The celebrated Erdős similarity problem asks if it is always possible to construct a set of positive Lebesgue measure that does not contain any (nontrivial) affine copy of a given infinite set. The problem remains widely open. In this talk, I will discuss an analogue of Erdős similarity problem ``in the large" and present our contributions. In particular, 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 $S \subseteq \mathbb{R}$ that contains no affine copy of that sequence, 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 answers a recent question of Kolountzakis and Papageorgiou. Joint work with with Xiang Gao and Yuveshen Mooroogen.