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arxiv: 2605.01682 · v1 · submitted 2026-05-03 · 🧮 math.NT

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Numbers in a Beatty sequence which are orders only of cyclic, abelian or nilpotent groups

Kang Shengyu

Pith reviewed 2026-05-09 17:06 UTC · model grok-4.3

classification 🧮 math.NT
keywords Beatty sequencescyclic numbersabelian numbersnilpotent numbersasymptotic formulascounting functionsnumber theorygroup orders
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The pith

Cyclic, abelian and nilpotent numbers in Beatty sequences have counting functions asymptotic to one over alpha times the usual counts.

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

The paper studies the distribution of integers that can serve as the order of a group only if that group is cyclic, abelian or nilpotent, but now restricted to lie inside a Beatty sequence. Such sequences are formed by taking the integer parts of an arithmetic progression with irrational slope alpha greater than one and fixed offset beta. Adapting the analytic methods previously used by Pollack and Just for the unrestricted counting functions, the authors show that the restricted counts inside the sequence obey the same main-term asymptotics multiplied exactly by the factor 1/alpha. A sympathetic reader cares because Beatty sequences with irrational alpha partition the positive integers, so the result indicates that these special numbers are asymptotically equidistributed across the parts of any such partition.

Core claim

Let C(x), A(x) and N(x) denote the counting functions of cyclic, abelian and nilpotent numbers up to x. For the Beatty sequence B_alpha,beta with alpha irrational of finite type, the corresponding restricted counting functions #C^*(x), #A^*(x) and #N^*(x) inside the sequence satisfy the same asymptotic formulas as C(x), A(x) and N(x) multiplied by the constant factor 1/alpha.

What carries the argument

The adaptation of the analytic methods of Pollack and Just to Beatty sequences generated by an irrational alpha of finite type.

If this is right

  • The natural density of cyclic numbers inside the Beatty sequence is exactly 1/alpha times their overall density.
  • The same scaling by 1/alpha holds for the densities of abelian numbers and nilpotent numbers inside the sequence.
  • The result applies uniformly for any fixed real offset beta.
  • The asymptotic formulas remain valid for every irrational alpha of finite type.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the finite-type restriction on alpha can be removed, the same scaling would hold for almost every irrational alpha.
  • The uniformity across Beatty sequences suggests these group-order numbers are equidistributed with respect to other irrational rotations or mechanical words.
  • The same transfer of analytic methods could be attempted for counting these numbers inside other lacunary sequences such as Piatetski-Shapiro sequences.

Load-bearing premise

The analytic methods of Pollack and Just transfer directly to the Beatty-sequence setting without new error terms or convergence issues caused by the irrational rotation.

What would settle it

For the golden-ratio Beatty sequence, compute the exact count of cyclic numbers up to x = 10^12 and verify whether it equals (1/alpha) times the known main term for C(x) plus an error smaller than the secondary term.

read the original abstract

Let \(C(x)\), \(A(x)\), and \(N(x)\) denote the counting functions of cyclic, abelian, and nilpotent numbers not exceeding \(x\), respectively. Their asymptotic formulas have been established in recent work by Pollack and Just. In this paper, by adapting the methods of Pollack and Just, we study the distribution of these numbers in Beatty sequences \(\mathcal{B}_{\alpha,\beta} = ([\alpha n + \beta])_{n=1}^{\infty}\), where \(\alpha > 1\) is an irrational number of finite type and \(\beta\) is a fixed real number. We prove that the counting functions \(\#C^*(x)\), \(\#A^*(x)\), and \(\#N^*(x)\) for cyclic, abelian, and nilpotent numbers in Beatty sequences satisfy asymptotic formulas that differ from those of Pollack and Just only by a factor \(1/\alpha\).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript adapts the analytic methods of Pollack and Just to Beatty sequences B_{α,β} with α > 1 irrational of finite type. It claims that the counting functions #C^*(x), #A^*(x), and #N^*(x) for cyclic, abelian, and nilpotent numbers in the sequence satisfy the same asymptotic formulas as C(x), A(x), N(x) except for an extra factor of 1/α, with identical relative error terms.

Significance. If the error-term control carries over, the result shows that the distribution of these special group-order numbers is uniform in Beatty sequences under the finite-type hypothesis on α. This provides a concrete extension of the Pollack-Just theorems to a natural class of thin sets and demonstrates the robustness of the underlying tauberian or contour-integration arguments when the summation is restricted to an irrational rotation.

major comments (2)
  1. [Proof of the main asymptotic statements (likely the argument following the statement of the three main theorems)] The central claim requires that the restriction to the Beatty sequence introduces no new error larger than o(x/α). The manuscript asserts that the methods of Pollack and Just adapt verbatim, but the finite-type condition on α controls only certain Diophantine approximations; it is not immediate that every contribution from the irrational rotation (exponential sums, discrepancy in the orbit {αn + β}) remains absorbable in the contour integration or zero-density estimates used for the original asymptotics. A detailed comparison of the error bounds before and after the restriction is needed.
  2. [Introduction and statement of results] The abstract and introduction state that the counting functions differ from those of Pollack and Just 'only by a factor 1/α'. This requires that the main term is exactly (1/α) times the unrestricted main term and that the relative error is unchanged. Without an explicit verification that the tauberian remainder or the contribution from the poles/zeros is scaled by precisely 1/α while all other terms stay o(main term), the 'only by a factor 1/α' assertion is not fully substantiated.
minor comments (2)
  1. [Abstract] Notation: the symbols C^*(x), A^*(x), N^*(x) are introduced without an explicit definition in the abstract; a sentence clarifying that these count the relevant numbers lying in the Beatty sequence up to x would improve readability.
  2. [Introduction] The paper cites Pollack and Just for the unrestricted asymptotics; a brief reminder of the precise error terms obtained in those works (e.g., the exponent in the o-term) would help the reader assess whether the same exponent is claimed here.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and valuable suggestions. The comments highlight areas where additional clarification on error terms would improve the manuscript. We address each major comment below and will revise accordingly.

read point-by-point responses

  1. Referee: [Proof of the main asymptotic statements (likely the argument following the statement of the three main theorems)] The central claim requires that the restriction to the Beatty sequence introduces no new error larger than o(x/α). The manuscript asserts that the methods of Pollack and Just adapt verbatim, but the finite-type condition on α controls only certain Diophantine approximations; it is not immediate that every contribution from the irrational rotation (exponential sums, discrepancy in the orbit {αn + β}) remains absorbable in the contour integration or zero-density estimates used for the original asymptotics. A detailed comparison of the error bounds before and after the restriction is needed.

    Authors: We agree that an explicit comparison of error bounds strengthens the argument. The finite-type hypothesis on α ensures that the discrepancy of the sequence {αn + β} is O(1/N^δ) for some δ>0 in the relevant ranges, which is sufficient to bound the exponential sums arising from the indicator of the Beatty sequence. These bounds are absorbed into the zero-density estimates and contour shifts already present in Pollack-Just, without enlarging the o(x/α) term. In the revision we will insert a dedicated paragraph (or short subsection) immediately after the statements of the main theorems that compares the unrestricted error terms with the restricted ones term-by-term, citing the precise discrepancy bounds from the finite-type condition. revision: yes

  2. Referee: [Introduction and statement of results] The abstract and introduction state that the counting functions differ from those of Pollack and Just 'only by a factor 1/α'. This requires that the main term is exactly (1/α) times the unrestricted main term and that the relative error is unchanged. Without an explicit verification that the tauberian remainder or the contribution from the poles/zeros is scaled by precisely 1/α while all other terms stay o(main term), the 'only by a factor 1/α' assertion is not fully substantiated.

    Authors: The main-term scaling follows at once from the fact that the Beatty sequence has asymptotic density 1/α; the tauberian theorems and residue calculations in Pollack-Just are applied to a Dirichlet series summed over the subsequence, which multiplies the principal terms by exactly 1/α. The relative error remains unchanged because the additional approximation error from the irrational rotation is smaller than the existing o(main term) by the finite-type hypothesis. To make this fully explicit we will add a short remark after the statement of the main theorems that tracks the scaling of each component (main term, tauberian remainder, pole/zero contributions) under the restriction. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation adapts external analytic methods of Pollack and Just

full rationale

The paper states that it adapts the methods of Pollack and Just to obtain asymptotics for the Beatty-sequence counting functions that differ from the original ones only by the factor 1/α. This factor follows directly from the known density of the Beatty sequence B_{α,β}. No quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and the load-bearing asymptotic formulas are imported from the external cited work rather than derived internally by reduction to the paper's own inputs. The finite-type hypothesis on α is invoked to control Diophantine approximations in the adaptation, but this does not create a self-referential loop or force the result by construction. The argument is therefore self-contained against the external benchmark of Pollack and Just.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the analytic machinery of Pollack and Just together with standard properties of Beatty sequences for irrationals of finite type. No new free parameters or invented entities appear.

axioms (2)
  • domain assumption Standard analytic estimates for the distribution of cyclic/abelian/nilpotent numbers remain valid after restriction to a Beatty sequence.
    Invoked when the authors state that the same methods adapt directly.
  • standard math α irrational of finite type ensures the Beatty sequence has the necessary uniform distribution properties.
    Standard hypothesis in the theory of Beatty sequences.

pith-pipeline@v0.9.0 · 8301 in / 1144 out tokens · 71483 ms · 2026-05-09T17:06:14.772108+00:00 · methodology

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Reference graph

Works this paper leans on

27 extracted references · 3 canonical work pages

  1. [1]

    and Shparlinski, Igor E

    Banks, William D. and Shparlinski, Igor E. , booktitle=. Character Sums with

  2. [2]

    , title =

    Wannes, W. , title =. Indian Journal of Pure and Applied Mathematics , year =

  3. [3]

    and Yeager, A

    Banks, William D. and Yeager, A. M. , title =. Colloquium Mathematicum , year =

  4. [4]

    Some asymptotic formulas in number theory , journal =

    Erd. Some asymptotic formulas in number theory , journal =. 1948 , volume =

    1948

  5. [5]

    Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis , author=

  6. [6]

    The Hardy--Littlewood Method , author=

  7. [7]

    The k -fold divisor function over

    Qi, Jinyun and Guo, Victor Zhenyu , journal=. The k -fold divisor function over. 2025 , doi=

    2025

  8. [8]

    Bulletin of the American Mathematical Society , volume=

    Some extremal problems in Fourier analysis , author=. Bulletin of the American Mathematical Society , volume=

  9. [9]

    Begunts, A. V. , journal=. On an asymptotic formula for. 2001 , language=

    2001

  10. [10]

    1974 , address =

    Halberstam, Heini and Richert, Hans-Egon , title =. 1974 , address =

    1974

  11. [11]

    , title =

    Akbal, Y. , title =. Colloquium Mathematicum , volume =

  12. [12]

    Proceedings of the American Mathematical Society , year =

    Pollack, Paul , title =. Proceedings of the American Mathematical Society , year =. doi:10.1090/proc/15658 , url =

    work page doi:10.1090/proc/15658

  13. [13]

    Just, Matthew , title =. J. Th\'. 2023 , volume =

    2023

  14. [14]

    , year =

    Guloglu, Ahmet and Nevans, C. , year =. Sums of multiplicative functions over a. Bulletin of the Australian Mathematical Society , doi =

  15. [15]

    Commentarii Mathematici Helvetici , year =

    Szele, Tibor , title =. Commentarii Mathematici Helvetici , year =. doi:10.1007/BF02568132 , language =

    work page doi:10.1007/bf02568132

  16. [16]

    Archiv der Mathematik , year =

    Pazderski, Gerhard , title =. Archiv der Mathematik , year =. doi:10.1007/BF01240770 , language =

    work page doi:10.1007/bf01240770

  17. [17]

    Inventiones mathematicae , author =

    Exponential sums with multiplicative coefficients , volume =. Inventiones mathematicae , author =. 1977 , pages =

    1977

  18. [18]

    Pollack, ``Numbers which are orders only of cyclic groups,'' Proceedings of the American Mathematical Society , vol

    P. Pollack, ``Numbers which are orders only of cyclic groups,'' Proceedings of the American Mathematical Society , vol. 150, no. 2, pp. 515--524, 2021

    2021

  19. [19]

    Just, ``Numbers which are only orders of abelian or nilpotent groups,'' J

    M. Just, ``Numbers which are only orders of abelian or nilpotent groups,'' J. Th\' e or. Nombres Bordeaux , vol. 35, no. 2, pp. 453--466, 2023

    2023

  20. [20]

    U ber die endlichen O rdnungszahlen, zu denen nur eine G ruppe geh\

    T. Szele, ``\" U ber die endlichen O rdnungszahlen, zu denen nur eine G ruppe geh\" o rt,'' Commentarii Mathematici Helvetici , vol. 20, pp. 265--267, 1947

    1947

  21. [21]

    Pazderski, ``Die O rdnungen, zu denen nur G ruppen mit gegebener E igenschaft geh\" o ren,'' Archiv der Mathematik , vol

    G. Pazderski, ``Die O rdnungen, zu denen nur G ruppen mit gegebener E igenschaft geh\" o ren,'' Archiv der Mathematik , vol. 10, pp. 331--343, 1959

    1959

  22. [22]

    Qi and V

    J. Qi and V. Z. Guo, ``The k -fold divisor function over B eatty sequences,'' The Ramanujan Journal , vol. 66, 2025

    2025

  23. [23]

    Wannes, ``Biases amongst products of two B eatty primes in arithmetic progressions,'' Indian Journal of Pure and Applied Mathematics , 2024

    W. Wannes, ``Biases amongst products of two B eatty primes in arithmetic progressions,'' Indian Journal of Pure and Applied Mathematics , 2024

    2024

  24. [24]

    W. D. Banks and I. E. Shparlinski, ``Character sums with B eatty sequences on B urgess-type intervals,'' in Analytic Number Theory: Essays in Honour of Klaus Roth , (Cambridge), pp. 15--21, Cambridge University Press, 2009

    2009

  25. [25]

    H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis , vol. 84 of CBMS Regional Conference Series in Mathematics . Providence: American Mathematical Society, 1994

    1994

  26. [26]

    Guloglu and C

    A. Guloglu and C. Nevans, ``Sums of multiplicative functions over a B eatty sequence,'' Bulletin of the Australian Mathematical Society , vol. 78, 2008

    2008

  27. [27]

    J. D. Vaaler, ``Some extremal problems in fourier analysis,'' Bulletin of the American Mathematical Society , vol. 12, pp. 183--216, 1985

    1985