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 - epsilon fraction of the cosets, the nontrivial Fourier coefficients are bounded by epsilon, then Green shows that vertical bar G/H vertical bar is bounded by a tower of twos of height 1/epsilon(3). He also gives an example showing that a tower of height Omega(log 1/epsilon) is necessary. Here, we give an improved example, showing that a tower of height Omega(1/epsilon) is necessary.