Analytic, probabilistic and combinatorial number theory

一个项目针对算术动力学中的一个重要问题改进随机模型，即理解Collatz函数T的迭代的极端轨道，其中如果n为偶数，则T(N)=n/2，如果n为奇数，则T(N)=(3n+1)/2。分析涉及基于随机行走和分支随机行走的模型，这些模型是概率论的主题。目标是更好地理解数字数据的某些方面，并比较和对比不同模型的预测。第二个项目处理的问题是，有多少个不相交的算术级数可能具有不同的模小于给定界。这里有一些与组合学中有关两两相交集族的中心问题的联系。第三，作者将研究与数域相关的Euler-Kronecker常数的分布，以及它们是如何与素数构型(称为‘素数k元组’)以及与Euler的Phi函数的值相联系的。对于第四个项目，提出者将继续调查算术级数中素数分布的细微差异。主要的新问题是，如果扩展的黎曼假设为真，差异会有多大。以前，提出者在假设扩展的黎曼假设为假的情况下研究了这些差异。提出者将继续他对Pratt树的结构的研究，Pratt树是一种由素数建立的结构。他还将继续研究满足受限等距性质的矩阵的显式构造，并将其应用于稀疏信号恢复。关于正整数的性质的问题，特别是整数因数和素数的分布的方式，数千年来一直吸引着人们，最近在计算机科学、信息安全和稀疏信号恢复中发现了应用。这项建议涉及数论中的几个项目，强调与其他数学领域的联系，如概率和组合学，以及在其他领域的应用。例如，对整数上某个迭代函数的研究导致了概率论的前沿研究，而对不相交算术级数集合的研究则导致了组合学中关于集合族相交的基本问题。其他项目涉及算术级数中素数的分布，素数的特殊配置，以及在压缩感知中有用的矩阵的构造(使用数论)。

One project deals with refining stochastic models for an important problem in arithmetic dynamics, namely understanding the extreme orbits of iterates of the Collatz function T, where T(n)=n/2 if n is even and T(n)=(3n+1)/2 if n is odd. The analysis involves models based on random walks and branching random walks, subjects from probability theory. Goals are to better understand certain aspects of the numerical data and compare and contrast the predictions of different models. A second project deals with a problem of how many disjoint arithmetic progressions are possible with distinct moduli less than a given bound. Here there are connections with central questions in combinatorics concerning families of pairwise intersecting sets. Thirdly, the proposer will investigate the distribution of Euler-Kronecker constants associated with number fields, and how they are connected with configurations of prime numbers called ``prime k-tuples'' and with values of Euler's phi function. For the fourth project, the proposer will continue his investigations into subtle discrepancies in the distribution of prime numbers in arithmetic progressions. The main new topic of inquiry is how large the discrepancies can be if the Extended Riemann Hypothesis is true. Previously the proposer studied the discrepancies under the assumption that the Extended Riemann Hypothesis is false. The proposer will continue his investigations into the structure of Pratt trees, a structure built up from prime numbers. He will also continue research into explicit constructions of matrices satisfying a Restricted Isometry Property which have application to sparse signal recovery.Questions about properties of positive integers, especially the way in which integers factor and the distribution of prime numbers, have fascinated people for thousands of years and have recently found applications in computer science, information security and sparse signal recovery. This proposal concerns several projects in the theory of numbers, emphasizing connections with other areas of mathematics such as Probability and Combinatorics as well as applications to other fields. For example, the study of a certain iterated function on the integers leads into cutting edge research in probability theory, and the study of collections of disjoint arithmetic progressions leads to fundamental problems in combinatorics about intersecting families of sets. Other projects concern the distribution of prime numbers in arithmetic progressions, special configurations of prime numbers, and the construction of matrices (using number theory) which are useful in compressed sensing.