• Corpus ID: 249712526

Full Poissonian Local Statistics of Slowly Growing Sequences

  title={Full Poissonian Local Statistics of Slowly Growing Sequences},
  author={Christopher Lutsko and Niclas Technau},
Fix α > 0, then by Fejér’s theorem (α(logn) mod 1)n≥1 is uniformly distributed if and only if A > 1. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided A > 1. This is the first example of a deterministic sequence modulo one whose gap distribution, and all of whose correlations are proven to be Poissonian. The range of A is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of (log… 

Figures from this paper


Pair Correlation of the Fractional Parts of αn θ
Fix α, θ > 0, and consider the sequence (αn mod 1)n≥1. Since the seminal work of Rudnick– Sarnak (1998), and due to the Berry–Tabor conjecture in quantum chaos, the fine-scale properties of these
The two-point correlation function of the fractional parts of √ n is Poisson ∗
Elkies and McMullen [Duke Math. J. 123 (2004) 95–139] have shown that the gaps between the fractional parts of √ n for n = 1, . . . , N , have a limit distribution as N tends to infinity. The limit
Pair correlation densities of inhomogeneous quadratic forms
Under explicit diophantine conditions on (α, β) ∈ 2 ,w eprove that the local two-point correlations of the sequence given by the values (m − α) 2 + (n−β) 2 , with (m, n) ∈ 2 , are those of a Poisson
The Pair Correlation Function of Fractional Parts of Polynomials
Abstract: We investigate the pair correlation function of the sequence of fractional parts of αnd, n=1,2,…,N, where d≥ 2 is an integer and α an irrational. We conjecture that for badly approximable
2 1 D ec 2 02 1 Correlations of the Fractional Parts of αn θ
Let m ≥ 3, we prove that (αn mod 1)n>0 has Poissonian m-point correlation for all α > 0, provided θ < θm, where θm is an explicit bound which goes to 0 as m increases. This work builds on the method
Gaps between logs
We calculate the limiting gap distribution for the fractional parts of log n, where n runs through all positive integers. By rescaling the sequence, the proof quickly reduces to an argument used by
Pair correlation for fractional parts of αn2
  • D. R. Heath-Brown
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2010
It was proved by Weyl [8] in 1916 that the sequence of values of αn2 is uniformly distributed modulo 1, for any fixed real irrational α. Indeed this result covered sequences αnd for any fixed
A pair correlation problem, and counting lattice points with the zeta function
The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair
Gaps in √n mod 1 and ergodic theory
Cut the unit circle S = R/Z at the points { √ 1}, { √ 2}, . . . , { √ N }, where {x} = x mod 1, and let J1, . . . , JN denote the complementary intervals, or gaps, that remain. We show that, in
Distribution of mass of holomorphic cusp forms
We prove an upper bound for the L^4-norm and for the L^2-norm restricted to the vertical geodesic of a holomorphic Hecke cusp form of large weight. The method is based on Watson's formula and