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On the fractal structure of attainable probability measures

Volume 66 / 2018

Andrea Sartori Bulletin Polish Acad. Sci. Math. 66 (2018), 123-133 MSC: Primary 11Z05. DOI: 10.4064/ba8161-9-2018 Published online: 26 October 2018


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.


  • Andrea SartoriDepartment of Mathematics
    King’s College London
    Strand, London WC2R 2LS
    England, UK

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