On the Eigen-Falconer theorem in $\mathbb{R}^d$
Wenxia Li, Zhiqiang Wang, Jiayi Xu
TL;DR
This work extends the Erdős similarity problem to higher dimensions by proving that sequences of nonzero vectors in $\,R^d$ shrinking to zero with successive norm ratios tending to 1 force the existence of a positive-measure set $E subseteq [0,1]^d$ containing no affine copies of the sequence. It generalizes Kolountzakis's results using a δ(A_n) condition on finite subsets of a bounded infinite set and constructs a probabilistic, finite-approximation framework within $ ext{GL}_d(R)$ to produce E. A key proposition is established via a hyperplane-partition reduction and random coverings, with careful expectation estimates ensuring a large E avoiding all affine copies of A. The paper also provides an example separating the Eigen–Falconer condition from the Kolountzakis criterion, clarifying the landscape of higher-dimensional Erdős-type results and strengthening the connection between geometric configurations and measure-theoretic obstructions.
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 $A$.