An Improved lower bound for arithmetic regularity

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemeredi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → [0, 1], there exists a subgroup H ≤ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for 1− fraction of the cosets, the nontrivial Fourier coefficients are bounded by , then Green shows that |G/H| is bounded by a tower of twos of height 1/ . He also gives an example showing that a tower of height Ω(log 1/ ) is necessary. Here, we give an improved example, showing that a tower of height Ω(1/ ) is necessary. Joint work with Shachar Lovett, Guy Moshkovitz, and Asaf Shapira. Host: Jacques Verstraete Tuesday, February 24, 2015 4:00 PM AP&M 6402 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *