On the correlations of $n^α$ mod 1

Abstract

A well known result in the theory of uniform distribution modulo one (which goes back to Fejér and Csillag) states that the fractional parts {n α } of the sequence (n α ) n≥1 are uniformly distributed in the unit interval whenever α > 0 is not an integer. For sharpening this knowledge to local statistics, the k-level correlation functions of the sequence ({n α }) n≥1 are of fundamental importance. We prove that for each k ≥ 2, the k-level correlation function R k is Poissonian for almost every α > 4k 2 − 4k − 1.

Correlations of the Fractional Parts of $αn^θ$

“…For m ≥ 3 there are hardly any results on the probabilistic theory for m-point correlation functions and even fewer deterministic results. An exception is the work of Yesha and the second named author [TY20], who showed that (n α mod 1)n has Poissonian m-point correlation, for almost all α > 4m 2 − 4m−1. Moreover, for lacunary sequences we refer to Rudnick and Zaharescu [RZ99, RZ02], for dilations of lacunary integer sequences; and Chaubey and Yesha [CY21] where this is extended to dilations of real-valued sequences.…”

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

“…Almost sure convergence. Having proved the variance bound (13), the almost sure convergence of the k-level correlation sums to R k−1 f (x) dx follows from a standard argument, as formulated in a general setting in the following proposition taken from [13].…”

“…Indeed, we can clearly assume that α ∈ J where J is a fixed finite interval and take ρ such that ρ ≥ 1 J . Let ϑ n (α) = αa n and c k (N ) = C k (N ); the bound (17) follows from (13), since…”

Poissonian correlations of higher orders

“…However, the subject has also gained significant interest on a purely mathematical level, where the correlations of general sequences of arithmetic origin were studied; see for example [2,3,20]. Most results only concern the case of correlations of order k = 2 (known as pair correlations), while correlations of higher order are combinatorially and analytically more difficult to study and often out of reach; among the relatively few results in that direction are for example [22] and [28].…”