On metric approximate subgroups

Journal article

Let 𝐺 be a group with a metric invariant under left and right translations, and let 𝔻𝑟 be the ball of radius r𝑟 around the identity. A (𝑘,𝑟)-metric approximate subgroup is a symmetric subset 𝑋 of 𝐺 such that the pairwise product set 𝑋𝑋 is covered by at most 𝑘 translates of 𝑋𝔻𝑟. 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 𝑋 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 𝑋 replacing finiteness. In particular, if 𝐺 has bounded exponent, we show that any (𝑘,𝑟)-metric approximate subgroup is close to a (1,𝑟′)-metric approximate subgroup for an appropriate 𝑟′.

Authors: Hrushovski, E; Fanlo, AR

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: World Scientific Publishing
• Journal: Journal of Mathematical Logic
• Volume: 25
• Issue: 3
• Article number: 2450022
• Publication date: 2024-07-12
• Acceptance date: 2024-04-22
• DOI: 10.1142/s0219061324500223
• EISSN: 1793-6691
• ISSN: 0219-0613
• Language: English
• Keywords: Lie model; discretization numbers; metric approximate subgroup; product theorem