On the fractal structure of attainable probability measures

• Autorzy: Sartori A.

Identyfikatory

DOI

10.4064/ba8161-9-2018

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

lattice points; attainable probability measures

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 2
• Strony: 123--133
• Opis fizyczny: Bibliogr. 8 poz., wykr.

Twórcy

autor

Sartori A.

• Department of Mathematics, King's College London, Strand, London WC2R 2LS, England, UK

Bibliografia

• [1] J. Cilleruelo, The distribution of the lattice points on circles, J. Number Theory 43 (1993), 198-202.
• [2] P. Erdős and R. R. Hall, On the angular distribution of Gaussian integers with fixed norm, Discrete Math. 200 (1999), 87-94.
• [3] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I, Math. Z. 1 (1918), 357-376.
• [4] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. II, Math. Z. 6 (1920), 11-51.
• [5] I. Kátai and I. Környei, On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 19 (1976), 87-91.
• [6] M. G. Kreĭn and A. A. Nudel'man, The Markov Moment Problem and Extremal Problems, Amer. Math. Soc., Providence, RI, 1977.
• [7] M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves, Ann. of Math. 177 (2013), 699-737.
• [8] P. Kurlberg and I. Wigman, On probability measures arising from lattice points on circles, Math. Ann. 367 (2017), 1057-1098.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).