Communications in Number Theory and Physics

Volume 11 (2017)

Number 4

Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

Pages: 837 – 877



Tristan Freiberg (Department of Pure Mathematics, University of Waterloo, Ontario, Canada)

Pär Kurlberg (Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden)

Lior Rosenzweig (Department of Mathematics, ORT Braude College, Karmiel, Israel; and Unit of Mathematics, AFEKA Tel-Aviv Academic College of Engineering, Tel Aviv, Israel)


We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors–Keating, and Smilansky, we formulate an analog of the Hardy–Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Ueberschär, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.

Received 30 January 2017

Accepted 5 April 2017

Published 29 November 2017