On an Erdős similarity problem in the large

Abstract In a recent paper, Kolountzakis and Papageorgiou ask if for every , there exists a set such that for every interval 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 analog 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 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 with the required property — but with — that contains no affine copy of .