Falconer lattice sets and the Erdos similarity problem

TL;DR

This paper demonstrates that certain extremely thin sets, which do not contain affine copies of slowly decaying sequences, satisfy the Erdős similarity conjecture using Falconer lattice sets and Bourgain's theorem.

Contribution

It introduces new examples of thin sets satisfying the Erdős similarity conjecture outside previous coverage, leveraging Falconer lattice sets and Bourgain's theorem.

Findings

These sets have small logarithmic dimension.

They do not contain affine copies of slowly decaying sequences.

They contain sequences of rapid decay for which the conjecture remains open.

Abstract

We show that a family of extremely thin sets satisfy the Erd\H{o}s similarity conjecture. These examples lie outside the range covered by recent work of Shmerkin and Yavicoli \cite{ShmerkinYavicoli2025}. As we shall see, they have small logarithmic dimension. They do not contain affine copies of slowly decaying sequences, so the result does not follow from earlier work of Falconer and Eigen \cite{Falconer1984,Eigen}. On the other hand, they do contain sequences of rapid decay, for which the conjecture is still open in general. Our argument is based on Falconer lattice sets and a theorem of Bourgain \cite{Bourgain2003}.