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Abstract: A famous theorem of Roth states that for any \(\alpha > 0\) and \(n\) sufficiently large in terms of \(\alpha\), any subset of \(\{1, dots, n\}\) with density \(\alpha\) contains a 3-term arithmetic progression. Green developed an arithmetic regularity lemma and used it to prove that not only is there one arithmetic progression, but in fact there is some integer \(d > 0\) for which the density of 3-term arithmetic progressions with common difference \(d\) is at least roughly what is expected in a random set with density \(\alpha\). That is, for every \(\epsilon > 0\), there is some \(n(\epsilon)\) such that for all \(n > n(\epsilon)\) and any subset \(A\) of \(\{1, dots, n\}\) with density \(\alpha\), there is some integer \(d > 0\) for which the number of 3-term arithmetic progressions in \(A\) with common difference \(d\) is at least \((\alpha^3-\epsilon)n\). We prove that \(n(\epsilon)\) grows as an exponential tower of 2's of height on the order of \(\log(1/\epsilon)\). We show that the same is true in any abelian group of odd order \(n\). These results are the first applications of regularity lemmas for which the tower-type bounds are shown to be necessary.

 Joint work with Jacob Fox and Huy Tuan Pham.