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 \(E \subseteq G \times G\) 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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Acknowledgements
Research supported by a fellowship from the Clay Mathematics Institute. I am grateful to Vitaly Bergelson for sharing [7] with me, to Ben Green for pointing me to the references [11, 15], to Sean Eberhard for pointing me to the reference [20] and to Julia Wolf for suggesting some useful clarifications.
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Austin, T. (2016). Ajtai–Szemerédi Theorems over quasirandom groups. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_19
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DOI: https://doi.org/10.1007/978-3-319-24298-9_19
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