On metric approximate subgroups.

Let G be a group with a metric invariant under left and right translations, and let 픻r be the ball of radius r 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 X픻r. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, <italic>Combinatorica</italic>, <bold>28</bold>(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, <italic>J. Amer. Math. Soc.</italic> <bold>25</bold>(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′. [ABSTRACT FROM AUTHOR]