Analytic, probabilistic and combinatorial number theory

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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.