Additive Combinatorics

加法组合学

• 批准号: 261014-2013
• 负责人: Solymosi, Jozsef
• 金额: $ 2.11万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635568
• 关键词: Additive; Combinatorics

My primary field of research is additive combinatorics. This relatively new area of mathematics is concerned about the additive or multiplicative structure of subsets of groups with small sumset or product set. The sumset of a set is formed by the pairwise sums of elements of the set. (In a product set we take the pairwise products.) Additive combinatorics has strong connections to number theory, harmonic analysis, ergodic theory, and theoretical computer science. I am interested in the geometry behind some of the problems in additive combinatorics. It is often the case that a problem in additive combinatorics can be translated to a question about curve-point incidences in a plane over some field. A classical example is the so called "sum-product problem". An arithmetic progression has a small sumset and a geometric progression has small product set, but the arithmetic and geometric progressions have very different structure. Is it possible to find a set with small sumset and small product set at the same time? The answer very much depends on the underlying field or ring. A subfield of a finite field has very small sumset and product set but the same is not possible over fields with characteristic zero or when the set is "very far" from a subfield. The general problem is called the "sum-product phenomenon". Surprisingly the problem can be translated to bounding the number of incidences between points and lines in the plane defined over the field. Understanding such connections and giving efficient bounds on the number of point-curve incidences is a central part of this research proposal. Such results would have important applications not only to other fields of mathematics but to computer science as well.

我的主要研究领域是添加剂组合学。这个相对较新的数学领域关注的是具有较小的集合或产品集的组的添加或乘法结构。集合的集合由集合元素的成对总和形成。 （在产品集中，我们采用成对产品。）添加剂组合学与数字理论，谐波分析，千古理论和理论计算机科学有很强的联系。我对添加剂组合学中一些问题背后的几何形状感兴趣。通常情况下，可以将附加组合物中的问题转化为某个字段上平面上曲线点发生率的问题。一个经典的示例是所谓的“总产品问题”。算术进程的集合很小，几何进程的产物集较小，但是算术和几何发展的结构非常不同。是否可以同时找到带有小型集和小产品的集合？答案在很大程度上取决于基础场或环。有限字段的子场具有很小的集合和产品集，但在特征零或该集合“非常远”的字段上不可能相同。一般问题称为“总和现象”。出乎意料的是，问题可以转化为在场上定义的平面中点和线之间的发病率数。了解这种联系并在点曲线发生率的数量上提供有效的界限是本研究建议的核心部分。这样的结果不仅可以针对其他数学领域，而且还将针对计算机科学领域。