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Ellenberg and Gijswijt gave the best known asymptotic upper bound for thecardinality of subsets of $\mathbb F_q^n$ without 3-term arithmeticprogressions. We improve this bound by a factor $\sqrt{n}$. In the case $q=3$,we also obtain more explicit upper bounds for the Cap Set Problem.

Improved explicit upper bounds for the Cap Set Problem