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

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