The Distribution of Spacings between Fractional Parts of Lacunary Sequences

A primary example is to take an integer g ≥ 2 and set a(x) = g. As is true for any increasing sequence of integers, for almost every α the fractional parts αa(x) are uniformly distributed modulo 1. Moreover, for lacunary sequences, it has long been known that the fractional parts of αa(x) have strong randomness properties. For instance, the exponential sums 1 √ N ∑ x≤N cos(2παa(x)) have a Gaussian value distribution as N → ∞ (see the survey in [5]). In this paper, we show that lacunary sequences have additional features in common with those of random sequences, which is the asymptotic distribution of spacings between elements of the sequence: Given a sequence {θn} ⊂ [0, 1), the nearest-neighbor spacing distribution is defined by ordering the first N elements of the sequence: θ1,N ≤ θ2,N ≤ · · · ≤ θN,N , and then defining the normalized spacings to be δ n := N(θn+1,N − θn,N) . The asymptotic distribution function of {δ n }n=1 is level spacing distribution P1(s), that is for each interval [a, b] we require that