Some problems in additive combinatorics

他计划研究加法组合学中的各种问题，具体来说：继续他在Polymath 4项目上的工作，以产生一种确定性算法来快速找到大素数;探索他开发的突破性概率方法的后果，以解决某些加法组合问题。（如二维角点问题），并继续发展一些新的想法和产品的不平等。 一个很好的例子是提出者在Polymath 4项目中的工作：到目前为止，他和其他参与者已经产生了一个算法，可以比以前的任何方法更快地计算素数计数函数的奇偶性;此外，提出者还提出了该算法的"多项式环模拟“，这对完成该项目大有希望理解素数的结构（2、3、5、7、11等）是古希腊人的不朽遗产之一，近年来取得了巨大的进展。 Polymath 4是一个由Timothy Gowers、Gil Kalai和Terrence Tao发起的在线合作研究项目，它解决了一个相关的基本问题：一个人多快才能产生大素数？ 提议者是这一项目的主要参与者，迄今在解决这一问题方面取得了重大进展。 他计划继续与其他Polymath 4研究人员和他的一些学生一起研究。 此外，他计划继续他的工作，在一个相对较新的领域开发一些突破性的方法，称为“添加剂组合”，这已成为一个热门领域，最近由于高尔斯，绿色，陶，和其他人的作品。

The proposer plans to work on a variety of problems in additive combinatorics,specifically: Continuing his work on the Polymath4 project to produce a deterministic algorithm to find large prime numbers quickly; exploring the ramifications of a ground-breaking probabilistic approach that he has developed to attack certain additive combinatorial problems (such as the 2D corners problem); and continuing the development of some fresh ideas on sum-product inequalities. A good example here is the proposer's work in the Polymath4 project: To date he, and other participants, have produced an algorithm to compute the parity of the prime counting function faster than any previous method; and furthermore, the proposer has produced a ``polynomial ring analogue'' of this algorithm which shows great promise towards completing the project (when generalized further, perhaps).Understanding the structure of prime numbers (2,3,5,7,11, and so) is one of the enduring legacies of the ancient Greeks, and in recent years enormous progress has been made. Polymath4, an online collaborative research program initiated by Timothy Gowers, Gil Kalai, and Terrence Tao, addresses a related, fundamental question: How quickly can one even produce large prime numbers? The proposer has been a major participant in this project, and has thus far made significant strides in addressing this problem. He plans to continue working on it with other Polymath4 researchers and with some of his students. In addition, he plans to continue his work on developing some ground-breaking methods in a relatively new field called ``additive combinatorics'', which has become a hot area of late due to works of Gowers, Green, Tao, and others.