# 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

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topSanders, 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 -

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