On metric approximate subgroups - Hrushovski, E; Fanlo, AR (World Scientific Publishing, 2024)
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′.