An improved lower bound for arithmetic regularity

Kaave Hosseini, Shachar Lovett, Guy Moshkovitz, Asaf Shapira

Research output: Contribution to journalArticlepeer-review


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 - ϵ fraction of the cosets, the nontrivial Fourier coefficients are bounded by ϵ, then Green shows that |G/H| is bounded by a tower of twos of height 1/ϵ3. He also gives an example showing that a tower of height Ω(log 1/ϵ) is necessary. Here, we give an improved example, showing that a tower of height Ω(1/ϵ) is necessary.

Original languageEnglish
Pages (from-to)193-197
Number of pages5
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number2
StatePublished - 1 Sep 2016


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