On Falconer’s distance set problem in the plane

Abstract

If E Ă R 2 is a compact set of Hausdorff dimension greater than 5{4, we prove that there is a point x P E so that the set of distances t|x´y|u yPE has positive Lebesgue measure. 1 arXiv:1808.09346v1 [math.CA] 28 Aug 2018 the complications of replacing the upper Minkowski dimension by the Hausdorff dimension in the claim above.Falconer's distance problem can be thought of as a continuous analogue of a combinatorial problem raised by Erdős in [11]: given a set P of N points in R d , what is the smallest possible cardinality of ∆pP q. A grid is the best known example in all dimensions. In two dimensions, Guth and Katz [15] proved a lower bound for |∆pP q| which nearly matches the grid example (up to a factor of log 1{2 N ). In higher dimensions, there is a larger gap, and the best known result is due to Solymosi and Vu [35]. The Erdős distinct distance problem also makes sense for general norms and much less is known about it. In the planar case, if K is smooth and has strictly positive curvature, the best known bound says that if |P | " N , then |∆ K pP q| Á N 3{4 , with stronger estimates established by Garibaldi in special cases ([13]). There is a conversion mechanism to go from Falconer-type results to Erdőstype results that was developed by the second author together with Hoffman ([17]), Laba ([19]), and Rudnev and Uriarte-Tuero ([18]). It gives estimates for point sets that are fairly spread out. Applying the conversion mechanism to Theorem 1.3 we get the following corollary:Corollary 1.5. Let K be a symmetric convex body in R 2 whose boundary BK is C 8 smooth and has strictly positive curvature. Let P be a set of N points in r0, 1s 2 so that the distance between any two points is Á N´1 {2 . Then there exists x P P such that (1.1) |∆ K,x pP q| Ç N 4 5 .

Configuration Sets with Nonempty Interior

“…Similar questions have been studied in, or are accessible to the methods of, a number of works, e.g., [11,2,5,23,24,21]. (There is of course an extensive literature on the related Falconer distance problem [8], and its generalizations to configurations, where the question is what lower bound on dim H (E) ensures that ∆ Φ (E) has positive Lebesque measure [41,7,6,15].) We will show that a sufficient value of s 0 (Φ) can be expressed in terms of d, k and α Φ , the amount of smoothing on L 2 -based Sobolev spaces satisfied by the family of generalized Radon transforms R t defined by Φ.…”

Incidence Estimates for Well Spaced Tubes

“…Using that connection, Wolff [31] proved that dim H (E) > 4/3 suffices. Recently, using decoupling, the paper [11] proved that dim H (E) > 5/4 suffices. Falconer's conjecture is closely related to the following conjecture about finite sets of balls.…”

Hausdorff dimension of pinned distance sets and the 𝐿²-method

“…The first result is due to Bourgain [1], who found an absolutely ǫ 0 > 0 such that dim H (∆(E)) 1 2 +ǫ 0 whenever dim H (E) 1. The best currently known results are due to Keleti and Shmerkin [6], who proved [5] decompose µ E = µ E,good + µ E,bad and consider the L 2 -norm of ν x,good := d x * (µ E,good ). It is pointed out in the Appendix of [5] that neither ν x,good is supported on ∆ x (E), nor ν x,bad is negligible on energy integrals.…”

“…The best currently known results are due to Keleti and Shmerkin [6], who proved [5] decompose µ E = µ E,good + µ E,bad and consider the L 2 -norm of ν x,good := d x * (µ E,good ). It is pointed out in the Appendix of [5] that neither ν x,good is supported on ∆ x (E), nor ν x,bad is negligible on energy integrals. Therefore, although good estimates on I τ (ν x,good ) still follow naturally, it does not imply any result on dim H (∆ x (E)).…”

“…Remark 1.2. Shortly after this paper was made public, Shmerkin [10] plug Guth-Iosevich-Ou-Wang's estimate [5] into Keleti-Shmerkin's framework [6] and obtained dim H (∆(E)) 40 57 = 0.702 · · · ; dim H (∆ x (E)) > 29 42 = 0690 · · · , for some x ∈ E,…”