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An improved lower bound for arithmetic regularity

Published online by Cambridge University Press:  11 March 2016

KAAVE HOSSEINI ,
SHACHAR LOVETT ,
GUY MOSHKOVITZ  and
ASAF SHAPIRA
KAAVE HOSSEINI
Affiliation:
Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093, USA. e-mails: [email protected]; [email protected]
SHACHAR LOVETT
Affiliation:
Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093, USA. e-mails: [email protected]; [email protected]
GUY MOSHKOVITZ
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. e-mails: [email protected]; [email protected]
ASAF SHAPIRA
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. e-mails: [email protected]; [email protected]
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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 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 193 - 197
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Gowers, T. Lower bounds of tower type for Szemerédi's uniformity lemma. GAFA 7 (1997), 322–337.Google Scholar
[2] Green, B. A Szemerédi-type regularity lemma in abelian groups. GAFA 15 (2005), 340–376.Google Scholar
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[4] Szemerédi, E. Regular partitions of graphs. Proc. Colloque Inter. CNRS. (1978), 399–401.Google Scholar