Gaps between sums of two squares, modular forms and Cohen-Kuznetsov series

Problems on distribution of gaps between values of quadratic forms are among the most well-known open questions in number theory. In this talk, I will give an overview of some results on gaps between sums of two squares and discuss a connection between this area and theory of modular forms. More precisely, we are going to construct a certain function detecting the large gaps and discuss its automorphic properties. This function of two variables turns out to be a special case of Cohen-Kuznetsov series, it also produces a sequence of rational functions with remarkable properties.