Analytic, probabilistic and combinatorial number theory

其中一个项目涉及改进算术动力学中一个重要问题的随机模型，即理解Collatz函数T的迭代的极端轨道，其中T（n）=n/2，如果n是偶数，T（n）=（3 n +1）/2，如果n是奇数。 分析涉及基于随机游走和分支随机游走的模型，这些模型来自概率论。 目标是更好地理解数值数据的某些方面，并比较和对比不同模型的预测。第二个项目处理的问题，有多少不相交的算术级数是可能的不同的模小于一个给定的界限。这里有连接的中心问题，在组合学有关家庭的两两相交集。第三，提议者将研究与数域相关的欧拉-克罗内克常数的分布，以及它们如何与称为"素数k元组“的素数的配置以及欧拉的phi函数的值相联系。在第四个专题中，提议者将继续研究算术级数中素数分布的细微差异。 主要的新课题的调查是多大的差异可以如果扩展黎曼假设是真的。以前的提议者研究的差异的假设下，扩展黎曼假设是假的。提议者将继续他对普拉特树的结构的研究，普拉特树是一种由素数构成的结构。 他还将继续研究满足限制等距性质的矩阵的显式构造，这些矩阵可应用于稀疏信号恢复。关于正整数的性质，特别是整数因子和素数分布的方式，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得到了应用。该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其他领域的应用。例如，对整数上的某个迭代函数的研究导致了概率论中的前沿研究，对不相交算术级数集合的研究导致了组合学中关于相交集合族的基本问题。其他项目涉及素数在算术级数中的分布，素数的特殊配置，以及在压缩传感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.