A sum–product theorem in semi-simple commutative Banach algebras

Abstract

The following analogue of the Erdös–Szemerédi sum-product theorem is shown. Let A= f 1,⋯, f N be a finite set of N arbitrary distinct functions on some set. Then either the sum set f i + f j or the product set f i f j has at least N 1+ c elements, where c>0 is an absolute constant. We use Freiman's lemma and Balog–Szemerédi–Gowers Theorem on graphs and combinatorics. As a corollary, we obtain an Erdös–Szemerédi type theorem for semi-simple commutative Banach algebras R. Thus if A⊂ R is a finite set, | A| large enough, then |A+A|+|A.A|>|A| 1+c, where c>0 is an absolute constant. The result and method have various consequences, for instance decay estimates on the convolution powers of finite multiplication subgroups. Let H be a finite multiplicative subgroup of R (as above) and let N=| H|, v= 1 N ∑ x∈H δ x . Then, for all constant c, there is a k= k( c) such that max z∈ R v ( k) ( z) < N − c , where v ( k) = v⋯ v is the k-fold convolution.