Small tripling with forbidden bipartite configurations

Abstract: A finite subset $A$ of a group $G$ is said to have \emph{$k$-tripling} if $|AAA|\leq k|A|$. I will report on recent joint work with A. Pillay, in which we study the structure finite sets $A$ with $k$-tripling, under the additional assumption that the bipartite graph relation $xy\in A$ omits some induced subgraph of a fixed size $d$. In this case, we show that $A$ is approximately a union of a bounded number of translates of a coset nilprogression in $G$ of bounded rank and step (where ``bounded" is in terms of $k$, $d$, and a chosen approximation error $\epsilon>0$). Our methods combine the work of Breuillard, Green, and Tao on the structure of approximate groups, together with model-theoretic tools based on the study of groups definable in NIP theories.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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