (ii) the development of quantitative variants, and (iii) the study of nonlinear variants, in which the family of orthogonal projections is replaced by some other family of maps. We give a brief introduction to these rapidly developing research areas in Section 4.

In this article, we initiate further investigation into the comparatively less studied two‑projection theorem; the proof of this theorem is elementary and relies on the Lebesgue density theorem and the Rademacher differentiation theorem, see [2].

In particular, we obtain a nonlinear two-projection theorem for generalized projections. This has immediate applications to pinned distance problems and radial projections. Furthermore, motivated by the work of Tao [2], we obtain a quantitative two-projection theorem for generalized projections.

Our paper has three aims:

1. We extend the classic two-projection theorem to a nonlinear setting and show that the conclusion of this theorem still holds when the family of orthogonal projection maps is replaced by a family of nonlinear projections from a wide class. These results are presented in Section 2, and the main result is Theorem 7.

2. As an application, we verify that rectifiable \(1\)-sets have pinned distance sets with positive measure for all but possibly one exceptional pin; see Section 2.3. We obtain analogous results for curve projection operators in Section 2.2 and radial projections in Section 2.4.

3. We utilize a multiscale analysis to obtain a quantitative version of the two-projection theorem for a family of nonlinear projections in Theorem 24. This second aim extends a theorem of Tao, and it provides a mechanism to bound the rectifiability constant. This serves as the opposite direction of the nonlinear quantitative Besicovitch theorem obtained by Davey and the second listed author in [3]. These results are presented in Section 3, and further context is provided in Section 4.

These results highlight the robustness of the tools available to us. Our primary aim is to elucidate such tools through our new results and new applications. To put our results into context, we provide a summary of recent developments in the field in Section 4. The proof of Theorem 7 is given in Sections 5, and an alternative proof using the area formula is provided in Section 6. The proof of Theorem 24 is provided in Section 7.

Some basic notation used throughout the paper follows.

1.1 Acknowledgment↩︎

K.T. is supported in part by the Simons Foundation Grant GR137264. Part of this work was completed at the American Institute of Math during the SQuaRE Covering Fractals by Curves. The authors wish to thank Paige Bright for many helpful comments and discussions on this article.

2 Generalized Two Projection Theorems and Applications↩︎

In this section, we investigate how the two-projection theorem can be extended to more general projection families. We derive the classic two-projection theorem as a consequence, and we introduce additional conditions so that the original proof continues to hold in a broader setting.

The proof of Theorem 7 is found in Section 5.

We now consider several concrete applications of Theorem 7 to establish results for orthogonal projections, curved projections, pinned distance sets, and radial projections.

2.1 Application 1: Orthogonal Projections↩︎

As the first application, we recover the classic two-projection theorem.

2.2 Application 2: Curve Projections↩︎

Curve projections were originally introduced by Simon and the second listed author to study Minkowski sum sets of the form \(A+\Gamma\), where \(A\subseteq \mathbb{R}^2\) and \(\Gamma\) is a well-behaved curve [4]. Further applications and properties of these maps were later developed in [3], [5]–[7].

With this set-up, the following is a consequence of Theorem 7.

Observe that the hypothesis that \(\gamma'\) is injective is necessary; for instance the statement is no longer true when \(\Gamma=\{(t,t^3):t\in\mathbb{R}\}\). Corollary 10 has implications for large unions of curves; see the follow-up paper [8] for more details.

2.3 Application 3: Pinned Distance Sets↩︎

Let \(E\subseteq \mathbb{R}^2\) be a compact set. For each \(y\in \mathbb{R}^2\), the pinned distance set of \(E\) at \(y\) is defined as \[\Delta_y(E):=\{|x-y|:x\in E\}.\]

In fact, the conclusion still holds even without the constraint \(\mathcal{H}^1\left(E\cap\ell_{y,y'}\right)=0\). This can be seen by a simple geometric lemma below.

By Lemma 12, we obtain the following two-projection theorem for pinned distance sets. The proof of Corollary 13 is similar in nature to that of Theorem 7, and for this reason we include both proofs in Section 5.

We note that a higher dimensional variant of Corollary 13 is obtained in [8] for pinned distance sets corresponding to \(E\subseteq \mathbb{R}^d\) for \(d\geqslant 2\). While the proof of Corollary 13 relies on a direct modification of the original proof of the two-projection theorem, the proof in [8] depends on generalizing an inequality of Federer, which gives control of the \(\mathcal{H}^m\)-measure of rectifiable sets in terms of their projections onto orthogonal systems of \(m\)-flats. For further references on pinned distance sets, see [9]–[12].

2.4 Application 4: Radial Projections↩︎

Now we consider a variant of the two-projection theorem for radial projections, and we begin with an example that shows some caution and restrictions are required.

Given \(x\in\mathbb{R}^2\), the radial projection \(\pi_x:\mathbb{R}^2\setminus\{x\}\to S^1\) is defined as \[\pi_x(y):=\frac{y-x}{|y-x|},\quad\forall\,y\in \mathbb{R}^2\setminus\{x\}.\]

Orponen and Sahlsten [13] showed that the two-projection theorem holds for radial projections under an additional geometric constraint. In fact, they proved that the above counterexample is essentially the only setting in which two-projection theorem could fail.

Their proof is similar in spirit to the proof of Theorem 7, and it relies on the local structure of density points. In fact, the proof of Theorem 7 can be adapted to prove the above theorem. The key observation is that for \(a,b\in \mathbb{R}^2\), we have \[|\pi_x(a)-\pi_{x}(b)|\sim \theta_x(a,b),\] where \(\theta_x(a,b)\in[0,\pi]\) is the angle between the vectors \(a-x\) and \(b-x\).

3 Quantitative Version of Two-projection Theorem for Curve Projections↩︎

We now turn to our aim of quantifying the two-projection theorem for generalized projections. Before we state the main result of this section, Theorem 24, we require a quantified notion of rectifiability.

3.1 Rectifiability Constant↩︎

The rectifiability constant was introduced by Tao [2] in his study of asymptotic upper bounds for the Favard length of arbitrary unrectifiable sets. The rectifiability constant provides a framework that measures how closely a planar set can be approximated by Lipschitz graphs. In this paper, we adopt this notion of the rectifiability constant - with some minor modifications - as a basic tool.

We first give a simplified characterization of unrectifiability. By the implicit function theorem, Whitney’s extension theorem, and Lusin’s theorem, every Lipschitz graph \(\zeta\) can be locally expressed as a countable union of horizontal or vertical Lipschitz graphs, up to an \(\mathcal{H}^1\)-null set. Each of these graphs can be parametrized either with respect to the first coordinate, \(\{(x,F(x)):x\in I\}\), or with respect to the second coordinate, \(\{(F(x),x):x\in I\}\), where \(I\) is a small interval. As a result, it suffices to test the unrectifiability with respect to Lipschitz graphs parametrized with the standard two coordinate axes.

The rectifiability constant quantifies the extent to which a set exhibits rectifiable or unrectifiable behavior, as established in [2].

3.2 Quantitative Two-projection Theorem↩︎

We first require some basic assumptions on the curve \(\Gamma\).

Let \(\Gamma=\{(t,\gamma(t)):t\in \mathbb{R}\}\) satisfy this condition. Then \(\gamma'\) is injective, hence monotone, and \(\gamma''\) exists almost everywhere. Without loss of generality, we assume that \(\gamma'\) is strictly decreasing, so that \(\Gamma\) is concave down.

With this setup, we study the one-parameter family of projections \(\{\Phi_\alpha\}_{\alpha\in\Lambda}\) on unit cube \([0,1]^2\):

We now state our second main result, which can be compared to [2]. Recall that the restricted rectifiability constant and accompanying notation is defined in Definition 18.

It is straightforward to deduce the qualitative two-projection theorem from Theorem 24. We include the proof here for completeness.

Combined with Theorem 30 later, we obtain the following characterization of purely \(1\)-unrectifiable sets via curve projection.

4 Short Survey: Curve Projections and Favard Curve Length↩︎

The purpose of this section is to put the previous section into greater context by giving an introduction to the Favard problem and its nonlinear analogues. For a more in-depth survey on these topics by the second author, see [14].

A primary application of the classic two projection theorem is to verify that a given set \(E\subset \mathbb{R}^2\) is purely \(1\)-unrectifiable, which in turn allows one to apply the Besicovitch projection theorem and conclude that \(\mathop{\mathrm{Fav}}(E)=0\).

Likewise, a primary application of Tao’s quantitative two-projection theorem is to verify the hypothesis of his corresponding quantitative Besicovitch theorem, which, in turn, yields quantitative bounds on Favard length (see [2]).

It follows from the Besicovitch projection theorem, Theorem 3, that, if \(E\subseteq\mathbb{R}^2\) is a purely \(1\)-unrectifiable compact set with \(0<\mathcal{H}^1(E)<\infty\), then \[\mathop{\mathrm{Fav}}(E)=\lim_{r\to 0}\mathop{\mathrm{Fav}}(N_r(E))=0,\] where \(N_r(E)=\{x\in\mathbb{R}:d(x,E)\leqslant r\}\) is the closed \(r\)-neighborhood of \(E\).

It is natural to ask how fast \(\mathop{\mathrm{Fav}}(N_r(E))\) decays as \(r\to 0\). In recent years, there have been substantial efforts on establishing asymptotic bounds for the decay of Favard length. This is known as the Favard length problem.

Several works [15]–[18] focused on special self-similar sets, including the classic four-corner Cantor set. Beyond self-similar sets, estimates for more general sets were provided in [2], [19]. In particular, Tao [2] analyzed the proof of the Besicovitch projection theorem and quantified its main steps via multilinear analysis, to obtain upper bounds on the Favard length. Tao also gave a quantitative version of two-projection theorem to check the hypotheses of the quantitative Besicovitch projection theorem.

It is worth mentioning that the Besicovitch projection theorem can be stated as follows. If \(E\subseteq\mathbb{R}^2\) is a Borel set with \(0<\mathcal{H}^1(E)<\infty\) and \(\mathop{\mathrm{Fav}}(E)>0\), there exists a Lipschitz graph \(\zeta\subseteq\mathbb{R}^2\) such that \(\mathcal{H}^1\left(E\cap\zeta\right)>0\). In recent years, several works [20]–[24] have aimed to quantify this statement. These result study how large \(\mathop{\mathrm{Fav}}(E)\) must be to guarantee that \(E\) contains a big piece of Lipschitz graph, that is, there exists a Lipschitz graph \(\zeta\subseteq\mathbb{R}^2\) such that \(\mathcal{H}^1\left(E\cap \zeta\right)\gtrsim \mathcal{H}^1(E)\). This question is closely related to analytic capacity, in particular to Vitushkin’s conjecture. We refer to [25] for an introduction to this topic; see also [21] for recent progress.

4.1 Favard Curve Length↩︎

It is natural to formulate the Favard problem in a nonlinear setting. For instance, we can introduce a nonlinear variant of Favard length.

We can also state a nonlinear variant of the Besicovitch projection theorem.

See [4] for detailed proof. For further developments on nonlinear Favard length problem, we refer to [3]–[7], [14], [26].

4.2 A Quantitative Besicovitch Projection Theorem for Curved Projections↩︎

Following the approach introduced by Tao in [2], Davey and the second listed author used multiscale analysis to quantify the Besicovitch projection theorem for curve projections in [3]. We begin with the definition of Hausdorff content.

We now state the quantitative nonlinear Besicovitch projection theorem in [3]. Roughly speaking, they showed that if \(E\) is close to being purely \(1\)-unrectifiable, \(\mathop{\mathrm{Fav}}_\Gamma(E)\) is very small.

The qualitative version of Besicovitch projection theorem for curve projections follows as a direct application of Theorem 32; see [3] for a detailed proof.

In Section 3, we give a quantitative two-projection theorem for curve projections, which goes in the opposite direction of Theorem 32. More precisely, while Theorem 32 shows that quantitative unrectifiability forces curve projections to be small, our result shows that if two curve projections are small, the set must have small intersection with every Lipschitz graph.

5 Proof of Theorem 7 and Corollary 13↩︎

We now prove Theorem 7. The main ingredient of the proof is applying the Lebesgue differentiation theorem, the Rademacher’s differentiation theorem, and the intermediate value theorem to achieve a contradiction. Moreover, this proof will serve as a model to prove Theorem 24. Our proof is very similar to that in [2], but we provide additional details and treat a more general setting.

6 Alternative Proof of Theorem 7↩︎

We provide an alternative proof of contrapositive of two-projection theorem for \(1\)-rectifiable set. The proof still relies on a Rademacher-type theorem. Meanwhile, it uses the area formula in place of the Lebesgue differentiation theorem and the intermediate value theorem. The main idea is the same: we connect projections to differentiability.

For the introduction and proofs of proceeding lemmas, see [27].

7 Proof of Theorem 24↩︎

Our proof follows along the lines of Tao’s quantitative two-projection theorem in the linear setting with some necessary modifications. We first give the setup and outline the strategy for passing from qualitative to quantitative results.

7.1 Setup↩︎

By the trivial bound \(\mathcal{R}_{E,\text{res}}(r_N,1,N^{1/100})\leqslant 1\), we have \(N\lesssim 1\), so we may assume that \(N\) is sufficiently large. For convenience, we assume that \(N^{1/100}\in\mathbb{N}\) and \(N^{1/100}\geqslant 3\).

Since \(E\subseteq [0,1]^2\), it suffices to prove that for every Lipschitz function \(F:\mathbb{R}\to \mathbb{R}\) with \(\mathop{\mathrm{Lip}}(F)\leqslant N^{1/100}\), \[\label{eqn:case32XY} \mathcal{L}^1(\{x\in[0,1]:\exists\,y\in[-r_N,r_N]\;\text{such that}\;(x,F(x)+y)\in E\})\lesssim N^{-1/100},\tag{2}\] and \[\label{eqn:case32YX} \mathcal{L}^1(\{x\in[0,1]:\exists\,y\in[-r_N,r_N]\;\text{such that}\;(F(x)+y,x)\in E\})\lesssim N^{-1/100}.\tag{3}\] We prove (2 ). The proof of (3 ) is the same with some straightforward modifications. See Remark 65.

Fix such an \(F\). Let \[A:=\{x\in[0,1]:\exists\,y\in[-r_N,r_N]\;\text{such that}\;(x,F(x)+y)\in E\}.\] Observe that \(A\) is a compact set, because \(E\) is a compact. Also, if \(x\in A\), there exists \(y\in[-r_N,r_N]\) such that \((x,F(x)+y)\in E\). Note that \[|\Phi_\lambda(x,F(x)+y)-\Phi_\lambda(x,F(x))|=|F(x)+y+\gamma(\lambda-x)-F(x)-\gamma(\lambda-x)|=|y|\leqslant r_N,\] so \[\label{eqn:alpha32neighbor} \{\Phi_{\lambda}(x,F(x)):x\in A\}\subseteq \mathcal{N}_{r_N}(\Phi_{\lambda}(E)),\quad\forall\,\lambda\in \Lambda.\tag{4}\]

We will show (2 ), that is, \[\mathcal{L}^1(A)\lesssim N^{-1/100}\] More precisely, we prove that if 2 fails, 4 imply that the small projection assumption fails for at least one of \(\alpha\) and \(\alpha'\).

7.2 Proof Strategy↩︎

Recall that the two-projection theorem is proved by contradiction using Lebesgue differentiation theorem, Rademacher’s differentiation theorem, and intermediate value theorem. In the quantitative setting, we replace these pointwise statements with scale-dependent versions.

For the quantitative Lebesgue differentiation theorem, we select a dyadic interval that interests \(A\) and is stable across finer scales, so that the density remains high on all subintervals. Since the number of scales \(\{r_n\}_{n=1}^N\) is finite, we use the pigeonhole principle to isolate a suitable scale. See Section 7.3.

A similar idea applies to the quantitative Rademacher’s differentiation theorem. we approximate the projection map \(x\mapsto \Phi_\lambda(x, F(x))\) by piecewise linear functions across scales, and use an \(L^2\)-estimate to obtain good approximate differentiability at a chosen scale. In particular, unlike in the linear case, we must not only quantify the derivative of the relevant Lipschitz function \(F\) but also the function \(\gamma\). See Section 7.4.

Finally, we replace the intermediate value theorem with a counting argument. The main tool is the the Rising Sun Lemma, which allows us to track the total growth and decay on the well-behaved dyadic intervals and derive a contradiction with the small projection assumption. This step involves several technical difficulties that do not appear in the linear setting. See Section 7.5.

7.3 Quantitative Lebesgue Differentiation Theorem↩︎

We first prove the quantitative Lebesgue differentiation theorem, which requires the following version of the pigeonhole principle. Note that it is also used in Section 7.4.

Now we state the quantitative Lebesgue differentiation theorem.

Informally, most points in the coarse-scale set \(A_{n_0-N^{-3/100}N}\) remain density points at the finer scale \(A_{n_0+N^{-3/100}N}\). Although \(A_{n_0+N^{-3/100}N}\) is slightly larger than \(A\), it still serves as a good approximation for \(A\) in our analysis.

7.4 Quantitative Rademacher’s Differentiation Theorem↩︎

Recall that \(\Gamma=\{t,\gamma(t):t\in\mathbb{R}\}\). By the nonlinear nature of the curve projections, it is not sufficient to quantify the behavior of \(F\) alone. We also need to quantify the curve \(\gamma\) so that it admits a good linear approximation. The heuristic is to control the derivative of the function \(x \mapsto \Phi_\lambda(x,F(x))\) for \(\lambda\in \{\alpha,\alpha'\}\). This derivative takes the form \(F'(x)-\gamma'(\lambda-x)\).

For \(n\in\{1,2,\cdots,N\}\), recall the dyadic grid \(\{jr_n\in [0,1]:j\in\mathbb{N}\}\). Let \(F_n:[0,1]\to \mathbb{R}\) be the unique piecewise linear function that agrees with \(F\) on this dyadic grid, that is, \(F_n(jr_n)=F(jr_n)\), and is linear on each dyadic interval \([jr_n,(j+1)r_n]\). For \(\lambda\in\{\alpha,\alpha'\}\), let \(\gamma_\lambda:[0,1]\to \mathbb{R}\) be defined as \(\gamma_\lambda(x):=\gamma(\lambda-x)\). Then similarly, for \(\lambda\in\{\alpha,\alpha'\}\), let \(\gamma_{\lambda,n}:[0,1]\to \mathbb{R}\) be the unique piecewise linear function that agrees with \(\gamma_\lambda\) on this dyadic grid.

We start with a property of proceeding piecewise linear functions that will be used in Section 7.6.

Since \(\mathop{\mathrm{Lip}}(F)\leqslant N^{1/100}\) and \(\sup\limits_{t\in \mathbb{R}}|\gamma'(t)|\leqslant 1\), we have \(\mathop{\mathrm{Lip}}(F_n)\leqslant N^{1/100}\) and \(\mathop{\mathrm{Lip}}(\gamma_{\lambda,n})\leqslant 1\) for \(\lambda\in\{\alpha,\alpha'\}\). Also, since \(F_n'\) exists a.e., \(\|F_n'\|_{L^\infty([0,1])}\leqslant N^{1/100}\). This gives \(\|F_n'\|_{L^2([0,1])}^2\lesssim N^{2/100}\). Similarly, \(\|\gamma_{\lambda,n}'\|_{L^2([0,1])}^2\leqslant 1\leqslant N^{2/100}\) for \(\lambda\in\{\alpha,\alpha'\}\). Thus, \[\label{eqn:L232sum} \|F_n'\|_{L^2([0,1])}^2+\|\gamma_{\alpha,n}'\|_{L^2([0,1])}^2+\|\gamma_{\alpha',n}'\|_{L^2([0,1])}^2\lesssim N^{2/100}.\tag{5}\]

We now show that \(\|F_n'\|_{L^2([0,1])}^2,\;\|\gamma_{\alpha,n}'\|_{L^2([0,1])}^2\), and \(\|\gamma_{\alpha',n}'\|_{L^2([0,1])}^2\) are increasing functions in \(n\). To see this, we first prove \(F_{k+1}'-F_k'\), \(\gamma_{\alpha,k+1}'-\gamma_{\alpha,k}'\), and \(\gamma_{\alpha',k+1}'-\gamma_{\alpha',k}'\) are respectively pairwise orthogonal in the sense of \(L^2([0,1])\).

Now we are ready to state the quantitative Ramemacher’s differentiation theorem.

Informally, this shows that \(F\) has good approximate differentiability between the coarse scale \(n_1 - N^{-9/100}N\) and the finer scale \(n_1 + N^{-9/100}N\). This serves as a quantitative version of Rademacher’s differentiation theorem. We also impose quantitative control on \(\gamma_{\alpha}\) and \(\gamma_{\alpha'}\) so that they interact well with the behavior of \(F\).

In the rest of paper, we abbreviate \[n_0^{\pm}=n_0\pm N^{-3/100}N\;\text{and}\; n_1^{\pm}=n_1\pm N^{-9/100}N.\]

7.5 Rising Sun Lemma↩︎

In this section, we highlight the importance of the Rising Sun Lemma. Roughly speaking, the Rising Sun Lemma serves as a substitute for the intermediate value theorem. It allows us to track the cumulative growth and decay of the map \(\lambda\mapsto\Phi_\lambda(x,F(x))\), for \(\lambda\in\{\alpha,\alpha'\}\), on suitably chosen dyadic intervals where the functions \(A\), \(F\), and \(\gamma\) are well behaved. The key point is that, on each such interval, the linear approximation has well-controlled slope. The Rising Sun Lemma then guarantees that either only few intervals contribute to a given level set, or a positive proportion of the contributing intervals exhibit unfavorable behavior.

We first fix a dyadic interval \(I_0\subseteq A_{n_0^-}\) with length \(r_{n_1^-}\) such that the following two inequalities hold: \[\label{eqn:Delta32A32inter32I0} \mathcal{L}^1\left(\Delta A\cap I_0\right)\lesssim N^{-2/100}\mathcal{L}^1(I_0),\tag{6}\] and \[\label{eqn:L232sum32on32I0} \int_{I_0} \left|F_{n_1^+}'-F_{n_1^-}'\right|^2+\left|\gamma_{\alpha,n_1^+}'-\gamma_{\alpha,n_1^-}'\right|^2+\left|\gamma_{\alpha',n_1^+}'-\gamma_{\alpha',n_1^-}'\right|^2\,d\mathcal{L}^1\lesssim N^{-3/100}\mathcal{L}^1\left(I_0\right).\tag{7}\] We postpone the process to find such \(I_0\) in Section 7.6.

Write \(I_0=[a,b]\). Note that \(F_{n_1^-}'\) is constant on \(I_0\). Let \(c=F_{n_1^-}'\) on \(I_0\). Since \(\gamma'\) is injective, from the hypotheses on \(\alpha,\alpha'\), we have either \(|c-\gamma'(\alpha-a)|\gtrsim 1\) or \(|c-\gamma'(\alpha'-a)|\gtrsim 1\). Without loss of generality, we may assume \(|c-\gamma'(\alpha-a)|\gtrsim 1\). Let \[\label{eqn:fix32point32derivative} t=\nabla\Phi_\alpha(a,F(a))\cdot (1,c)=c-\gamma'(\alpha-a)\tag{8}\]

7.5.1 The Case of \(t>0\)↩︎

First suppose \(t>0\). Partition \(I_0\) into \(S=r_{n_1^-}/r_{n_1^+}\) dyadic intervals \(J_1,J_2,\cdots,J_S\) of length \(r_{n_1^+}\). Also, partition \(\mathbb{R}\) into disjoint intervals \(\{K_m:m\in\mathbb{Z}\}\) of length \(100N^{1/100}r_{n_1^+}\).

Next, we show that there are few bad intervals.

Before stating the Rising Sun Lemma, we need one final lemma to compare the length of \(J_j\) to its image, under the assumption that \(J_j\) is either good or bad.

Now we state the Rising Sun Lemma.

7.5.2 The Case of \(t<0\)↩︎

Next, we consider the case \(t<0\). Recall the definition of \(t\) in (8 ). The argument differs slightly from the case \(t>0\). It relies on the geometry of \(\Gamma\), which implicitly exhibits the transversality of the curve projections.

Recall we partition \(I\) into \(S=r_{n_1^-}/r_{n_1^+}\) dyadic intervals \(J_1,J_2,\cdots,J_S\) of length \(r_{n_1^+}\). We also partition \(\mathbb{R}\) into disjoint intervals \(\{K_m:m\in\mathbb{Z}\}\) of length \(100N^{1/100}r_{n_1^+}\).

Let’s brief discuss the geometry behind Lemma 57. Let \(\mathbf{a}=(a_1,a_2),\mathbf{b}=(b_1,b_2)\in\mathbb{R}^2\). We may assume \(b_1>a_1\) and \(\Phi_\alpha(\mathbf{a})-\Phi_\alpha(\mathbf{b})>0\). Note that \(\mathbf{a}+\Gamma\) and \(\mathbf{b}+\Gamma\) will either intersect at a point or be disjoint.

First, suppose \((\mathbf{a}+\Gamma)\cap(\mathbf{b}+\Gamma)\ne\varnothing\). That is, there exists \(s_0,t_0\in I\) such that \[\mathbf{x}:=(a_1,a_2)+(s_0,\gamma(s_0))=(b_1,b_2)+(t_0,\gamma(t_0)).\]

Comparing coordinates, we have \(x_1=s_0+a_1=t_0+b_1\) and \(x_2=a_2+\gamma(s_0)=b_2+\gamma(t_0)\). Set \[d_\alpha:=\text{dist}(\mathbf{x},\ell_\alpha)=|\alpha-x_1|.\] Since \(b_1-a_1>0\) and \(\Phi_\alpha(\mathbf{a})-\Phi_\alpha(\mathbf{b})>0\), by the convexity of \(\Gamma\), \(d_\alpha=x_1-\alpha\) and \(\mathbf{b}\) must lie to the right of \(\mathbf{a}\).

We can show that \[\Phi_\alpha(\mathbf{a})-\Phi_\alpha(\mathbf{b})\sim d_\alpha(b_1-a_1).\] This estimate suggests that \(d_\alpha\) captures the difference between \(\Phi_\alpha(\mathbf{a})\) and \(\Phi_\alpha(\mathbf{b})\).

Let \(J_j=[p_j,q_j]\) be a good interval, and consider the points \(\mathbf{a}=(p_j,F_{n_1^+}(p_j))\) and \(\mathbf{b}=(q_j,F_{n_1^+}(q_j))\). We do not know the exact value of \(|F_{n_1^+}(p_j)-F_{n_1^+}(q_j)|\). However, since \(J_j\) is a good interval and the length \(q_j-p_j=r_{n_1^+}\) is fixed, this quantity is bounded. Therefore, \(d_\alpha\) is controlled by \(|F_{n_1^+}(p_j)-F_{n_1^+}(q_j)|\). Geometrically, this can be interpreted as a vertical shift in a fixed range of the curves.

Next, suppose \((\mathbf{a}+\Gamma)\cap(\mathbf{b}+\Gamma)=\varnothing\). Then we will apply a vertical shift to \(\mathbf{a}+\Gamma\) to reduce to the intersection case. In fact, since \(\Lambda\) is compact, we may define the shift by \[d:=\min_{\lambda\in \Lambda} \left(\Phi_\lambda(\mathbf{a})-\Phi_\lambda(\mathbf{b})\right).\] In the case where \(\mathbf{a}=(p_j,F_{n_1^+}(p_j))\) and \(\mathbf{b}=(q_j,F_{n_1^+}(q_j))\), the quantity \(d\) is again controlled by \(q_j-p_j\). The reason is the same as in the intersection case.

In fact, from the discussion above, we see that the curve projection behaves similar to the orthogonal projection in the sense that it separates points at a predictable rate. Furthermore, we can deduce that \[\mathcal{L}^1(\{\lambda\in\Lambda:|\Phi_\lambda(\mathbf{a})-\Phi_\lambda(\mathbf{b})|\leqslant\delta|\mathbf{a}-\mathbf{b}|\})\lesssim\delta,\quad\forall\,\mathbf{a},\mathbf{b}\in E,\delta>0,\] which is the transversal condition. Geometrically, it means that for any two distinct points \(\mathbf{a},\mathbf{b}\in E\), the set of parameters \(\lambda\) for which their projections are nearly aligned has small measure. See [5] and detailed proof in [5].

Back to the proof, we now state the Rising Sun Lemma in this case, which is symmetric to Lemma 51.

7.6 Conclusion by Contradiction↩︎

We proceed by contradiction. Assume that the conclusion in Theorem 24 does not hold. Then \(\mathcal{L}^1(A)\gtrsim N^{-1/100}\), so \[\mathcal{L}^1\left(A_{n_0^-}\right)\gtrsim N^{-1/100}.\] We first localize \(I_0\subseteq A_{n_0^-}\) with length \(r_{n_1^-}\) satisfying 6 and 7 .

We aim to derive a contradiction by showing the small projection assumption fails for \(\alpha\). To achieve this goal, we will apply the Rising Sun Lemma. We only prove the case of \(t>0\), because the argument for the case of \(t<0\) is essentially the same.

Recall we partition \(\mathbb{R}\) into \(K_m\), and use \(K_m\) to study the image of the linear approximation of projection. We will argue that a large proportion of \(K_m\) is in \(\mathcal{N}_{r_{n_1}}(\Phi_\alpha(E))\). We start by classifying \(K_m\) by the Rising Sun Lemma.

We claim that there are only few high-multiplicity intervals \(K_m\). To make this precise, we relate the multiplicity of the \(K_m\) to the behavior of the intervals \(J_j\) in the Rising Sun Lemma. For this purpose, we introduce the notions of typical and atypical intervals.

Heuristically, the number of typical intervals is relatively large.

Since a \(J_j\) is typical if it only reaches low multiplicity \(K_m\), we can shift our focus to such low multiplicity intervals. However, we need to distinguish between \(J_j\)’s that intersect \(A\) and those that do not. If \(A\cap J_j=\varnothing\), \(J_j \subseteq \Delta A\). Note that by 6 , at most \(O(N^{-2/100}S)\) of intervals \(J_j\) do not intersect \(A\). Thus, all but \(O(N^{-2/100}S)\) of intervals \(J_j\) that intersects \(A\) are typical.

Furthermore, if \(J_j\) is typical and \(A\cap J_j\ne\varnothing\), we fix a \(x_j\in A\cap J_j\) such that \(F_{n_1^+}(x_j)+\gamma_{\alpha,n_1^+}\) lies in a low-multiplicity interval \(K_m\). We denote the collection of all such low-multiplicity intervals \(K_m\) as \(\mathscr{K}\). Recall we claimed that the size of \(\mathscr{K}\) is relatively large. We verify that claim now.

Next, we verify \(K_m\in \mathcal{N}_{r_{n_1}}(\Phi_\alpha(E))\) for \(K_m\in\mathscr{K}\).

Now we are ready to reach a contradiction. By Lemma 63 and 64, \[\mathcal{L}^1(\mathcal{N}_{r_{n_1}}(\Phi_\alpha(E)))\gtrsim N^{-1/100}SN^{1/100}r_{n_1^+}=r_{n_1^-}.\] This contradicts the assumption \(\mathcal{L}^1(\mathcal{N}_{r_{n+1}}(\Phi_\alpha(E)))\leqslant r_n\) for sufficiently large \(N\).

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