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

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 $A$.