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There is no Khintchine threshold for metric pair correlations. (English) Zbl 1466.11048

If \((x_n)_{n \geq 1}\) is a sequence of real numbers in \([0,1)\), the distribution of its pair correlations is captured by the function defined for \(N\) integer and \(s \geq 0\) by \[ R(s,N) := \frac{1}{N} \sharp\left\{(i,j), \ 1 \leq i,j \leq N, \ i \neq j, \ ||x_i - x_j|| \leq s/N\right\} \] where \(|| \cdot ||\) is the distance to the nearest integer. The distribution of pair correlations is said to be (asymptotically) Poissonian if \(\lim_{N \to \infty} R(s,N) = 2s\) for all \(s \geq 0\). Now, an increasing sequence of positive integers \((a_n)_{n \geq 1}\) is said to have the metric pair correlation property if the sequence \((\{a_n \alpha\})_{n \geq 1}\) is asymptotically Poissonian for Lebesgue-almost all \(\alpha \in [0,1]\), where \(\{x\}\) is the fractional part of the real number \(x\): for example it is known that \((n^d)_{n \geq 1}\) has the metric pair correlation property if \(d\) is an integer \(\geq 2\), and that \((a_n)_{n \geq 1}\) has the metric pair correlation property if it is an exponentially growing sequence of integers.
In a 2017 paper the first author et al. [Isr. J. Math. 222, No. 1, 463–485 (2017; Zbl 1388.11043)] proved that a sequence has the metric pair correlation property if its additive energy has order at most \(N^{3 - \varepsilon}\) for some \(\varepsilon > 0\), while the sequence does not have the metric pair correlation property if its additive energy is larger than \(c N^3\) for infinitely many \(N\), for some constant \(c > 0\). Here the additive energy of the sequence \((a_n)_{n \geq 1}\) is defined for \(N \geq 1\) by: \[ E(N) := \sharp\left\{(n_1, n_2, n_3, n_4), \ n_i \leq N, \ a_{n_1} + a_{n_2} = a_{n_3} + a_{n_4}\right\}. \] Since \(E(N)\) is between \(N^2\) and \(N^3\), it is natural to ask for a sharp threshold below \(N^3\) for the additive energy that implies the metric pair correlation property. In particular it was speculated in [T. F. Bloom et al., Mathematika 64, No. 3, 679–700 (2018; Zbl 1441.11185)], that, if \(E(N) \sim N^3 \psi(N)\), for some decreasing function \(\psi\), then the sequence has the metric pair correlation property. The authors of the paper under review construct a subtle counter-example to this speculation.
Please note that Reference [5] has appeared [T. F. Bloom and A. Walker, Isr. J. Math. 235, No. 1, 1–11 (2020; Zbl 1450.11094)].

MSC:

11K06 General theory of distribution modulo \(1\)
11K60 Diophantine approximation in probabilistic number theory
11B30 Arithmetic combinatorics; higher degree uniformity
05A18 Partitions of sets
11B25 Arithmetic progressions
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References:

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