On metric approximate subgroups
Let G𝐺 be a group with a metric invariant under left and right translations, and let Dr𝔻𝑟 be the ball of radius r𝑟 around the identity. A (k,r)(𝑘,𝑟)-metric approximate subgroup is a symmetric subset X𝑋 of G𝐺 such that the pairwise product set XX𝑋𝑋 is covered by at most k𝑘 translates of XDr𝑋𝔻𝑟. This...
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| Format: | Journal article |
| Language: | English |
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World Scientific Publishing
2024
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| _version_ | 1811140284194488320 |
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| author | Hrushovski, E Fanlo, AR |
| author_facet | Hrushovski, E Fanlo, AR |
| author_sort | Hrushovski, E |
| collection | OXFORD |
| description | Let G𝐺 be a group with a metric invariant under left and right translations, and let Dr𝔻𝑟 be the ball of radius r𝑟 around the identity. A (k,r)(𝑘,𝑟)-metric approximate subgroup is a symmetric subset X𝑋 of G𝐺 such that the pairwise product set XX𝑋𝑋 is covered by at most k𝑘 translates of XDr𝑋𝔻𝑟. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc.25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of X𝑋 finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on X𝑋 replacing finiteness. In particular, if G𝐺 has bounded exponent, we show that any (k,r)(𝑘,𝑟)-metric approximate subgroup is close to a (1,r')(1,𝑟′)-metric approximate subgroup for an appropriate r'𝑟′. |
| first_indexed | 2024-09-25T04:19:32Z |
| format | Journal article |
| id | oxford-uuid:d5b94673-3c7a-451c-a51e-f83b648ddaa0 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:19:32Z |
| publishDate | 2024 |
| publisher | World Scientific Publishing |
| record_format | dspace |
| spelling | oxford-uuid:d5b94673-3c7a-451c-a51e-f83b648ddaa02024-07-30T12:03:51ZOn metric approximate subgroupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d5b94673-3c7a-451c-a51e-f83b648ddaa0EnglishSymplectic ElementsWorld Scientific Publishing2024Hrushovski, EFanlo, ARLet G𝐺 be a group with a metric invariant under left and right translations, and let Dr𝔻𝑟 be the ball of radius r𝑟 around the identity. A (k,r)(𝑘,𝑟)-metric approximate subgroup is a symmetric subset X𝑋 of G𝐺 such that the pairwise product set XX𝑋𝑋 is covered by at most k𝑘 translates of XDr𝑋𝔻𝑟. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc.25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of X𝑋 finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on X𝑋 replacing finiteness. In particular, if G𝐺 has bounded exponent, we show that any (k,r)(𝑘,𝑟)-metric approximate subgroup is close to a (1,r')(1,𝑟′)-metric approximate subgroup for an appropriate r'𝑟′. |
| spellingShingle | Hrushovski, E Fanlo, AR On metric approximate subgroups |
| title | On metric approximate subgroups |
| title_full | On metric approximate subgroups |
| title_fullStr | On metric approximate subgroups |
| title_full_unstemmed | On metric approximate subgroups |
| title_short | On metric approximate subgroups |
| title_sort | on metric approximate subgroups |
| work_keys_str_mv | AT hrushovskie onmetricapproximatesubgroups AT fanloar onmetricapproximatesubgroups |