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表征加性或乘性结构。 我对乘法和加法结构的“不相容性”特别感兴趣。Erdos和Szemeredi的一个旧猜想指出，如果A是一个有限的整数集，那么和集或积集应该是大的。A的和集定义为A+A={a+B| a，B \in A}，乘积集为A*A={ab| a，B \in A}。Erdos和Szemeredi证明了和集或积集在A的大小上几乎是二次的，即|A+A| +| A*A|~|一||一|.最著名的数量下界|A+A| +| A*A|从我早期的工作，但它仍然是非常远离猜想。同样的问题被认为是布尔甘，卡茨和陶，有限域的情况下。他们的结果有重要的应用，不仅数论，但计算机科学，拉姆齐理论和密码学。事实证明，离散几何的方法不仅可以有效地用于真实的/复杂的情况下，但有限域的情况下，以及。

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.