Problems in additive combinatorics

加法组合数学中的问题

• 批准号: 261014-2008
• 负责人: Solymosi, Jozsef
• 金额: $ 2.33万
• 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: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=503790
• 关键词: Problems; additive; combinatorics

My main field of research is in Additive Combinatorics. The central problem of the area is to characterize additive or multiplicative structures. I am especially interested in the "incompatibility" of multiplicative and additive structures. An old conjecture of Erdos and Szemeredi states that if A is a finite set of integers then the sum-set or the product-set should be large. The sum-set of A is defined as A+A={a+b | a,b \in A} and the product set is A*A={ab | a,b \in A}. Erdos and Szemeredi conjectured that the sum-set or the product set is almost quadratic in the size of A, i.e. |A+A|+|A*A|~|A||A|. The best known lower bound on the quantity |A+A|+|A*A| follows from an earlier work of mine, however it is still very far from the conjecture. The same problem was considered by Bourgain, Katz, and Tao, for the finite field case. Their result has important applications , not only to number theory, but to computer science, Ramsey theory, and cryptography. It turned out that methods from discrete geometry can be used efficiently not only for the real/complex case but for the finite field case as well.

我的主要研究领域是加法组合学。该领域的中心问题是描述加法或乘法结构。我对乘法和加性结构的“不相容”特别感兴趣。Erdos和Szemeredi的一个老猜想指出，如果A是一个有限整数集，那么和集或积集应该是大的。定义A的和集为A+A={A+ b | A,b \in A}，乘积集为A*A={ab | A,b \in A}。Erdos和Szemeredi推测和集或积集在A的大小上几乎是二次的，即|A+A|+|A*A|~|A||A|。最著名的数量|A+A|+|A*A|的下界来自我早期的工作，但它离猜想还很远。对于有限域的情况，Bourgain、Katz和Tao也考虑了同样的问题。他们的结果不仅对数论，而且对计算机科学、拉姆齐理论和密码学都有重要的应用。结果表明，离散几何方法不仅可以有效地应用于实/复情况，而且可以有效地应用于有限域情况。