On the fractal structure of attainable probability measures

The set of representations of an integer as a sum of two squares gives rise to a probability measure on the unit circle in a natural way. Given the sequence of such measures we call its weak$^{\ast }$ limit points attainable probability measures. Kurlberg and Wigman (2016) studied the set of attainable measures and discovered that its projection onto the first two non-trivial Fourier coefficients has a peculiar structure, visibly reproducing itself in a “fractal”-looking manner near the $y$-axis. They conjectured that one can describe this picture using analytic functions. We show that this is indeed true.