Philadelphia Area Number Theory Seminar

Underlying a lot of modern number theory is the philosophy that arithmetic quantities for which no obvious reason for correlation exists should indeed be uncorrelated in a precise quantitative sense. A classical example is provided by the square-root cancellation in exponential sums such as the quadratic Gauss sums (which feature in the proof of quadratic reciprocity) or Kloosterman sums. Polygonal paths traced by their normalized incomplete sums give a fascinating insight into their chaotic formation. In this talk, we will present our recent results describing the limiting shape distribution in two ensembles of Gauss and Kloosterman sum paths as well as related results on sums of products of Kloosterman sums.

Dubi Kelmer (Boston College and Princeton)

Following Margulis's proof of the Oppenheim conjecture we know that integer values of an irrational indefinite quadratic form in n >= 3 variables are dense on the real line. The same is true for an inhomogeneous form obtained by shifting values by a fixed vector if either the form or the shift is irrational.  In this talk, I will describe several approaches to this problem that give effective results that hold for a fixed rational form Q and almost all shifts, by reducing it to the density of certain orbits of a discrete group acting on the torus. I will then describe different approaches using dynamics, representation theory, and estimates on exponential sums for this problem.