UBC Harmonic Analysis and Fractal Geometry: Kyle Yip

The celebrated Erdõs 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 Erdõs 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⊆ℝ that contains no affine copy of that sequence, such that |S∩I|≥1−ϵ for every interval I⊂ℝ with unit length, where ϵ>0 is arbitrarily small. This answers a recent question of Kolountzakis and Papageorgiou. Joint work with with Xiang Gao and Yuveshen Mooroogen.