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

Abstract: In this paper, we study the analogous Erdős 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 $A$.