Combinatorial Number Theory

Freiman定理指出，给定一个集合a，如果其和集（即a中两个元素的和的集合）不太大，则a可以包含在具有整数坐标的d维方框的整数集合的同态像中。我们建议把盒子的尺寸限制得更细一些。我们还想改进Erdos和Szemerediin组合数论中关于和集和积集不能都小的一些猜想的结果。我们想把一些结果推广到图上的h倍和积和积问题。近年来，由于这些问题与计算机科学和谐波分析问题的关系，这些问题和类似问题的重要性变得更加明显。费尔曼和高尔斯在这方面作出了重要贡献。当前数学和应用数学发展的特点之一是出现了具有组合色彩的问题，有时在看似不相关的问题中。其中一些问题在早期就已经有不同动机的研究。这种现象的著名例子是在计算机科学的推动下，图论和计算代数等数学主题的复兴，特别是复杂性理论和算法理论。我在组合数论方面的工作有这个特点。但除了部分计算机科学界的兴趣外，它主要属于谐波分析和微分方程的活跃研究方向。

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.