Poisson spacing statistics for lattice points on circles by Pär Kurlberg & Stephen Lester ISBN 101007/S00222025013241 instant download

Poisson spacing statistics for lattice points on circles by Pär Kurlberg & Stephen Lester ISBN 101007/S00222025013241 instant download

AbstractWe show that along a density one subsequence of admissible radii, the nearest neighbor spacing between lattice points on circles is Poissonian.Mathematics Subject Classification Primary 11K36 · Secondary 11E251 IntroductionFor a sequence of real numbers (an) and integer N ≥ 1 the probability measure givenbyνN = 1N∑︂δ({an}),1≤n≤Nwhere δ(·) denotes the Dirac delta function and {an} is the fractional part of an,describes the distribution of the points (an)Nn=1 within R/Z and we say that ({an})is uniformly distributed modulo 1 if νN weakly converges to the Lebesgue measureλ on R/Z as N → ∞. A classical result of Weyl provides the following criterion:({an}) is uniformly distributed if and only if the Fourier coefficients of νN convergepointwise to those of λ as N → ∞.A more refined study of the behavior of a sequence modulo 1 examines the local spacing statistics of the sequence, that is how the sequence is distributed at thescale 1/N, which provides insight into how the elements of the sequence are spacedtogether. Computing the local spacing statistics of a given sequence is in general achallenging problem. Two important examples include the nearest neighbor spacingstatistics of the sequence consisting of the ordinates of zeros of the Riemann zetafunction as well as the sequence of energy levels of quantized Hamiltonians; both