Combinatorial Number Theory

Freiman定理说，给定一个集合A，如果这个和集(即A中两个元素的和的集合)不太大，那么A可以包含在具有整数坐标的d维盒子的整数集中的同态映象中。我们建议把盒子的大小限制得更大一些。我们还想改进Erdos和Szmeredied在组合数论中提出的一些猜想的结果，即和集和积集不能都是小的。我们想把一些结果推广到图上的h重和或积和积问题。这些问题和类似问题的重要性在最近几年变得更加明显，因为它们与计算机科学和调和分析中的问题有关。费弗曼和高尔斯在这方面做出了重要贡献。当前数学和应用数学的几个发展的特征之一是出现具有组合色彩的问题，有时出现在看似无关的问题中。事实证明，其中一些问题在早期就已经被研究过，动机不同。这种现象的著名例子是在计算机科学，特别是复杂性理论和算法理论的推动下，图论和计算代数等数学主题的复兴。我在组合数论方面的工作就有这个特点。但除了部分计算机科学界的兴趣外，它主要属于调和分析和微分方程式的一个活跃的研究方向。

Freiman's Theorem says that given a set A, if the sumset (i.e. the set of sums of two elements in A) is not too big,then A can be included in the homomorphicimage in the set of integers of a d-dimensional box with integercoordinates. We proposeto sharpen our bound on the size of the box. We would also like to improve ourresults on some conjectures made by Erdos and Szemerediin combinatorial number theory,which says that the sum set and product set cannot both be small.We would like to generalize some of the results to h-fold sum orproduct and sum-product problems along graphs.The importance of these and similar issues became moreapparent in recent years because of their relation to issues incomputer science and harmonic analysis. Important contributionsalong this line were made by Fefferman and Gowers. One of the features of several current developments in mathematicsand applied mathematics is the emergence of issues with acombinatorial flavor, sometimes in seemingly unrelated questions.Some of these issues turned out to have already been studied inearlier days with different motivations. Well-known examples ofthis phenomenon is the revitalization of mathematical topics suchas graph theory and computational algebra under impetus ofcomputer science, in particular complexity theory and the theoryof algorithms. My work in combinatorial number theory has thisfeature. But besides an interest from part of the computer sciencecommunity, it does primarily belong to an active researchdirection in harmonic analysis and differential equations.