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arXiv · 2209.10823

Sets of full measure avoiding Cantor sets

Mihail N. Kolountzakis
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Abstract

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are not universal in measure, i.e. they satisfy the above conjecture. These are symmetric Cantor sets $C$ which can be quite thin: the length of the $n$-th generation intervals defining the Cantor set is decreasing almost doubly exponentially. Further, we achieve to construct a set, not just of positive measure, but of \textit{full measure} not containing any affine copy of $C$. Our method is probabilistic.

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Mihail N. Kolountzakis. 2022-09-22. Sets of full measure avoiding Cantor sets. https://arxiv.org/abs/2209.10823

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