Research Projects

An important theme that connects probability and number theory is that of comparing the spacing statistics of a uniformly distributed sequence on the unit interval with that of a sequence of random points chosen uniformly and independently on the unit interval. This comparison is done with the help of explicitly defined notions such as pair correlation, $k$-level correlation and level spacing distribution. These are called ``small-scale statistics" of a uniformly distributed sequence, and have been extensively studied for various types of arithmetic sequences, such as fractional parts of integer multiples of an irrational number, sequences arising from fractional parts of $(P(n))$, where $P(n)$ is a polynomial with irrational coefficients, and sequences of fractional parts of $a_n r$, where $r$ is an irrational number, and $(a_n)$ is a lacunary sequence. Small scale statistics have also been considered for sequences arising from interesting arithmetic objects such as the zeroes of the Riemann zeta function, zeta zeroes of curves over finite fields, and Hecke angles arising from the theory of modular forms. Some of these sequences have to be rescaled to fit a uniform distribution model. This project has threefold goals. 1) Our first goal is to consider analogues of the $k$-level correlation functions and the level spacing distribution in ``mesoscopic regimes", that is, intervals of short length around a fixed point inside the unit interval. This study is inextricably linked with the behaviour of exponential sums associated to these sequences. We want to systematically develop general principles about the relationship of these exponential sums with the existence of spacing distribution laws, in the same line of thought as Weyl's criterion and the Erdos-Turan inequality. 2) Our second goal is to consider sequences which may not be uniformly distributed, but are equidistributed with respect to some continuous distribution function. In order to study spacing statistics, we first need to straighten these sequences into uniformly distributed sequences. If we are considering distribution and small-scale statistics in mesoscopic regimes, then straightening such a sequence is essentially equivalent to suitably rescaling it. This gives us a large set of tools to address small scale statistics of a much larger class of arithmetic sequences, namely those that are equidistributed with respect to some continuous distribution function. 3) Our third major goal is to apply the theory and tools developed above to sequences of Hecke angles arising from the theory of modular forms. These are equidistributed with respect to the continuous function $\mu(t) = 2sin²(\pi t)$ (also called the Sato-Tate distribution function). Pertinent conjectures have been made about the small scale statistics of straightened Hecke angles by Katz and Sarnak. We want to study these conjectures as well as their analogues in mesoscopic regimes.