A Note on Iterated Beatty Sequences

Afshin Zarei; Ali N. Valizadeh; Mohsen Khani

A Note on Iterated Beatty Sequences

T0 review · 2 major / 2 minor · reviewed 2026-07-15 · grok-4.5

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Pith's one-line read For every large enough irrational modulus, an elementary test decides membership in every iterate of the Beatty sequence.

desk verdict Abstract-only claim of an elementary N&S membership test for nth Beatty iterates when α>(3+√5)/2; nothing checkable yet. read the letter →

arxiv 2607.12817 v1 pith:PXEFDQBQ submitted 2026-07-14 math.NT

Mohsen Khani , Ali N. Valizadeh , Afshin Zarei This is my paper · ORCID

classification math.NT MSC 11B8311A5511B75

keywords Beattysequencesiteratedirrationalmoduluselementarymembershipcriterionfloorfunctionsgolden-ratiobound

topics The Collatz Conjecture

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

Beatty sequences are the integer sequences obtained by taking successive multiples of an irrational number and rounding down. This note claims that once the irrational modulus α is larger than the golden-ratio conjugate (3+√5)/2, one can decide, by purely elementary arithmetic conditions, whether a given natural number lies in the nth iterate of that sequence for any fixed n. The result therefore supplies an explicit membership test that works for every depth of iteration, without needing generating functions or advanced Diophantine machinery. A sympathetic reader cares because iterated Beatty sequences appear in partition problems, mechanical words, and non-standard numeration systems; an elementary criterion would make membership checks routine rather than recursive.

What carries the argument

The elementary membership condition itself: an explicit arithmetic predicate, built from floor functions and modular comparisons, that characterises the nth iterate once α exceeds the stated golden-ratio threshold.

What would settle it

Exhibit a single irrational α>(3+√5)/2, a depth n, and a natural number x for which the claimed elementary condition disagrees with direct computation of the nth Beatty iterate.

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Extended reading notes

Core claim

For every irrational α greater than (3+√5)/2 and every positive integer n there exists a necessary and sufficient elementary-number-theoretic condition that decides whether a natural number x belongs to the nth iterate of the Beatty sequence of modulus α.

Load-bearing premise

The modulus must be larger than (3+√5)/2; the paper presents this lower bound as essential for the elementary characterisation to hold.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is an abstract-only note. The one thing to know is that the authors claim an elementary necessary-and-sufficient condition deciding whether a natural number sits in the nth iterate of the Beatty sequence of modulus α, but only for irrational α larger than (3+√5)/2. They give neither the condition nor any sketch of why that threshold appears.

If the full paper actually writes down a clean, elementary criterion and proves it works for all n, that is honest progress inside a narrow corner of combinatorial number theory. Beatty iterates are a real object; a membership test that stays elementary is useful to people who already work with them. The abstract frames the result as new relative to the existing literature on single and iterated Beatty sequences, and nothing in the abstract smells circular or parameter-fitted.

The soft spots are exactly what you would expect from an abstract: we cannot see the formula, cannot check edge cases near the bound, and cannot tell whether the lower bound is sharp or merely convenient. The stress-test concern is fair—if the same elementary characterization already works for the golden ratio or other smaller irrationals, the stated range is too narrow; if it fails below the bound, the paper needs to show the failure. Neither direction is visible here. Significance is modest: this does not reorganize the field or settle a famous conjecture; it is a short technical note.

Who it is for: specialists who already care about Beatty sequences and their iterates, and who want an elementary decision procedure rather than generating-function or mechanical-word machinery. A serious referee should see the full text if the criterion is actually written down and the proofs are elementary as claimed. On the abstract alone I would not bring it to reading group and I would not cite it yet. Send it to peer review rather than desk-reject; the claim is concrete enough to deserve a look, even if the bound turns out to be non-sharp or the argument has gaps.

Referee Report

2 major / 2 minor

Summary. The manuscript asserts that for any irrational α > (3+√5)/2 ≈ 2.618 and any positive integer n, elementary number theory supplies a necessary and sufficient condition deciding whether a natural number x belongs to the nth iterate of the Beatty sequence of modulus α. The abstract states the existence of such a criterion but does not display the condition, any derivation, lemma, or explicit formula.

Significance. An elementary membership test for iterated Beatty sequences would be a useful addition to the classical theory of Beatty sequences and related combinatorial number theory. The threshold α > (3+√5)/2 is a natural scale in this literature (linked to the golden ratio and Sturmian dynamics). If the claimed characterization is correct, parameter-free, and sharp, the note would be of genuine interest. Significance cannot be fully assessed from the abstract alone, as neither the form of the condition nor the role of the lower bound is exhibited.

major comments (2)

[Abstract] The abstract asserts the existence of a necessary-and-sufficient elementary condition for membership in the nth Beatty iterate, yet neither states the condition nor sketches any argument. Without the body of the paper it is impossible to verify that the criterion is elementary, that it is necessary and sufficient, or that it is free of hidden non-elementary ingredients.
[Abstract] The lower bound α > (3+√5)/2 is presented as essential for the elementary characterization. The abstract supplies no information on sharpness, no counter-example or obstruction for smaller irrational moduli (e.g., the golden ratio), and no indication of how the bound enters the argument. This is a load-bearing hypothesis that must be justified or shown to be necessary.

minor comments (2)

[Abstract] The abstract is extremely terse; a more informative abstract that at least indicates the shape of the membership condition (e.g., involving floor functions, continued-fraction data, or Beatty duals) would aid readers and referees.
[Abstract] Notation for the nth iterate of the Beatty sequence is used without definition in the abstract; a brief parenthetical clarification would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detectable: abstract-only note claims an elementary membership criterion for Beatty iterates under an external irrationality bound, with no fitted parameters or self-referential reductions visible.

full rationale

Only the abstract is available. It asserts that for any irrational α > (3+√5)/2 and any positive integer n there exists a necessary and sufficient elementary-number-theoretic condition deciding membership of a natural number x in the nth iterate of the Beatty sequence of modulus α. The bound is a fixed external constant (related to the golden ratio), not a quantity fitted from data or defined in terms of the claimed condition. No equations, no intermediate lemmas, no self-citations, and no uniqueness theorems appear in the supplied text. Consequently none of the six enumerated circularity patterns can be exhibited by quotation and reduction. The derivation, whatever its internal details, is presented as self-contained elementary number theory against an externally given class of irrationals; residual uncertainty about the missing proof is a correctness concern, not circularity. Score 0 is therefore the only finding consistent with the hard rules.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Abstract-only review. The claim rests on the standard definition of Beatty sequences (floor(kα)), the irrationality of α, and the explicit lower bound α>(3+√5)/2. No free parameters are fitted; no new entities are postulated. The elementary-number-theory toolkit is treated as background.

assumptions (3)

domain assumption α is irrational and strictly larger than (3+√5)/2
Stated in the abstract as the hypothesis under which the characterization holds; the bound is load-bearing for the claim.
standard math Beatty sequence of modulus α is the set {floor(kα) | k∈ℕ}
Classical definition assumed throughout; iterates are successive applications of this map.
domain assumption Elementary number-theoretic arguments (floor-function identities, inequalities) suffice to decide membership
The abstract asserts that the condition is obtained by elementary methods; this is an unproved methodological claim until the full proof is seen.

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Cite this review

Pith. "Pith review of A Note on Iterated Beatty Sequences." pith.science (2026). https://pith.science/paper/PXEFDQBQ

@misc{pith2026260712817,
  author       = {Pith},
  title        = {Pith review of: A Note on Iterated Beatty Sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXEFDQBQ}},
  note         = {Machine review of arXiv:2607.12817}
}

abstract

For any irrational number $ \alpha>\frac{3+\sqrt{5}}{2}\approx2.618$ and given a positive $ n\in\mathbb{N} $, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $ x $ to be in the $n$th iterate of the Beatty sequence of modulus $\alpha$.

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Reviewed July 15, 2026 · model on record in the stance chip above.