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arXiv · 2008.10009

Direct and inverse results for popular differences in trees of positive dimension

Alexander Fish·Leo Jiang·with a joint appendix with Ilya D. Shkredov
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Abstract

We establish analogues for trees of results relating the density of a set $E \subset \mathbb{N}$, the density of its set of popular differences, and the structure of $E$. To obtain our results, we formalise a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting we use a recent result of the first author with Bj\"orklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.

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Alexander Fish, Leo Jiang, with a joint appendix with Ilya D. Shkredov. 2020-08-23. Direct and inverse results for popular differences in trees of positive dimension. https://arxiv.org/abs/2008.10009

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