On an Erdős similarity problem in the large

Xiang Gao, Yuveshen Mooroogen, Chi Hoi Yip

Published 2023-11-12Version 1

In a recent paper, Kolountzakis and Papageorgiou ask whether there exist large sets that do not contain any affine copy of a given increasing sequence of exponential growth. Here, a large set means 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, where $\epsilon>0$ is arbitrarily small. 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 a large set 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.