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Thursday 16 Feb 2017Solving xz=y^2 in certain subsets of finite groups

Tom Sanders - University of Oxford

H101 14:30-16:30

One of Klaus Roth’s less famous number-theoretic results asserts that a subset of $\Z/N\Z$ without any (proper) three-term arithmetic progressions has size $O(N/\log \log N)$. This bound has since been much improved and a cute model problem where subsets of $\F_3^n$ are considered has recently received a beautiful resolution by the polynomial method. In this talk we shall be interested in the other direction: not what happens in the model setting but what happens in the non-Abelian setting. In particular we shall discuss an argument to show that if $G$ is a finite group then any subset with no triples $(x,y,z)$ having $xz=y^2$ and $x \neq z$ has size at most $|G|/(\log \log |G|)^{\Omega(1)}$. (The asymptotic are as $|G| \rightarrow \infty$.)

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