New Bounds on the Dimensions of Planar Distance Sets

• Published in: Geometric and functional analysis Vol. 29; no. 6; pp. 1886 - 1948
• Main Authors: Keleti, Tamás, Shmerkin, Pablo
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2019; Springer Nature B.V
• Subjects: Analysis; Combinatorial analysis; Mathematics; Mathematics and Statistics; Secondary 26A16; Pinned distance sets; Hausdorff dimension; Distance sets; Packing dimension; Primary 28A75; 49Q15; Lipschitz functions; 28A80; Falconer’s problem
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-019-00500-9

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 + 3 s ( 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 multi-scale 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.