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

Abstract

In a recent paper, Chan, Łaba, 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 $E\subset {\mathbb{R}}^d$, $d\ge 2$, 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 $|x^{i+1}-x^i|=t_j$, $1\le i\le k$. In other words, $E$ contains vertices of a chain of arbitrary length with prescribed gaps.

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