The distribution of spacings between the fractional parts of $n^2\alpha$

Abstract

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.

Substantial revisions in the exposition. Accepted for publication in Inventiones mathematicae

Cited by in corpus (29)

• Pair Correlation for Fractional Parts of $\alpha n^2$
• The primes are not metric Poissonian
• There is no Khintchine threshold for metric pair correlations
• Gaps between logs
• Pair correlation for quadratic polynomials mod 1
• Spacings and pair correlations for finite Bernoulli convolutions
• The Champernowne constant is not Poissonian
• Gap statistics and higher correlations for geometric progressions modulo one
• Effective multi-scale approach to the Schrödinger cocycle over a skew shift base
• On the spectrum of skew-shift Schrödinger operators
• On a random walk with memory and its relation to Markovian processes
• Pair correlation of sequences $(\{a_n\alpha \}{)}_{n\in \mathbb{N}}$ with maximal order of additive energy
• The twisted second moment of modular half integral weight $L$--functions
• Pair correlations and equidistribution
• Universality of smoothness of Density of States in arbitrary higher-dimensional disorder under non-local interactions I. From Viéte--Euler identity to Anderson localization
• Remark on a result of Bourgain on poissonian pair correlation
• Pair correlation densities of inhomogeneous quadratic forms II
• Spacings between integers having typically many prime factors
• Global eigenvalue distribution of matrices defined by the skew-shift
• Continuous Distribution Arising From the Three Gaps Theorem
• The distribution of rational points and polynomial maps on an affine variety over a finite field on average
• Pair correlation of roots of rational functions with rational generating functions and quadratic denominators
• Equidistribution of Kronecker sequences along closed horocycles
• On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential
• On the correlations of directions in the Euclidean plane
• On the Restricted Divisor Function in Arithmetic Progressions
• Poissonian pair correlation on manifolds via the heat kernel
• A Local Law for Singular Values from Diophantine Equations
• Equidistribution results for sequences of polynomials