New Bounds on the Dimensions of Planar Distance Sets

Abstract

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R 2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff dimension at least 37/54 ≈ 0.685. Moreover, if s ∈ (1, 3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x − y| : x ∈ A} has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1 + s + 3s(2 − s)) ≥ 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres-Schlag and Iosevich-Liu for sets of Hausdorff dimension > 1. Our proof uses a multiscale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.2010 Mathematics Subject Classification. Primary: 28A75, 28A80; Secondary: 26A16, 49Q15.

Incidence Estimates for Well Spaced Tubes

“…In this section, we consider a distinct distances type problem for δ-balls in R 2 , which is related to the Falconer distance problem in R 2 . As we mentioned in the introduction, Orponen [16] and Keleti-Shmerkin [23][24] [14] essentially solved the Falconer distance problem for sets that are close to Ahlfors-David regular. Here we consider the opposite type of set -Ahlfors-David regular sets of a given dimension are packed as tightly as possible, and we consider here sets that are as spread out as possible.…”

“…Theorem 1.4 proves this conjecture up a factor of δ ǫ for sets E that are as widely spaced as possible. In the other direction, there has been some remarkable work by Orponen [16] and Keleti-Shmerkin [14] on the case when E is tightly spaced. We say that E is an Ahlfors-David regular set of δ-balls if, for each ball of E, the concentric Sδ ball contains ≈ S α balls of E. Orponen's paper [16] implies that this conjecture holds up to a factor of δ ǫ for Ahlfors-David regular sets.…”

On Falconer’s distance set problem in the plane

“…This result was striking because in previous work on the problem, there was no evidence that the Ahlfors-David case would be any easier than the general case. This approach was further developed by Keleti and Shmerkin [25]. They proved very strong estimates for sets that are even roughly like Ahlfors-David regular sets.…”

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.…”

“…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,…”