On an Erdős similarity problem in the large

In a recent paper, Kolountzakis and Papageorgiou ask if for every $\epsilon \in (0,1]$, there exists a set $S\subseteq R$ such that $|S\cap I|⩾1-\epsilon$ for every interval $I\subset R$ 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 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 $S$ 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 $\left\{2^n\right\}$.