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On the fractal structure of attainable probability measures

Sartori A.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 2

123--133

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 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 coeffcients 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.