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The bad and rough rotation is Poissonian

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Pith's one-line read Dilating the set of rough numbers by any badly approximable angle produces a sequence with Poissonian correlations of every order, the first explicit sequence of the form $\{a_n\alpha\}$ with any such property, and a matching converse…

desk verdict The paper has a genuinely new idea and a plausible triangular-array theorem, but the sequence-level main theorem rests on an unproved regularity assertion in Section 5.3. read the letter →

arxiv 2506.01736 v1 pith:G42E6XCF submitted 2025-06-02 math.NT

Manuel Hauke This is my paper
classification math.NT MSC 11K3111K3611N3511J7011K55
keywords roughnumbersPoissoniancorrelationsgapsbadlyapproximableBohrsetsOstrowskiexpansionssievemethodsfine-scalestatistics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that a number-theoretic sequence, the dilates of the rough integers by a badly approximable angle, behaves statistically like independent random points on the circle. Rough integers are those whose smallest prime factor exceeds a slowly shrinking threshold $f(x)=x^{r(x)}$ with $r(x)\to 0$; the main theorem states that for every badly approximable $\alpha$, the sequence $(a_n^{(f)}\alpha \bmod 1)$ has Poissonian correlations of every order and therefore Poissonian gaps. This is the first known explicit sequence of the form $\{a_n\alpha\}$ with any Poissonian correlation. The paper also proves a converse: for Lebesgue-almost every $\alpha$, the same sequence fails to have Poissonian pair correlations, which disproves a 2018 conjecture about such sequences. The mechanism combines a sieve-theoretic count of rough-number tuples with a new equidistribution theorem for residue classes inside Bohr sets.

What carries the argument

The argument is carried by three objects working in sequence. The first is the rough-number triangular array $\{n\le x: P^-(n)>f(x)\}$ with counting function $\Phi(x,z)$, approximated by the Fundamental Lemma of sieve theory. The second is the singular series $S_{k-1}(h)=\prod_{p\le f(x)}(1-g_h(p)/p)$, where $g_h(p)$ counts collisions among $0,h_1,\dots,h_{k-1}$ modulo $p$; Lemma 4 turns this product into an asymptotic for the number of rough $k$-tuples with difference vector $h$. The third is the diophantine Bohr set $B(\alpha,x)=\{h\le x:\{h\alpha\}\in[0,\rho_x]\}$, on which the singular series must be averaged; the decisive new input is Theorem 3, which proves that residue classes modulo $d$ are equidistributed inside such Bohr sets, established through Ostrowski expansions and a Markov-chain convergence argument. A step-function procedure in Section 5.2 converts the averaged Euler products into the rectangle volume $\mathrm{Vol}(R)$, and Section 5.3 passes from the triangular array to the actual sequence.

What would settle it

Take $r(x)=(\log\log x)^A/\log x$ on most of each very long interval and set $r(x)=\frac{1}{2}(\log\log x)^A/\log x$ on the final segment $[x/(\log x)^{k+1},x]$, with interval endpoints growing fast enough to keep $r$ monotone. This $r$ satisfies the theorem's lower bound yet makes $\log f(x)/\log f(x/(\log x)^{k+1})\to 2$, so the Section 5.3 equivalence $\Phi(x,f(x))\sim\Phi(x,f(x/(\log x)^{k+1}))$ fails; that would show the sequence-level theorem does not follow from the triangular-array theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1: for $f(x)=x^{r(x)}$ with $r(x)$ decreasing monotonically to $0$ and $r(x) \gg_A (\log\log x)^A/\log x$ for every $A>0$, and for any badly approximable $\alpha$, the sequence $(a_n^{(f)}\alpha \bmod 1)$ has Poissonian correlations of all orders $k\ge 2$, hence Poissonian gaps, and the same holds for the triangular array of uniformly rough numbers up to $x$. The matching negative result, Theorem 2, says that if $\alpha$ satisfies $\liminf_{n\to\infty} n\|n\alpha\|\log f(n)=0$, then the sequence does not even have Poissonian pair correlations; since that Diophantine condition holds for Lebesgue-almost every $\alpha$, the positive result genuinely depends on bad approximability. The two theorems together refute a 2018 conjecture predicting that non-Poissonian behavior for almost all $\alpha$ would rule out any Poissonian $\alpha$.

Load-bearing premise

The transfer from the uniformly rough triangular array to the true sequence assumes that replacing the cutoff $f(x)$ by $f(x/(\log x)^{k+1})$ changes neither the counting function nor the correlation limit, because the two cutoffs are treated as asymptotic powers of one another; the stated hypotheses on decreasing $r(x)$ do not force that ratio $\log f(x/(\log x)^{k+1})/\log f(x)$ to tend to $1$.

Editorial extensions

If this is right

  • For every badly approximable $\alpha$, the rough-number rotation has Poissonian correlations of every order and therefore Poissonian gaps, giving the first known explicit sequence of the form $\{a_n\alpha\}$ with any Poissonian correlation.
  • The gap distribution between successive points is exponential, and for every fixed $k$ the distribution of $k$-th neighbour spacings is determined by the same correlation functions.
  • For Lebesgue-almost every $\alpha$, the same sequence fails even to have Poissonian pair correlations, so the positive result cannot be extended to a metric statement; this refutes a 2018 conjecture claiming that non-Poissonian behavior for almost all $\alpha$ would force non-Poissonian behavior for every $\alpha$.
  • The equidistribution theorem for residue classes inside Bohr sets is flexible enough to average multiplicative functions over Bohr sets, which the paper identifies as a tool of independent interest.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Repairing the Section 5.3 regularity gap would make the method a general template: any sieve-amenable integer set whose singular series averages over a Bohr set should inherit Poissonian correlations for badly approximable $\alpha$, and the paper already notes one can sift only primes in chosen congruence classes without changing the proof.
  • The theorem sharpens the expected dichotomy: for a fixed slowly growing sequence, the set of $\alpha$ with Poissonian correlations can be a measure-zero, full-Hausdorff-dimension set, so the interesting question is not 'almost all $\alpha$' but which Diophantine null sets work.
  • A direct numerical test is available: for the golden ratio $\alpha=(\sqrt5-1)/2$ and a roughness exponent like $r(x)=1/\sqrt{\log x}$, the $k$-point correlations of the rough-number rotation should converge to the rectangle volume, though the implied convergence has no rate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies fine-scale statistics of the sequence (a_n^{(f)} α) mod 1, where a_n^{(f)} is the n-th integer whose least prime factor exceeds f(n). For f(x)=x^{r(x)} with r decreasing to 0 and r(x) ≫_A (log log x)^A / log x for every A>0, Theorem 1 claims that for every badly approximable α the sequence has Poissonian correlations of all orders, hence Poissonian gaps; the same is claimed for the associated triangular arrays. Theorem 2 asserts a converse for α satisfying liminf n ∥nα∥ log f(n)=0, and in particular that the result fails for Lebesgue-almost every α, disproving a conjecture of Larcher and Stockinger. The proof combines a sieve estimate (Lemma 4), an equidistribution theorem for Bohr sets modulo d (Theorem 3), proved via Ostrowski expansions and Markov chains, a key averaging lemma (Lemma 16), and a final reduction from triangular arrays to the actual sequence in Section 5.3.

Significance. If the main theorem were established, it would be a notable advance: it would give the first explicit sequence of the form {a_n α} with Poissonian correlations of all orders, confirm a rough-number analogue of the Rudnick–Sarnak heuristic, and settle the Larcher–Stockinger conjecture in the negative. The paper also contains a substantial and partly novel technical apparatus: the Bohr-set divisibility theorem (Theorem 3), the sieve estimate Lemma 4, and the averaging machinery of Section 5.1 are developed in considerable detail and are plausibly correct. The Markov-chain approach to equidistribution in Bohr sets is a genuine methodological contribution that may be of independent interest. However, the advertised sequence-level result is not proven as stated because the passage from triangular arrays to the sequence in Section 5.3 contains a load-bearing unproved assertion.

major comments (2)
  1. [Section 5.3] The reduction from triangular arrays to the actual sequence is invalid under the stated hypotheses. The text asserts: 'Observe that Φ(x,z)≤N(x,f)≤Φ(x,z−) and Φ(x,z)∼Φ(x,z−) since log z / log z− → 1.' The implication 'log z / log z− → 1' does not follow, and is in fact false for admissible r. Let L=(log x)^{k+1}. The lower-bound hypothesis r(x) ≫_A (log log x)^A / log x for all A does not prevent r from dropping sharply. For example, set g(t)=exp(−√(log log t)) and choose x_n so that x_n/L_n > x_{n−1} and g(x_{n−1})/g(x_n)=n, where L_n=(log x_n)^{k+1}; define r(x)=g(x_n) on [x_n,x_{n+1}). This r decreases monotonically to 0 and satisfies the lower bound, but at x=x_n we have log f(x_n/L_n)/log f(x_n)=n(1+o(1))→∞. Hence Φ(x,f(x)) and Φ(x,f(x/L)) need not be asymptotically equal, and the two triangular-array limits in the squeeze are normalized by different, incompatible quantities. The proof of Theorem 1 therefore does not establish the claimed sequence result under the stated assumptions. A regularity condition such as r(x)/r(x/(log x)^{k+1})→1 appears to be necessary, and must be added to the theorem or proved from a strengthened hypothesis.
  2. [Section 6] Theorem 2 is first proved for triangular arrays, and the transition to the actual sequence is explicitly delegated to 'the step to move to the sequence can be established as in Section 5.3.' Since the Section 5.3 reduction is not valid under the stated assumptions on r, the sequence version of Theorem 2 is not established either. The statement of Theorem 2, as well as the assertion that it disproves the Larcher–Stockinger conjecture for the sequence (a_n^{(f)} α), is therefore currently unsupported.
minor comments (5)
  1. [Definition 1] The definition of Φ(x,z) reads '#{1≤n≤x : P^−(x)>z}'; the argument of P^− should be n, not x.
  2. [Lemma 11] The displayed statement '#B(x,I_x)∼λ(I_x) I_x' should be '#B(x,I_x)∼λ(I_x) x'; the proof and the surrounding text use λ(I_x)x.
  3. [Notation, Section 2.0.1] The sentence 'We denote by {x} the integer part of x' should read 'the fractional part of x'; the standard notation {x} is used later with that meaning.
  4. [Section 5.3] The sentence 'since log z / log z− → 1' should also specify that z and z− depend on x and that the limit is taken as x→∞; as written it is ambiguous.
  5. [Proposition 13] The proof of Proposition 13 labels the second inequality as '(ii)' and then refers to '(iii)' for the digit form; the labels should be rechecked for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Poissonian correlation result is derived from independent sieve estimates, a new Bohr-set equidistribution theorem, and elementary averaging arguments; the self-citations are not load-bearing in a circular way.

full rationale

The paper's central claim in Theorem 1 is not obtained by fitting a parameter and then renaming it as a prediction, nor by defining an object in terms of the target conclusion. The proof proceeds through a uniform sieve estimate (Lemma 4), a new equidistribution result modulo d in badly approximable Bohr sets (Theorem 3, proved via Ostrowski expansions and Markov chains), an averaging argument over multiplicative functions (Lemma 16), and a step-function/Riemann-sum approximation. At no point does the proof presuppose the Poissonian value Vol(R) that it derives; the limit emerges only after taking N to infinity and then M to infinity in the Euler-product argument. The two self-citations are to [20] for a Bohr-set counting bound used in tail estimates and to [21] for an elementary integral evaluation; neither imports a uniqueness theorem, neither defines the rough-number sequence in terms of the correlation function, and neither has assumptions containing the target Poissonian statement. Under the stated rules these are independent supports rather than circular ones. There is a potential analytic gap in Section 5.3, where the transfer from triangular arrays to the actual sequence uses 'Φ(x, z) ∼ Φ(x, z−) since log z / log z− → 1'; the stated monotonicity and lower-bound assumptions on r(x) may not force that ratio to tend to 1. This is a correctness/falsifiability concern and would need a regularity condition such as r(x)/r(x/(log x)^(k+1)) → 1, but it is not circularity: the contested ratio is not an input disguised as the output, and no equation in the chain is equivalent by construction to the claimed correlation limit. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard analytic number theory tools plus the domain assumptions that alpha is badly approximable and that the roughness parameter decays slowly. No new particles, forces, dimensions, or fitted constants are introduced. The proof is not circular in the sense of defining its target quantity into existence.

assumptions (7)
  • standard math Fundamental Lemma of Sieve Theory as stated in Lemma 5
    Used in Lemma 4 to count k-tuples of rough numbers with prescribed differences; quoted from Koukoulopoulos.
  • standard math Denjoy-Koksma inequality for rotations
    Used in Lemma 11 to estimate the cardinality of Bohr sets; a classical discrepancy result.
  • standard math Ostrowski expansion existence, uniqueness, and cylinder-set correspondence
    Used throughout Section 4 to relate Bohr sets to Ostrowski digits and to set up the Markov chain argument.
  • standard math Markov chain convergence theorem for finite irreducible aperiodic chains
    Used in Corollary 10 and Lemma 14 to prove equidistribution of sums of Ostrowski digits modulo d.
  • standard math Khintchine's theorem in metric Diophantine approximation
    Used in Theorem 2 to show condition (1) holds for Lebesgue almost every alpha; the text does not verify the monotonicity hypothesis.
  • domain assumption Alpha is badly approximable, so its partial quotients are bounded
    This is the standing hypothesis of Theorem 1 and Theorem 3; it is used repeatedly for discrepancy estimates and Bohr-set cardinalities.
  • domain assumption Roughness function r(x) decreases to 0 and r(x) >>_A (log log x)^A / log x for every A
    This is part of the statement of Theorem 1 and is used to ensure the rough numbers are sufficiently sparse for the averaging analysis.

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Pith. "Pith review of The bad and rough rotation is Poissonian." pith.science (2026). https://pith.science/paper/G42E6XCF

@misc{pith2026250601736,
  author       = {Pith},
  title        = {Pith review of: The bad and rough rotation is Poissonian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G42E6XCF}},
  note         = {Machine review of arXiv:2506.01736}
}
abstract

Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ with $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{N} $ and $\alpha$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \in \mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $\alpha$, $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ with these properties. Further, we show that this result is false for Lebesgue almost every $\alpha$, 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.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution

    math.NT 2026-07 reject novelty 7.0 of 10

    Weak γ-PPC implies neither γ-PPC nor equidistribution, and weak γ1-PPC does not imply weak γ2-PPC for distinct γ in (0,1/2].

Reference graph

Works this paper leans on

61 extracted references · 58 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aistleitner, M

    C. Aistleitner, M. Hauke, and A. Zafeiropoulos. On sequences with exponentially dis- tributed gaps.Random Structures Algorithms, 66(1):Paper No. e21265, 15, 2025

    work page 2025

  2. [2]

    Aistleitner, T

    C. Aistleitner, T. Lachmann, and F. Pausinger. Pair correlations and equidistribution. J. Number Theory, 182:206–220, 2018

    work page 2018

  3. [3]

    Aistleitner, T

    C. Aistleitner, T. Lachmann, and N. Technau. There is no Khintchine threshold for metric pair correlations.Mathematika, 65(4):929–949, 2019

    work page 2019

  4. [4]

    Aistleitner, G

    C. Aistleitner, G. Larcher, and M. Lewko. Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems.Israel J. Math., 222(1):463–485,

    work page

  5. [5]

    Allouche and J

    J.-P. Allouche and J. Shallit.Automatic sequences. Cambridge University Press, Cam- bridge, 2003. Theory, applications, generalizations

    work page 2003

  6. [6]

    Beresnevich, A

    V. Beresnevich, A. Haynes, and S. Velani. Sums of reciprocals of fractional parts and multiplicative Diophantine approximation.Mem. Amer. Math. Soc., 263(1276):vii + 77, 2020. 27

    work page 2020

  7. [7]

    Berry and M

    M. Berry and M. Tabor. Level clustering in the regular spectrum.Proc. of the Royal Soc. of London. A. Math. and Phy. Sciences, 356(1686):375–394, 1977

    work page 1977

  8. [8]

    T. F. Bloom, S. Chow, A. Gafni, and A. Walker. Additive energy and the metric Poisso- nian property.Mathematika, 64(3):679–700, 2018

    work page 2018

Show all 61 references
  1. [9]

    T. F. Bloom and A. Walker. GCD sums and sum-product estimates.Israel J. Math., 235(1):1–11, 2020

    2020

  2. [10]

    J. R. Chen. On the representation of a larger even integer as the sum of a prime and the product of at most two primes.Sci. Sinica, 16:157–176, 1973

    1973

  3. [11]

    S. Chow. Bohr sets and multiplicative Diophantine approximation.Duke Math. J., 167(9), 2018

    2018

  4. [12]

    Chow and N

    S. Chow and N. Technau. Littlewood and Duffin-Schaeffer-type problems in Diophantine approximation.Mem. Amer. Math. Soc., 296(1475):v+74, 2024

    2024

  5. [13]

    El-Baz, J

    D. El-Baz, J. Marklof, and I. Vinogradov. The two-point correlation function of the fractional parts of√nis Poisson.Proc. Amer. Math. Soc., 143(7):2815–2828, 2015

    2015

  6. [14]

    N. D. Elkies and C. T. McMullen. Gaps in√nmod 1and ergodic theory.Duke Math. J., 123(1):95–139, 2004

    2004

  7. [15]

    Freiberg, P

    T. Freiberg, P. Kurlberg, and L. Rosenzweig. Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers.Commun. Number Theory Phys., 11(4):837–877, 2017

    2017

  8. [16]

    D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim. Small gaps between primes or almost primes.Trans. Amer. Math. Soc., 361(10):5285–5330, 2009

    2009

  9. [17]

    Grepstad and G

    S. Grepstad and G. Larcher. On pair correlation and discrepancy.Arch. Math. (Basel), 109(2):143–149, 2017

    2017

  10. [18]

    G. H. Hardy and J. E. Littlewood. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes.Acta Math., 44(1):1–70, 1923

    1923

  11. [19]

    G. Harman. On the distribution ofαpmodulo one.J. London Math. Soc. (2), 27(1):9–18, 1983

    1983

  12. [20]

    Hauke and E

    M. Hauke and E. Kowalski. Rational approximation with chosen numerators. Preprint, arXiv:2502.08335 [math.NT] (2025), 2025

    2025

  13. [21]

    Hauke and A

    M. Hauke and A. Zafeiropoulos. Poissonian correlations of higher orders.J. Number Theory, 243:202–240, 2023

    2023

  14. [22]

    D. R. Heath-Brown. Pair correlation for fractional parts ofαn2.Math. Proc. Cambridge Philos. Soc., 148(3):385–407, 2010

    2010

  15. [23]

    A. J. Irving. Diophantine approximation with products of two primes.J. Lond. Math. Soc. (2), 89(2):581–602, 2014

    2014

  16. [24]

    American Mathematical Society, Providence, RI, 2004

    H.IwaniecandE.Kowalski.Analytic number theory,volume53ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004

    2004

  17. [25]

    Khintchine

    A. Khintchine. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen.Math. Ann., 92(1-2):115–125, 1924

    1924

  18. [26]

    Koukoulopoulos.The distribution of prime numbers, volume 203 ofGraduate Studies in Mathematics

    D. Koukoulopoulos.The distribution of prime numbers, volume 203 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, [2019]©2019

    2019

  19. [27]

    Kuipers and H

    L. Kuipers and H. Niederreiter.Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974

    1974

  20. [28]

    Kurlberg

    P. Kurlberg. The distribution of spacings between quadratic residues. II.Israel J. Math., 120:205–224, 2000

    2000

  21. [29]

    Kurlberg and Z

    P. Kurlberg and Z. Rudnick. The distribution of spacings between quadratic residues. Duke Math. J., 100(2):211–242, 1999

    1999

  22. [30]

    Lachmann and N

    T. Lachmann and N. Technau. On exceptional sets in the metric Poissonian pair corre- lations problem.Monatsh. Math., 189(1):137–156, 2019

    2019

  23. [31]

    Larcher and W

    G. Larcher and W. Stockinger. Pair correlation of sequences({anα})n∈N with maximal additive energy.Math. Proc. Cambridge Philos. Soc., 168(2):287–293, 2020. 28

    2020

  24. [32]

    C. Lutsko. Long-range correlations of sequences modulo 1.J. Number Theory, 234:333– 348, 2022

    2022

  25. [33]

    Lutsko, A

    C. Lutsko, A. Sourmelidis, and N. Technau. Pair Correlation of the Fractional Parts of αnθ.J. Eur. Math. Soc., online first, 2021

    2021

  26. [34]

    Lutsko and N

    C. Lutsko and N. Technau. Correlations of the Fractional Parts ofαnθ.Amer. J. Math., to appear. Preprint, arXiv:2112.11524 [math.NT] (2021)

    2021 arXiv

  27. [35]

    Lutsko and N

    C. Lutsko and N. Technau. Full Poissonian local statistics of slowly growing sequences. Compos. Math., 161(1):148–180, 2025

    2025

  28. [36]

    J. Marklof. The Berry-Tabor conjecture. InEuropean Congress of Mathematics, Vol. II (Barcelona, 2000), volume 202 ofProgr. Math., pages 421–427. Birkhäuser, Basel, 2001

    2000

  29. [37]

    J. Marklof. Pair correlation and equidistribution on manifolds.Monatsh. Math., 191(2):279–294, 2020

    2020

  30. [38]

    Marklof and A

    J. Marklof and A. Strömbergsson. Gaps between logs.Bull. Lond. Math. Soc., 45(6):1267– 1280, 2013

    2013

  31. [39]

    Matomäki, M

    K. Matomäki, M. Radziwiłł, and T. Tao. Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges.Proc. Lond. Math. Soc. (3), 118(2):284–350, 2019

    2019

  32. [40]

    Meyn and R

    S. Meyn and R. L. Tweedie.Markov chains and stochastic stability. Cambridge University Press, Cambridge, second edition, 2009. With a prologue by Peter W. Glynn

    2009

  33. [41]

    H. Mikawa. On prime twins.Tsukuba J. Math., 15(1):19–29, 1991

    1991

  34. [42]

    Ostrowski

    A. Ostrowski. Bemerkungen zur Theorie der Diophantischen Approximationen.Abh. Math. Sem. Univ. Hamburg, 1(1):77–98, 1922

    1922

  35. [43]

    Radziwiłł and A

    M. Radziwiłł and A. Shubin. Poissonian pair correlation forαnθ mod 1.Int. Math. Res. Not. IMRN, (9):7654–7679, 2024

    2024

  36. [44]

    F. A. Ramírez. A note on sequences not having metric Poissonian pair correlations.Arch. Math. (Basel), 116(6):659–665, 2021

    2021

  37. [45]

    A. M. Rockett and P. Szüsz.Continued fractions. World Scientific Publishing Co., Inc., River Edge, NJ, 1992

    1992

  38. [46]

    Z. Rudnick. What is. . .quantum chaos?Notices Amer. Math. Soc., 55(1):32–34, 2008

    2008

  39. [47]

    Rudnick and P

    Z. Rudnick and P. Sarnak. The pair correlation function of fractional parts of polynomials. Comm. Math. Phys., 194(1):61–70, 1998

    1998

  40. [48]

    Rudnick, P

    Z. Rudnick, P. Sarnak, and A. Zaharescu. The distribution of spacings between the fractional parts ofn2α.Invent. Math., 145(1):37–57, 2001

    2001

  41. [49]

    Rudnick and A

    Z. Rudnick and A. Zaharescu. The distribution of spacings between fractional parts of lacunary sequences.Forum Math., 14(5):691–712, 2002

    2002

  42. [50]

    A. Rényi. On the representation of an even number as the sum of a single prime and single almost-prime number.Izv. Akad. Nauk SSSR Ser. Mat., 12:57–78, 1948

    1948

  43. [51]

    W. M. Schmidt.Diophantine approximations and Diophantine equations, volume 1467 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991

    1991

  44. [52]

    M. M. Skriganov. Ergodic theory onSL(n), Diophantine approximations and anomalies in the lattice point problem.Invent. Math., 132(1):1–72, 1998

    1998

  45. [53]

    Steinerberger

    S. Steinerberger. Poissonian pair correlation and discrepancy.Indag. Math. (N.S.), 29(5):1167–1178, 2018

    2018

  46. [54]

    T. Tao. Continued fractions, Bohr sets, and the Littlewood conjecture. https://terrytao.wordpress.com/2012/01/03/continued-fractions-bohr-sets-and -the-littlewood-conjecture/

    2012

  47. [55]

    Technau and A

    N. Technau and A. Walker. On the triple correlations of fractional parts ofn2α.Canad. J. Math., 74(5):1347–1384, 2022

    2022

  48. [56]

    Technau and M

    N. Technau and M. Widmer. Counting lattice points and weak admissibility of a lattice and its dual.Israel J. Math., 240(1):99–117, 2020

    2020

  49. [57]

    J. L. Truelsen. Divisor problems and the pair correlation for the fractional parts ofn2α. Int. Math. Res. Not. IMRN, (16):3144–3183, 2010. 29

    2010

  50. [58]

    A. Walker. The primes are not metric Poissonian.Mathematika, 64(1):230–236, 2018

    2018

  51. [59]

    H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins.Math. Ann., 77(3):313–352, 1916

    1916

  52. [60]

    Woess.Denumerable Markov chains

    W. Woess.Denumerable Markov chains. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2009. Generating functions, boundary theory, ran- dom walks on trees. 30

    2009

  53. [2017]

    With an appendix by Jean Bourgain

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