Combinatorial Number Theory

Freiman定理说，给定一个集合A，如果和集（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维盒的整数集合的同态像中。我们建议把箱子的尺寸缩小。我们还想改进Erdos和Szemerediin在组合数论中关于和集和积集不能都小的一些结果，并想把其中的一些结果推广到沿着图的h重和或积以及和积问题，这些问题和类似问题的重要性在近年来变得更加明显，因为它们与计算机科学和调和分析中的问题有关.沿着这条线的重要贡献是由Febriman和Gowers。当前数学和应用数学发展的一个特点是出现了一些具有组合特征的问题，有时候这些问题看起来并不相关。其中一些问题在早期已经被研究过，但研究的动机却不尽相同。著名的例子，这种现象是振兴的数学主题，如图论和计算代数的推动下，计算机科学，特别是复杂性理论和理论的算法。我在组合数论方面的工作就有这个特点。但它除了引起部分计算机科学界的兴趣外，主要属于调和分析和微分方程中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.