  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 $\{2^n\}$.
