Sequences with almost Poissonian Pair Correlations

Weiß, Christian; Skill, Thomas

doi:10.48550/arxiv.1905.02760

Cited by 2 publications

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“…If for each ε > 0 we have A N (α) = It is well known (also with a higher-dimensional generalisations due to Beck [3]) that for almost every α ∈ [0, 1] the discrepancy of the Kronecker sequence is ≪ (log N) 1+ε , for each ε > 0. Thus the above corollary generalizes and sharpens a result of Skill and Weiß [26] stating that (φn) ∞ n=1 has Poissonian pair correlations on each scale β < 1 where φ denotes the Golden ratio √ 5+1…”

Section: Appendix B Discrepancy and K-point Correlation Functions At ...supporting

confidence: 72%

On the triple correlations of fractional parts of $n^2α$

Technau,

Walker

2020

Preprint

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For fixed α ∈ [0, 1], consider the set S α,N of dilated squares α, 4α, 9α, . . . , N 2 α modulo 1. Rudnick and Sarnak conjectured that for Lebesgue almost all such α the gapdistribution of S α,N is consistent with the Poisson model (in the limit as N tends to infinity). In this paper we prove a new estimate for the triple correlations associated to this problem, establishing an asymptotic expression for the third moment of the number of elements of S α,N in a random interval of length L/N , provided that L > N 1/4+ε . The threshold of 1/4 is substantially smaller than the threshold of 1/2 (which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations (αa n mod 1) ∞ n=1 for a non-lacunary sequence (a n ) ∞ n=1 of increasing integers. This is partially due to the fact that second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick-Sarnak-Zaharescu and Heath-Brown, which connects the triple correlation function to some modular counting problems.In an appendix we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.3 We should remark that Rudnick-Sarnak phrased their result in terms of the pair correlation function, rather than explicitly mentioning W α,L,N , but the results are equivalent.

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Section: Appendix B Discrepancy and K-point Correlation Functions At ...supporting

confidence: 72%

On the triple correlations of fractional parts of $n^2α$

Technau,

Walker

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…If, for each ε > 0, we have It is well known (also with a higher-dimensional generalization due to Beck [3]) that, for almost every α ∈ [0, 1], the discrepancy of the Kronecker sequence is ≪ (log N) 1+ε , for each ε > 0. Thus, the above corollary generalizes and sharpens a result of Weiß and Skill [28], stating that (ϕn) ∞ n=1 has Poissonian pair correlations on each scale β < 1, where ϕ denotes the Golden ratio √ 5+1…”

Section: Discussionsupporting

confidence: 74%

On the triple correlations of fractional parts of

Technau¹,

Walker

2021

Can. J. Math.-J. Can. Math.

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For fixed $\alpha \in [0,1]$ , consider the set $S_{\alpha ,N}$ of dilated squares $\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $ modulo $1$ . Rudnick and Sarnak conjectured that, for Lebesgue, almost all such $\alpha $ the gap-distribution of $S_{\alpha ,N}$ is consistent with the Poisson model (in the limit as N tends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{\alpha ,N}$ in a random interval of length $L/N$ , provided that $L> N^{1/4+\varepsilon }$ . The threshold of $\tfrac {1}{4}$ is substantially smaller than the threshold of $\tfrac {1}{2}$ (which is the threshold that would be given by a naïve discrepancy estimate). Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations $(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $ for a nonlacunary sequence $(a_n)_{n=1}^{\infty } $ of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems. In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

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Sequences with almost Poissonian Pair Correlations

Cited by 2 publications

References 9 publications

On the triple correlations of fractional parts of $n^2α$

On the triple correlations of fractional parts of $n^2α$

On the triple correlations of fractional parts of

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