Ajtai–Szemerédi Theorems over quasirandom groups

Abstract

Two versions of the Ajtai-Szemerédi Theorem are considered in the Cartesian square of a finite non-Abelian group G. In case G is sufficiently quasirandom, we obtain strong forms of both versions: if E ⊆ G × G is fairly dense, then E contains a large number of the desired patterns for most individual choices of 'common difference'. For one of the versions, we also show that this set of good common differences is syndetic.

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