We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on three-term arithmetic progressions and the existence of long arithmetic progressions in sumsets A+B in Zp. The bounds we obtain for these results are not the best one...

We study the maximal mutual information about a random variable Y (representing non-private information) displayed through an additive Gaussian channel when guaranteeing that only ε bits of information is leaked about a random variable X (representing private information) that is correlated with Y . Denoting this quantity by gε(X,Y ), we show that for perfect privacy, i.e., ε = 0, one has g0(X,...

Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Mi...