Problems in additive combinatorics

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。我特别感兴趣的是乘法结构和加法结构的“不相容”。Erdos和Szmeredi的一个古老猜想指出，如果A是有限整数集，则和集或乘积集应该是大的。A的和集定义为A A={ab|a，b\in A}，乘积集为A*A={ab|a，b\in A}。Erdos和Szmeredi猜想和集或乘积集在A的大小上几乎是二次的，即|A A||A*A|~|A||A|。最广为人知的量|A|A*A|的下界来自于我以前的工作，但它仍然与猜想相去甚远。对于有限域的情况，Bourain、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用于实/复数情况，而且可以有效地用于有限域情况。

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.