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An improved lower bound for arithmetic regularity

Published online by Cambridge University Press:  11 March 2016

KAAVE HOSSEINI
Affiliation:
Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093, USA. e-mails: skhossei@cse.ucsd.edu; slovett@cse.ucsd.edu
SHACHAR LOVETT
Affiliation:
Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093, USA. e-mails: skhossei@cse.ucsd.edu; slovett@cse.ucsd.edu
GUY MOSHKOVITZ
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. e-mails: guymosko@tau.ac.il; asafico@tau.ac.il
ASAF SHAPIRA
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. e-mails: guymosko@tau.ac.il; asafico@tau.ac.il

Abstract

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 HG 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

[1] Gowers, T. Lower bounds of tower type for Szemerédi's uniformity lemma. GAFA 7 (1997), 322337.Google Scholar
[2] Green, B. A Szemerédi-type regularity lemma in abelian groups. GAFA 15 (2005), 340376.Google Scholar
[3] Moshkovitz, G. and Shapira, A. A short proof of Gowers' lower bound for the regularity lemma. Combinatorica, to appear.Google Scholar
[4] Szemerédi, E. Regular partitions of graphs. Proc. Colloque Inter. CNRS. (1978), 399–401.Google Scholar
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