A Freĭman-type theorem for locally compact abelian groups

Tom Sanders[1]

  • [1] University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WA (England)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 4, page 1321-1335
  • ISSN: 0373-0956

Abstract

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Suppose that G is a locally compact abelian group with a Haar measure μ . The δ -ball B δ of a continuous translation invariant pseudo-metric is called d -dimensional if μ ( B 2 δ ′ ) ≤ 2 d μ ( B δ ′ ) for all δ ′ ⊂ ( 0 , δ ] . We show that if A is a compact symmetric neighborhood of the identity with μ ( n A ) ≤ n d μ ( A ) for all n ≥ d log d , then A is contained in an O ( d log 3 d ) -dimensional ball, B , of positive radius in some continuous translation invariant pseudo-metric and μ ( B ) ≤ exp ( O ( d log d ) ) μ ( A ) .

How to cite

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Sanders, Tom. "A Freĭman-type theorem for locally compact abelian groups." Annales de l’institut Fourier 59.4 (2009): 1321-1335. <http://eudml.org/doc/10429>.

@article{Sanders2009,
abstract = {Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu $. The $\delta $-ball $B_\{\delta \}$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $ \mu (B_\{2\delta ^\{\prime\}\}) \le 2^d\mu (B_\{\delta ^\{\prime\}\})$ for all $\delta ^\{\prime\} \subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu (nA) \le n^d\mu (A)$ for all $n \ge d \log d $, then $A$ is contained in an $O(d \log ^3 d)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu (B) \le \exp (O(d\log d))\mu (A)$.},
affiliation = {University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WA (England)},
author = {Sanders, Tom},
journal = {Annales de l’institut Fourier},
keywords = {Freĭman’s theorem; Fourier transform; balls in pseudo- metrics; polynomial growth; Freiman's theorem; balls in pseudometrics},
language = {eng},
number = {4},
pages = {1321-1335},
publisher = {Association des Annales de l’institut Fourier},
title = {A Freĭman-type theorem for locally compact abelian groups},
url = {http://eudml.org/doc/10429},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Sanders, Tom
TI - A Freĭman-type theorem for locally compact abelian groups
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 4
SP - 1321
EP - 1335
AB - Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu $. The $\delta $-ball $B_{\delta }$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $ \mu (B_{2\delta ^{\prime}}) \le 2^d\mu (B_{\delta ^{\prime}})$ for all $\delta ^{\prime} \subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu (nA) \le n^d\mu (A)$ for all $n \ge d \log d $, then $A$ is contained in an $O(d \log ^3 d)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu (B) \le \exp (O(d\log d))\mu (A)$.
LA - eng
KW - Freĭman’s theorem; Fourier transform; balls in pseudo- metrics; polynomial growth; Freiman's theorem; balls in pseudometrics
UR - http://eudml.org/doc/10429
ER -

References

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  1. G. Birkhoff, A note on topological groups, Compositio Math. 3 (1936), 427-430 Zbl0015.00702MR1556955
  2. J. Bourgain, On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999), 968-984 Zbl0959.11004MR1726234
  3. M.-C. Chang, A polynomial bound in Freĭman’s theorem, Duke Math. J. 113 (2002), 399-419 Zbl1035.11048MR1909605
  4. B. J. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005), 340-376 Zbl1160.11314MR2153903
  5. B. J. Green, I. Z. Ruzsa, Freĭman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2) 75 (2007), 163-175 Zbl1133.11058MR2302736
  6. B. J. Green, Tom Sanders, A quantitative version of the idempotent theorem in harmonic analysis, Ann. of Math. (2) 168 (2008), 1025-1054 Zbl1170.43003MR2456890
  7. I. Z. Ruzsa, An analog of Freiman’s theorem in groups, Astérisque (1999), xv, 323-326 Zbl0946.11007MR1701207
  8. T. Sanders, Three term arithmetic progressions and sumsets, (2007) Zbl1221.11029
  9. Tomasz Schoen, The cardinality of restricted sumsets, J. Number Theory 96 (2002), 48-54 Zbl1043.11010MR1931192
  10. I. D. Shkredov, On a generalization of Szemerédi’s theorem, Proc. London Math. Soc. (3) 93 (2006), 723-760 Zbl1194.11024MR2266965
  11. I. D. Shkredov, On sets with small doubling, (2007) Zbl1219.11019MR2492806
  12. T. C. Tao, V. H. Vu, Additive combinatorics, 105 (2006), Cambridge University Press, Cambridge Zbl1127.11002MR2289012

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