Improved dimension bounds for Falconer's distance problem in the plane
Abstract
Let E⊂ℝ2 be analytic with t=dimHE>1. We prove dimHΔE≥ 4/5+η0 for an absolute, non-explicit constant η0>0, and dimHΔE≥ (t2−2t+5)/(2t2−7t+10) for 1<t≤(11−√(41))/4, which reaches 7/8 at the right endpoint. For pinned distances, dimHΔxE≥ q*=0.7813… for every x∈ℝ2 outside an exceptional set of Hausdorff dimension at most one, with stronger bounds depending on t and further exceptional-set estimates. If E is compact and supports an s-Frostman measure μ, s>1, then underlinedimBΔxE≥ max{4/5+η0,min{(3s+1)/5,7/8}} for μ-almost every x. The same bound holds for pinned Hausdorff dimension under a common dyadic mass law, allowing oscillation with scale. We retain information about both endpoints in the Kolmogorov complexity estimates of Stull and Fiedler--Stull. Applying the partition estimates with the endpoints reversed yields 4/5 and the rational bound. Equality forces both information profiles to have excess information on a common interval of scales. Encoding one source measure for all pins converts this constraint into a configuration of points and circles with many rich perpendicular bisectors. Lower bounds of Katz--Tardos and Ren--Wang contradict the upper bound of Fu--Gan--Ren; compactness then gives η0>0. This rigidity argument has a precedent in the work of Héra, Shmerkin and Yavicoli on Furstenberg sets, using Bourgain's projection theorem. A separate finite-scale selection proves the pinned lower box bound.
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BibTeX
@misc{iosevich2026improved,
title = {{Improved dimension bounds for Falconer's distance problem in the plane}},
author = {Alex Iosevich and Alexia Yavicoli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05500},
eprint = {2610.05500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05500},
}
RIS
TY - UNPB
TI - Improved dimension bounds for Falconer's distance problem in the plane
AU - Alex Iosevich
AU - Alexia Yavicoli
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05500
UR - https://arxiv.org/abs/2610.05500
AB - Let E⊂ℝ^2 be analytic with t=dim_HE>1. We prove dim_HΔE≥ 4/5+η_0 for an absolute, non-explicit constant η_0>0, and dim_HΔE≥ (t^2−2t+5)/(2t^2−7t+10) for 1<t≤(11−√(41))/4, which reaches 7/8 at the right endpoint. For pinned distances, dim_HΔ_xE≥ q_*=0.7813… for every x∈ℝ^2 outside an exceptional set of Hausdorff dimension at most one, with stronger bounds depending on t and further exceptional-set estimates. If E is compact and supports an s-Frostman measure μ, s>1, then underlinedim_BΔ_xE≥ max{4/5+η_0,min{(3s+1)/5,7/8}} for μ-almost every x. The same bound holds for pinned Hausdorff dimension under a common dyadic mass law, allowing oscillation with scale. We retain information about both endpoints in the Kolmogorov complexity estimates of Stull and Fiedler--Stull. Applying the partition estimates with the endpoints reversed yields 4/5 and the rational bound. Equality forces both information profiles to have excess information on a common interval of scales. Encoding one source measure for all pins converts this constraint into a configuration of points and circles with many rich perpendicular bisectors. Lower bounds of Katz--Tardos and Ren--Wang contradict the upper bound of Fu--Gan--Ren; compactness then gives η_0>0. This rigidity argument has a precedent in the work of Héra, Shmerkin and Yavicoli on Furstenberg sets, using Bourgain's projection theorem. A separate finite-scale selection proves the pinned lower box bound.
ER -