Analytic, probabilistic and combinatorial number theory

解析数论、概率数论和组合数论

• 批准号: 1201442
• 负责人: Kevin Ford
• 金额: $ 28.82万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-15 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201442&HistoricalAwards=false
• 关键词: Analytic; probabilistic; combinatorial; number; theory

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.

其中一个项目涉及为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T(n)=n/2如果n是偶数，T(n)=(3n+1)/2如果n是奇数。该分析涉及基于随机漫步和分支随机漫步的模型，这些模型都来自概率论。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是有多少不相交的等差数列可能具有不同的模小于给定的界。这里与组合学中关于两两相交集族的中心问题有联系。第三，作者将研究与数域相关的欧拉-克罗内克常数的分布，以及它们如何与素数组（称为“素数k元组”）和欧拉函数的值相联系。对于第四个项目，提案人将继续研究算术数列中素数分布的微妙差异。研究的主要新课题是，如果扩展黎曼假设成立，差异会有多大。在此之前，作者在扩展黎曼假设不成立的假设下研究了这些差异。提议者将继续研究普拉特树的结构，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，并将其应用于稀疏信号恢复。关于正整数的性质的问题，特别是整数因子和素数分布的方式，几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。该提案涉及数论中的几个项目，强调与其他数学领域的联系，如概率论和组合学，以及在其他领域的应用。例如，对整数上的某个迭代函数的研究导致了概率论的前沿研究，对不相交等差数列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等差数列中的分布，素数的特殊构型，以及在压缩感知中有用的矩阵的构造（使用数论）。