The sum-product phenomenon in various groups, expanding maps and applications

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The proposal belongs to a research activity in combinatorial number theory, in particular 'additive combinatorics'. Many of the problems go back quite a while, to the work of Erdos-Szemeredi and Freiman. But the revitalization of part of this subject appears mainly during the last decade, with motivations from harmonic analysis such as the Kakeya set problem in higher dimension, the work on arithmetic progressions of Gowers and Green-Tao, the 'sum-product theorem' in finite fields and its many applications (to computer science, the theory of exponential sums and to the spectral theory of Heckeoperators) . The problems in the proposal are mainly related to the sum-product and product phenomena in various rings and groups, continuing a line of research (in particular by the PI) that turned out to have rather unexpected applications besides their intrinsic interest from the combinatorial point of view. Indeed, purely combinatorial (and basically elementary) techniques brought progress on issues that had stalled for quite some time, such as on the expander properties of certain 'thin' SL2 Caley graphs, and the 'expanding properties' of algebraic functions on finite fields.Recent results around the 'sum-product' and 'product-phenomena' in various fields, rings and groups lead to progress on an amazing number of issues, ranging from computer science to representation theory. The purpose of the proposal is to explore further several types of questions that underly these applications, including those are of importance to quantum computation and nanotechnology.