The bad and rough rotation is Poissonian

2025/06/02 by Manuel Hauke, Hauke, Manuel · 2 citations
Engineering · #11J71 11N25 11N80 11K45 11K60 11J83 #FOS: Mathematics #Number Theory (math.NT) #Robotic Mechanisms and Dynamics

paper · pdf · doi:10.48550/arxiv.2506.01736

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences (a_nα)_{n ∈ ℕ} \mod 1 with (a_n)_{n ∈ ℕ} ⊆ ℕ and α irrational have been extensively studied in the last decades. In this article, we prove that for (a_n)_{n ∈ ℕ} arising from the set of rough numbers with explicit roughness parameters and any badly approximable α, (a_nα)_{n ∈ ℕ} \mod 1 has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence (a_nα)_{n ∈ ℕ} \mod 1 with these properties. Further, we show that this result is false for Lebesgue almost every α, thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod d in diophantine Bohr sets which might be of independent interest.

Citations

• On Sequences with Exponentially Distributed Gaps
• Rational approximation with chosen numerators
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• Long-Range Correlations of Sequences Modulo 1
• On the triple correlations of fractional parts of n²α
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• GCD sums and sum-product estimates
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• Additive Energy and the Hausdorff dimension of the exceptional set in\n metric pair correlation problems
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• The two-point correlation function of the fractional parts of \√(n)\n is Poisson
• Small gaps between primes or almost primes
• The distribution of spacings between fractional parts of lacunary sequences
• The distribution of spacings between quadratic residues, II
• Level clustering in the regular spectrum

Cited by

• On the behavior of the colored Jones polynomial of the figure-eight knot under modular transformations
• Twisted approximation with restricted denominators

Related

• Additive energy, discrepancy and Poissonian k-level correlation
• Pair Correlation of the Fractional Parts of αn^θ
• Gap statistics and higher correlations for geometric progressions modulo\n one
• The distribution of spacings of real-valued lacunary sequences modulo one
• Remark on a result of Bourgain on poissonian pair correlation