The distribution of spacings of real‐valued lacunary sequences modulo one

Abstract

Let (𝑎 𝑛 ) ∞ 𝑛=1 be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all 𝛼 ∈ ℝ, the pair correlation of (𝛼𝑎 𝑛 ) ∞ 𝑛=1 mod 1 is Poissonian. We show that all higher correlations and hence the nearest-neighbour spacing distribution are Poissonian as well, thereby extending a result of Rudnick and Zaharescu to real-valued sequences.

Intermediate-scale statistics for real-valued lacunary sequences

“…n=1 is an integer-valued lacunary sequence; this was recently extended to real-valued lacunary sequences by Chaubey and the author [2]. At the other extreme, Lutsko and Technau recently proved [10] Poissonian gap statistics for the slowly growing sequence x n = α( log n) A (A > 1) -remarkably, this holds for any α > 0, and not only in the metric sense -see also the closely related results [8,9] about Poissonian correlations for the sequence x n = αn θ where θ is small.…”

“…We then unsmooth along the subsequence, and finally deduce the result along the full sequence. We would like to use the results of [2], and for that it would be more convenient to work with a "transformed" correlation function: for k ≥ 2 and for a smooth, compactly supported function ψ : R k−1 → R (which may depend on N), we denote the smoothed k-level correlation function…”

“…while the term ψ 2 r,1 is not explicitly stated there, it follows from the proof, which we now sketch (for the full details we refer the reader to [2]): for x = (x 1 , . .…”

On the number of gaps of sequences with Poissonian pair correlations

“…Rudnick, Sarnak and Zaharescu [22] gave a Diophantine criterion for α which guarantees that the sequence ({n 2 α}) n∈N has exponential gaps (without providing an explicit example of a number α satisfying the condition), and Lutsko and Technau proved that the sequence ({α(log n) A }) n∈N for α > 0 and A > 1 has exponential gaps [16]. Among metric results, it is known that the sequence ({q n α}) n∈N for a lacunary (q n ) n∈N has exponential gap distribution for almost all α [7,23], and that ({α n }) n∈N has exponential gaps for almost all α > 1 [1].…”