On metric approximate subgroups

Abstract

Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset [Formula: see text] of [Formula: see text] such that the pairwise product set [Formula: see text] is covered by at most [Formula: see text] translates of [Formula: see text]. 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 [Formula: see text] 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 [Formula: see text] replacing finiteness. In particular, if [Formula: see text] has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].