event
Thursday 29 Sep 2022: Remarks about inhomogeneous pair correlations
Felipe Ramirez - Wesleyan
Harrison 103 14:30-15:30
A sequence a_n of natural numbers is said to have metric Poissonian pair correlations (MPPC) if for almost every real number x the associated sequence x*a_n (mod 1) on the circle has asymptotically Poissonian pair correlations. Informally speaking, this means that the points of the sequence clump together to the same degree that they would if they had been picked randomly. For example, the sequence of natural numbers does not have MPPC, while the sequence of square numbers does. Generally, if a sequence has too much additive structure, like the natural numbers, then it will not have MPPC. If it has very little additive structure, like the squares or the powers of two, then it will have MPPC. But there is a zone in between "too much" and "very little" additive structure where the picture is not so clear, and there has been a lot of work devoted to finding a threshold separating sequences with MPPC from those without, based on what is called the sequence's "additive energy." I will survey this work, and I will discuss an associated inhomogeneous problem where the corresponding questions seem to be easier to answer.
