Small doubling implies small tripling for balls of large radius
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912567406886912 |
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| author | Tessera, Romain Tointon, Matthew |
| author_facet | Tessera, Romain Tointon, Matthew |
| contents | We show that if $K\ge1$ is a parameter and $S$ is a finite symmetric subset of a group containing the identity such $|S^{2n}|\le K|S^n|$ for some integer $n\ge2K^2$, then $|S^{3n}|\le\exp(\exp(O(K^2)))|S^n|$. Such a result was previously known only under the stronger assumption that $|S^{2n+1}|\le K|S^n|$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20500 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Small doubling implies small tripling for balls of large radius Tessera, Romain Tointon, Matthew Combinatorics We show that if $K\ge1$ is a parameter and $S$ is a finite symmetric subset of a group containing the identity such $|S^{2n}|\le K|S^n|$ for some integer $n\ge2K^2$, then $|S^{3n}|\le\exp(\exp(O(K^2)))|S^n|$. Such a result was previously known only under the stronger assumption that $|S^{2n+1}|\le K|S^n|$. 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. |
| title | Small doubling implies small tripling for balls of large radius |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.20500 |