arxiv: 2605.05784 · v1 · submitted 2026-05-07 · 🧮 math.NT

Recognition: unknown

The quotient problem for linear recurrence sequences

Parvathi S Nair, S. S. Rout

Pith reviewed 2026-05-08 05:37 UTC · model grok-4.3

classification 🧮 math.NT

keywords linear recurrence sequencesquotient problemDiophantine approximationSchmidt subspace theoremmulti-recurrencemoving hyperplanesfinitely generated ringspolynomial factors

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The pith

For unequal indices m and n in linear recurrence sequences U and V, a polynomial P exists making the scaled quotient d U(m)/V(n) a multi-recurrence while V(n)/P(n) becomes a linear recurrence.

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

The paper studies the Diophantine finiteness question for pairs (m,n) where d_{m,n} U(m)/V(n) lies in a finitely generated subring of the complex numbers and log|d| grows slower than linearly in n. It proves an explicit structural theorem that controls all such pairs through recurrence relations. When m differs from n, multiplication by a suitable polynomial P turns the adjusted quotient into a multi-recurrence sequence and forces the adjusted denominator to satisfy a linear recurrence. When the indices coincide, both the adjusted numerator and denominator satisfy ordinary linear recurrences. The argument proceeds by applying Schmidt's subspace theorem to suitably chosen moving hyperplanes and moving polynomials.

Core claim

Let U(m) and V(n) be linear recurrence sequences. For every pair (m,n) admitting a positive integer d with log|d|=o(n) such that d U(m)/V(n) belongs to a finitely generated subring of C, there exists a polynomial P with the following properties: when m ≠ n, the sequence d P(n) U(m)/V(n) is a multi-recurrence while V(n)/P(n) is a linear recurrence; when m = n, both d P(n) U(m)/V(n) and V(n)/P(n) are linear recurrences. The proof is carried out by combining Schmidt's subspace theorem with the techniques of moving hyperplanes, moving polynomials, and moving points.

What carries the argument

The polynomial P(n) that factors the denominator V(n) so that the scaled ratio d U(m)/V(n) reduces to a multi-recurrence sequence, obtained by applying Schmidt's subspace theorem to moving hyperplanes indexed by the variable n.

If this is right

Every admissible pair (m,n) must satisfy a system of linear recurrence relations after a polynomial adjustment of bounded degree.
The denominator sequence V(n) is forced to be divisible by a polynomial factor whose coefficients are independent of the numerator index m.
When m equals n the problem collapses to the ordinary theory of linear recurrence sequences taking values in a fixed ring.
The growth restriction log|d|=o(n) is compatible with the order of the resulting multi-recurrence.
Finiteness of the set of such pairs (m,n) follows once the multi-recurrence and linear recurrence conditions are solved explicitly.

Where Pith is reading between the lines

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

The same moving-hyperplane construction may apply verbatim to ratios of sequences satisfying higher-order recurrences or to sequences taking values in number fields of fixed degree.
The explicit polynomial P supplies a reduction that could turn the original finiteness question into a finite check against the integer points of certain recurrence varieties.
The result supplies a uniform way to handle variable-index Diophantine problems for any family of sequences to which Schmidt's theorem applies.

Load-bearing premise

That Schmidt's subspace theorem together with moving-hyperplane methods apply directly to the given linear recurrence sequences without extra conditions on the characteristic roots or their linear independence.

What would settle it

Two explicit linear recurrence sequences U and V together with indices m ≠ n and a d satisfying log|d|=o(n) such that d U(m)/V(n) lies in a finitely generated subring but no polynomial P exists making d P(n) U(m)/V(n) a multi-recurrence.

read the original abstract

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study the finiteness problem for the set $(m, n)\in \mathbb{N}^2$ such that there exist non-zero positive integers $d_{m, n}$ satisfying $\log |d_{m, n}|=o(n)$, and $d_{m, n}U(m)/V(n)$ is an element from a finitely generated subring of $\C$. In particular, we prove that for $m\neq n $, there exists a polynomial $P$ such that $d_{m, n}P(n)U(m)/V(n)$ is a multi-recurrence and $V(n)/P(n)$ is a linear recurrence and for $m=n$ both $d_{m, n}P(n)U(m)/V(n)$ and $V(n)/P(n)$ are linear recurrences. To prove our results, we employ Schmidt's subspace theorem, and the concept of moving hyperplanes, moving polynomials, and moving points.

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.

Desk Editor's Note private letter to a colleague

The paper gives a structural result turning adjusted quotients of linear recurrences into recurrences via a polynomial factor under an o(n) growth condition on d, but the Schmidt subspace theorem application may need unstated independence conditions on the roots.

read the letter

The central claim is that for linear recurrences U and V, when log|d| = o(n) and d U(m)/V(n) lies in a finitely generated subring, there is a polynomial P such that d P(n) U(m)/V(n) becomes a multi-recurrence (or linear when m equals n) while V(n)/P(n) is linear. This is the new piece relative to earlier finiteness statements on integer quotients. The approach uses Schmidt's subspace theorem together with moving hyperplanes, polynomials, and points to produce the adjustment. That formulation is a concrete step forward for anyone studying integrality questions in recurrence sequences, since it supplies an explicit conversion rather than just bounding the solutions. The abstract states the result cleanly and names the tools without obvious circularity or invented entities. The o(n) condition is handled by the moving techniques, which is a reasonable way to absorb the slow growth. The stress-test worry about characteristic root independence is worth checking in the full text. Standard Schmidt applications in this area often require multiplicative independence of the roots or linear independence of their logs to control the approximations; if the paper does not derive or assume that for arbitrary U and V, the reduction to the recurrence property could fail in degenerate cases like repeated roots. The abstract gives no indication of extra conditions, so that is the main soft spot. The rest of the argument looks proportionate to the claim. This paper is for number theorists working on Diophantine problems with linear recurrences and S-unit type equations. A reader already familiar with Schmidt's theorem and moving hyperplane methods will see the value in the polynomial adjustment and can judge whether the root conditions are handled. It is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee to verify the details and any missing independence hypotheses. I would send it to review rather than desk reject.

Referee Report

1 major / 1 minor

Summary. The manuscript studies the finiteness problem for pairs (m, n) of natural numbers such that there exists a positive integer d_{m,n} with log |d_{m,n}| = o(n) and d_{m,n} U(m)/V(n) lies in a finitely generated subring of C, where U(m) and V(n) are linear recurrence sequences. The main theorem asserts that there exists a polynomial P such that, when m ≠ n, the adjusted term d_{m,n} P(n) U(m)/V(n) is a multi-recurrence sequence and V(n)/P(n) is a linear recurrence sequence, while for m = n both adjusted quantities are linear recurrences. The proof is based on Schmidt's subspace theorem together with moving hyperplanes, moving polynomials, and moving points.

Significance. If the result holds in the claimed generality for arbitrary linear recurrences, it would extend classical finiteness theorems on integer-valued quotients of recurrence sequences to a broader Diophantine setting involving controlled denominators and finitely generated rings. The combination of Schmidt's theorem with moving techniques is a methodological strength that could enable more flexible reductions than standard S-unit or logarithmic-form approaches. The paper supplies no machine-checked proofs or explicit parameter-free derivations, but the existence statement is falsifiable in principle via explicit recurrence examples.

major comments (1)

[Abstract] Abstract and the statement of the main result: the existence of P is asserted for arbitrary linear recurrence sequences U and V without any stated hypotheses on the multiplicative independence of their characteristic roots or the linear independence over Q of the logarithms of those roots. Standard applications of Schmidt's subspace theorem to ratios in finitely generated rings (via moving hyperplanes or points) require such conditions to control the exceptional subspaces; the manuscript does not indicate whether these are assumed, derived from the o(n) growth of log |d|, or rendered unnecessary by the moving techniques. This is load-bearing for the central claim, as counterexamples with repeated roots or commensurable logs could invalidate the reduction to multi-recurrence or linear-recurrence form.

minor comments (1)

The abstract packs the main theorem, the auxiliary polynomial P, and the proof methods into a single dense sentence; separating the precise statement of the result from the tools employed would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about the hypotheses underlying the main theorem. We address the comment in detail below.

read point-by-point responses

Referee: [Abstract] Abstract and the statement of the main result: the existence of P is asserted for arbitrary linear recurrence sequences U and V without any stated hypotheses on the multiplicative independence of their characteristic roots or the linear independence over Q of the logarithms of those roots. Standard applications of Schmidt's subspace theorem to ratios in finitely generated rings (via moving hyperplanes or points) require such conditions to control the exceptional subspaces; the manuscript does not indicate whether these are assumed, derived from the o(n) growth of log |d|, or rendered unnecessary by the moving techniques. This is load-bearing for the central claim, as counterexamples with repeated roots or commensurable logs could invalidate the reduction to multi-recurrence or linear-recurrence form.

Authors: We agree that the manuscript does not explicitly address the role of independence conditions on the characteristic roots. In the proof, Schmidt's subspace theorem is applied in conjunction with moving hyperplanes, moving polynomials, and moving points. These moving techniques are designed to absorb any exceptional subspaces that arise from multiplicative relations among the roots or linear dependence over Q of their logarithms (including cases of repeated roots). Such exceptional loci are incorporated into the choice of the polynomial P, which adjusts the sequences so that the stated recurrence properties hold. The hypothesis log |d_{m,n}| = o(n) supplies the necessary height control to ensure that the moving framework remains effective without a priori independence assumptions. We will add a clarifying paragraph in the introduction explaining this aspect of the argument. We do not claim the result holds without the moving techniques, but we maintain that the combination renders the standard independence hypotheses unnecessary for the stated conclusion. revision: partial

Circularity Check

0 steps flagged

No circularity; central claim rests on external Schmidt subspace theorem

full rationale

The derivation applies Schmidt's subspace theorem together with moving hyperplanes/points/polynomials to establish existence of P such that the stated recurrence properties hold. These tools are independent of the paper's own results. No self-citations appear load-bearing, no parameters are fitted to data and then relabeled as predictions, and the statements about d_{m,n}P(n)U(m)/V(n) and V(n)/P(n) are not defined in terms of each other. The log|d|=o(n) and finitely-generated-subring conditions are explicit inputs, not smuggled via self-reference. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the definition of linear recurrence sequences and the applicability of Schmidt's subspace theorem from Diophantine approximation.

axioms (1)

standard math Schmidt's subspace theorem applies to the linear forms arising from the recurrence quotients
Invoked to bound approximations and control dependencies in the proof.

pith-pipeline@v0.9.0 · 8701 in / 1210 out tokens · 69713 ms · 2026-05-08T05:37:23.770811+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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