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

该提案属于组合数论的研究活动，特别是“加性组合”。许多问题可以追溯到很久以前，Erdos-Szemeredi和Freiman的工作。但这一学科的部分复兴主要是在过去十年中出现的，其动机来自谐波分析，如高维中的Kakeya集问题，高尔斯和格林-道对等差数列的研究，有限域中的“和积定理”及其许多应用（计算机科学，指数和理论和Heckeoperators的谱理论）。提案中的问题主要与各种环和群中的和积和积现象有关，继续进行一系列研究（特别是PI），结果证明，除了从组合的角度来看，它们的内在利益之外，还有相当意想不到的应用。事实上，纯粹的组合（基本上是基本的）技术在停滞了很长一段时间的问题上取得了进展，比如某些“薄”SL2 Caley图的展开性质，以及有限域上代数函数的展开性质。最近在各个领域、环和组中围绕“和-积”和“产品现象”的研究结果导致了从计算机科学到表示理论等大量问题的进展。该提案的目的是进一步探索这些应用背后的几种类型的问题，包括那些对量子计算和纳米技术很重要的问题。

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.