On an Erdős similarity problem in the large

Abstract

In a recent paper, Kolountzakis and Papageorgiou ask if for every $\hat{I}\mu \in (0,1]$, there exists a set $S\subseteq \mathbb{R}$ such that $|S\cap I|\ge 1-\hat{I}\mu$ for every interval $I\subset \mathbb{R}$ with unit length, but that does not contain any affine copy of a given increasing sequence of exponential growth or faster. This question is an analogue of the well-known Erdős similarity problem. In this paper, we show that for each sequence of real numbers whose integer parts form a set of positive upper Banach density, one can explicitly construct such a set $S$ that contains no affine copy of that sequence. Since there exist sequences of arbitrarily rapid growth that satisfy this condition, our result answers Kolountzakis and Papageorgiou's question in the affirmative. A key ingredient of our proof is a generalization of results by Amice, Kahane, and Haight from metric number theory. In addition, we construct a set $S$ with the required property -- but with $\hat{I}\mu \in (1/2,1]$ -- that contains no affine copy of $\{2^n\}$.

17 pages, revised based on referee comments