Minimal and nearly minimal measure expansions in connected unimodular groups

Abstract

Let $G$ be a connected unimodular group equipped with a (left and hence right) Haar measure ${\mu }_G$, and suppose $A,B\subseteq G$ are nonempty and compact. An inequality by Kemperman gives us ${\mu }_G(AB)\ge \min \{{\mu }_G(A)+{\mu }_G(B),{\mu }_G(G)\}.$ Our first result determines the conditions for the equality to hold, providing a complete answer to a question asked by Kemperman in 1964. Our second result characterizes compact and connected $G$, $A$, and $B$ that nearly realize equality, with quantitative bounds having the sharp exponent. This can be seen up-to-constant as a $(3k-4)$-theorem for this setting and confirms the connected case of conjectures by Griesmer and by Tao. As an application, we get a measure expansion gap result for connected compact simple Lie groups. The tools developed in our proof include an analysis of the shape of minimally and nearly minimally expanding pairs of sets, a bridge from this to the properties of a certain pseudometric, and a construction of appropriate continuous group homomorphisms to either $\mathbb{R}$ or $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ from the pseudometric.

84 pages; typos corrected, some proofs in Section 8 are simplified

References in corpus (3)

• Near Equality in the Brunn-Minkowski Inequality
• Expansion in simple groups
• Approximate subgroups with bounded VC-dimension

Cited by in corpus (4)

• A nonabelian Brunn-Minkowski inequality
• On the small measure expansion phenomenon in connected noncompact nonabelian groups
• An inequality for the compositions of convex functions with convolutions and an alternative proof of the Brunn-Minkowski-Kemperman inequality
• A Cauchy-Davenport theorem for locally compact groups