"Improved lower bounds for Arithmetic Regularity Lemma"
Monday, November 24th, 2014, 2:00 pm
EBU3B, Room 4258
The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemerédi 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 − eps fraction of the cosets, the nontrivial Fourier coefficients are bounded by eps, then Green shows that |G/H| is bounded by a tower of twos of height 1/eps^3. He also gives an example showing that a tower of height Ω(log 1/eps) is necessary. Here, we give an improved example, showing that a tower of height Ω(1/eps) is necessary.
Joint work with Shachar Lovett, Guy Moshkovitz, and Asaf Shapira.