On the Eigen-Falconer theorem in $\mathbb{R}^d$

TL;DR

This paper extends the Eigen-Falconer theorem to higher dimensions, showing that certain vector sequences do not necessarily generate affine copies within sets of positive measure, advancing understanding of the Erdős similarity conjecture.

Contribution

The paper generalizes the Eigen-Falconer theorem to $ ext{R}^d$, demonstrating the existence of measure-positive sets avoiding affine copies of specific vector sequences.

Findings

Existence of measure-positive sets avoiding affine copies of certain sequences

Generalization of Eigen-Falconer theorem to higher dimensions

Insights into the Erdős similarity conjecture in $ ext{R}^d$

Abstract

In this paper, we study the analogous Erd\H{o}s similarity conjecture in higher dimensions and generalize the Eigen-Falconer theorem. We show that if $A=\{\boldsymbol{x}_n\}_{n=1}^\infty \subseteq \mathbb{R}^d$ is a sequence of non-zero vectors satisfying \[ \lim_{n \to \infty} \|\boldsymbol{x}_n\| =0 \quad \text{and} \quad \lim_{n \to \infty} \frac{\|\boldsymbol{x}_{n+1}\|}{\|\boldsymbol{x}_n\|} = 1, \] then there exists a measurable set $E \subseteq \mathbb{R}^d$ with positive Lebesgue measure such that $E$ contains no affine copies of $A$.