Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pinned distances of planar sets with low dimension

Jacob B. Fiedler, D. M. Stull

TL;DR

This work advances the pinned distance problem in the plane by deriving improved lower bounds for the dimension of pinned distance sets when the base set has small Hausdorff dimension and, simultaneously, by developing a robust framework of universal pin sets. The authors fuse effective-dimensional techniques—Kolmogorov complexity, the Point-to-Set Principle, and radial projection results—with geometric measure theory to bound the complexity of distances and distances conditioned on projections. They prove an explicit effective bound on dim^{x,A}(|x-y|) in terms of the ambient and regularity parameters, then reduce these to classical bounds via thin-tubes results, producing unconditional lower bounds for analytic sets and establishing the existence of universal pin sets in several regimes. Moreover, they demonstrate that weak regularity and AD-regularity of pin sets yield strong universality statements: compact AD-regular sets of dimension above one serve as universal pins for all Borel Y, enabling plane Falconer-type conclusions with a fixed pin-set.

Abstract

In this paper, we give improved bounds on the Hausdorff dimension of pinned distance sets of planar sets with dimension strictly less than one. As the planar set becomes more regular (i.e., the Hausdorff and packing dimension become closer), our lower bound on the Hausdorff dimension of the pinned distance set improves. Additionally, we prove the existence of small universal sets for pinned distances. In particular, we show that, if a Borel set $X\subseteq\mathbb{R}^2$ is weakly regular ($\dim_H(X) = \dim_P(X)$), and $\dim_H(X) > 1$, then \begin{equation*} \sup\limits_{x\in X}\dim_H(Δ_x Y) = \min\{\dim_H(Y), 1\} \end{equation*} for every Borel set $Y\subseteq\mathbb{R}^2$. Furthermore, if $X$ is also compact and Ahlfors-David regular, then for every Borel set $Y\subseteq\mathbb{R}^2$, there exists some $x\in X$ such that \begin{equation*} \dim_H(Δ_x Y) = \min\{\dim_H(Y), 1\}. \end{equation*}

Pinned distances of planar sets with low dimension

TL;DR

This work advances the pinned distance problem in the plane by deriving improved lower bounds for the dimension of pinned distance sets when the base set has small Hausdorff dimension and, simultaneously, by developing a robust framework of universal pin sets. The authors fuse effective-dimensional techniques—Kolmogorov complexity, the Point-to-Set Principle, and radial projection results—with geometric measure theory to bound the complexity of distances and distances conditioned on projections. They prove an explicit effective bound on dim^{x,A}(|x-y|) in terms of the ambient and regularity parameters, then reduce these to classical bounds via thin-tubes results, producing unconditional lower bounds for analytic sets and establishing the existence of universal pin sets in several regimes. Moreover, they demonstrate that weak regularity and AD-regularity of pin sets yield strong universality statements: compact AD-regular sets of dimension above one serve as universal pins for all Borel Y, enabling plane Falconer-type conclusions with a fixed pin-set.

Abstract

In this paper, we give improved bounds on the Hausdorff dimension of pinned distance sets of planar sets with dimension strictly less than one. As the planar set becomes more regular (i.e., the Hausdorff and packing dimension become closer), our lower bound on the Hausdorff dimension of the pinned distance set improves. Additionally, we prove the existence of small universal sets for pinned distances. In particular, we show that, if a Borel set is weakly regular (), and , then \begin{equation*} \sup\limits_{x\in X}\dim_H(Δ_x Y) = \min\{\dim_H(Y), 1\} \end{equation*} for every Borel set . Furthermore, if is also compact and Ahlfors-David regular, then for every Borel set , there exists some such that \begin{equation*} \dim_H(Δ_x Y) = \min\{\dim_H(Y), 1\}. \end{equation*}
Paper Structure (16 sections, 42 theorems, 125 equations)

This paper contains 16 sections, 42 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Universal sets for pinned distances
  3. Preliminaries
  4. Kolmogorov complexity and effective dimension
  5. The Point-to-Set Principle
  6. Projection theorem
  7. Projection lemmas
  8. Admissible partitions
  9. Main projection theorem
  10. Effective dimension of pinned distances
  11. Constructing a better partition
  12. Main effective theorem
  13. Reduction to the classical bounds
  14. Universal pin sets
  15. Weakly regular universal sets
  16. ...and 1 more sections

Key Result

Theorem 2

Suppose $E\subseteq\mathbb{R}^2$ is analytic, $d:= \dim_H(E) \leq 1,$ and $D:= \dim_P(E)$. Then, where $\alpha = \min\{1 + d, D\}$.

Theorems & Definitions (71)

  • Conjecture 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Lemma 7: LutStu20
  • Lemma 8: LutStu20
  • Lemma 9: LutStu20
  • Theorem 10: Point-to-set principle LutLut18
  • ...and 61 more