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 correlations of sequences of the form $$(a_n \alpha )_{n \ge 1}$$ ( a n α ) n ≥ 1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here $$\alpha $$ α is a real parameter, and $$(a_n)_{n \ge 1}$$ ( a n ) 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 $$\alpha $$ α , in terms of the additive energy of the integer sequence $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 . In the present paper we develop a similar framework for the case when $$(a_n)_{n \ge 1}$$ ( a n ) 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 $$\theta >1$$ θ > 1 , the sequence $$(n^\theta \alpha )_{n \ge 1}$$ ( n θ α ) n ≥ 1 has Poissonian pair correlation for almost all $$\alpha \in {\mathbb {R}}$$ α ∈ R .

A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS

We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if$A\subseteq \mathbb{F}_{p}$satisfies$|A|\leq p^{64/117}$then$\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018,arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy$E^{+}(P)$of some subset$P\subseteq A+A$. Our main novelty comes from reducing the estimation of$E^{+}(P)$to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’,Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.

Pair correlation of sequences with maximal additive energy

We show for sequences $\left(a_{n}\right)_{n \in \mathbb N}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb N}$ does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.

Additive correlation and the inverse problem for the large sieve

Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$. We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$}, in the sense that we have the additive energy estimate $$ E(A,S)\gg N\log N. $$ This is, in a sense, optimal.

BOUNDS FOR TRIPLE EXPONENTIAL SUMS WITH MIXED EXPONENTIAL AND LINEAR TERMS

We establish bounds for triple exponential sums with mixed exponential and linear terms. The method we use is by Shparlinski [‘Bilinear forms with Kloosterman and Gauss sums’, Preprint, 2016, arXiv:1608.06160] together with a bound for the additive energy from Roche-Newton et al. [‘New sum-product type estimates over finite fields’, Adv. Math.293 (2016), 589–605].

Additive Energy and Irregularities of Distribution

We consider strictly increasing sequences (an)n≥1 of integers and sequences of fractional parts ({anα})n≥1 where α ∈ R. We show that a small additive energy of (an)n≥1 implies that for almost all α the sequence ({anα})n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.