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

Université Paris Cité
University of Bristol

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

Résumé

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.

langue originale	Anglais
Numéro d'article	9
journal	Discrete Analysis
Volume	2025
Les DOIs	https://doi.org/10.19086/da.143426
état	Publié - 1 janv. 2025
Modification externe	Oui

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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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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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Small Doubling Implies Small Tripling for Balls of Large Radius

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