TY - JOUR

T1 - An improved lower bound for arithmetic regularity

AU - Hosseini, Kaave

AU - Lovett, Shachar

AU - Moshkovitz, Guy

AU - Shapira, Asaf

N1 - Publisher Copyright:
Copyright © 2016 Cambridge Philosophical Society.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84960379103&partnerID=8YFLogxK

U2 - 10.1017/S030500411600013X

DO - 10.1017/S030500411600013X

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AN - SCOPUS:84960379103

SN - 0305-0041

VL - 161

SP - 193

EP - 197

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 2

ER -