2015/07/05 by Terence Tao, Tao, Terence · 2 citations
Mathematics · #11B30 #60G50 #Abelian group #Bounded function #Combinatorics #Combinatorics (math.CO) #Coset #Discrete mathematics #FOS: Mathematics #Geometry #Graph theory and applications #Inverse #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Polynomial #Probability (math.PR) #Structured program theorem #math.CO #math.PR #msc:11B30 #msc:60G50
paper · pdf · doi:10.48550/arxiv.1507.01276
published in arXiv (Cornell University) (Cornell University) · 45 pages, no figures. Referee suggestions and corrections implemented
openalex publication_date 2015/07/05 · arxiv created 2015/10/01 · arxiv updated 2015/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We give a structural description of the finite subsets A of an arbitrary group G which obey the polynomial growth condition |An| ≤ nd |A| for some bounded d and sufficiently large n, showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of |Am| fairly explicitly for m ≥ n, at least when A is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures μ whose n-fold convolutions μ*n obey the condition ‖ μ*n ‖ℓ2-2 ≤ nd ‖μ‖ℓ2-2. In the abelian case, this description recovers the inverse Littlewood-Offord theorem of Nguyen-Vu, and gives a variant of a recent nonabelian inverse Littlewood-Offord theorem of Tiep-Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.