A pair correlation problem, and counting lattice points with the zeta function

Abstract

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 correlations of sequences of the form (anα) n≥1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and (an) n≥1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number α, in terms of the additive energy of the integer sequence (an) n≥1 . In the present paper we develop a similar framework for the case when (an) n≥1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number θ > 1, the sequence (n θ α) n≥1 has Poissonian pair correlation for almost all α ∈ R.

Correlations of the Fractional Parts of $αn^θ$

“…In 1998 Rudnick and Sarnak [RS98], showed that the 2-point correlation (or pair correlation) of (1.3) is Poissonian for any integer θ ≥ 2, and (Lebesgue) almost every α > 0. Two decades later [AEBM21] and [RT21] proved the same statement for all non-integer θ > 1, and 0 < θ < 1 respectively. However, excluding these metric results, very little is known about sequences on the unit interval growing with polynomial rate.…”

On some applications of GCD sums to Arithmetic Combinatorics

“…We follow the method from [12], [4], [1] to give the proof of Lemma 8 below. Generally speaking, our bound ( 19) is close to the optimal one, see [5].…”

“…On the other hand, GRH implies [2] that n p = O(log 2 p) (the best unconditional bound belongs to Burgess [7] who proved n p ≪ p 1 4 √ e +o (1) ). Thus in the integer case our Theorem 1 gives upper bound (3) of a comparable quality.…”

“…The method of the proofs of Theorems 1, 2 uses so-called GCD sums (see, e.g., [1], [5], [4], [12]), which are connected with a series of questions of the Uniform Distribution, as well as Number Theory in particular, with large values of the zeta function. In our paper we follow beautiful exposition of random zeta functions approach from [12].…”

Pair Correlation of the Fractional Parts of $αn^θ$

“…For θ ∈ N >1 , Rudnick and Sarnak [RS98] proved the metric Poissonian pair correlation of (1.3) in the late 90s. The case of non-integer θ > 1 was only recently settled by Aistleitner, El-Baz, and Munsch [AEBM21]. The regime θ < 1 will be addressed in a forthcoming work of Rudnick and the second named author, see [RT].…”