Finite Chains inside Thin Subsets of ${\Bbb R}^d$

TL;DR

This paper proves that any subset of Euclidean space with Hausdorff dimension greater than (d+1)/2 contains chains of arbitrary length with prescribed gaps, extending geometric configuration results to more general thin sets.

Contribution

It establishes the existence of chains with arbitrary prescribed gaps in sets of sufficiently large Hausdorff dimension without additional structural assumptions.

Findings

Sets with Hausdorff dimension > (d+1)/2 contain chains of arbitrary length with prescribed gaps.

The result generalizes previous work requiring Fourier decay conditions.

Existence of such chains is guaranteed for a broad class of thin sets.

Abstract

In a recent paper, Chan, \L aba, and Pramanik investigated geometric configurations inside thin subsets of the Euclidean set possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside thin sets of a given Hausdorff dimension without any additional assumptions on the structure. We prove that if the Hausdorff dimension of $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then there exists a non-empty interval $I$ such that given any sequence $\{t_1, t_2, \dots, t_k; t_j \in I\}$, there exists a sequence ${\{x^j\}}_{j=1}^{k+1}$, such that $x^j \in E$ and $|x^{i+1}-x^i|=t_j$, $1 \leq i \leq k$. In other words, $E$ contains vertices of a chain of arbitrary length with prescribed gaps.