Small Doubling Implies Small Tripling for Balls of Large Radius

Romain Tessera; Matthew Tointon

doi:10.19086/da.143426

Small Doubling Implies Small Tripling for Balls of Large Radius

Romain Tessera
, Matthew Tointon

Research output: Contribution to journal › Article › peer-review

Abstract

We show that if K ≥ 1 is a parameter and S is a finite symmetric subset of a group containing the identity such |S²ⁿ| ≤ K|Sⁿ| for some integer n ≥ 2K², then |S³ⁿ| ≤ exp(exp(O(K²)))|Sⁿ|. Such a result was previously known only under the stronger assumption that |S²ⁿ+¹| ≤ K|Sⁿ|. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.

Original language	English
Article number	9
Journal	Discrete Analysis
Volume	2025
DOIs	https://doi.org/10.19086/da.143426
Publication status	Published - 1 Jan 2025
Externally published	Yes

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abstract = "We show that if K ≥ 1 is a parameter and S is a finite symmetric subset of a group containing the identity such |S2n| ≤ K|Sn| for some integer n ≥ 2K2, then |S3n| ≤ exp(exp(O(K2)))|Sn|. Such a result was previously known only under the stronger assumption that |S2n+1| ≤ K|Sn|. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.",

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N2 - We show that if K ≥ 1 is a parameter and S is a finite symmetric subset of a group containing the identity such |S2n| ≤ K|Sn| for some integer n ≥ 2K2, then |S3n| ≤ exp(exp(O(K2)))|Sn|. Such a result was previously known only under the stronger assumption that |S2n+1| ≤ K|Sn|. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.

AB - We show that if K ≥ 1 is a parameter and S is a finite symmetric subset of a group containing the identity such |S2n| ≤ K|Sn| for some integer n ≥ 2K2, then |S3n| ≤ exp(exp(O(K2)))|Sn|. Such a result was previously known only under the stronger assumption that |S2n+1| ≤ K|Sn|. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.

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SN - 2397-3129

VL - 2025

JO - Discrete Analysis

JF - Discrete Analysis

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ER -

Small Doubling Implies Small Tripling for Balls of Large Radius

Abstract

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