On an Erd\H{o}s similarity problem in the large

Abstract

In a recent paper, Kolountzakis and Papageorgiou ask if for every $\epsilon \in (0,1]$, there exists a set $S \subseteq \mathbb{R}$ such that $\vert S \cap I\vert \geq 1 - \epsilon$ 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 Erd\H{o}s 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 $\epsilon \in (1/2, 1]$ -- that contains no affine copy of $\{2^n\}$.