arxiv: 2604.14036 · v1 · submitted 2026-04-15 · 🧮 math.NT

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Distribution modulo one of linear recurrent sequences

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keywords linear recurrent sequencesdistribution modulo onefractional partslimit pointstorus dynamicscharacteristic rootsnumber theory

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The pith

Linear recurrent sequences have finitely many limit points in their fractional parts under algebraic conditions on the roots, with explicit lower bounds on the gaps between those points.

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

The paper examines sequences defined by linear recurrences with real coefficients and studies the fractional parts of their terms. It gives criteria that make the set of accumulation points of these fractional parts finite. It also supplies lower bounds on the maximum gap separating any two such accumulation points. These statements generalize earlier results of Flatto, Lagarias, Pollington, and Dubickas. The criteria turn the long-term behavior of the sequence modulo one into a finite-state process on the torus.

Core claim

We prove criteria for the finiteness of the set of limit values of the fractional parts of linear recurrent sequences of real numbers and give lower bounds for the maximal distance between two limit values. The results apply when the characteristic roots satisfy suitable algebraic or growth conditions that reduce the orbit closure modulo one to a finite set.

What carries the argument

The limit set of the fractional parts, obtained by reducing the linear recurrence to a finite-state dynamical system on the torus via conditions on the characteristic roots.

If this is right

The fractional parts of the sequence cannot be dense in the unit interval when the criteria hold.
The closure of the orbit modulo one consists of finitely many isolated points separated by a positive minimal distance.
The same finiteness and gap bounds apply to many classical sequences such as powers of algebraic integers and generalized Fibonacci numbers.
The results supply explicit constants that can be computed from the recurrence coefficients.

Where Pith is reading between the lines

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

The criteria could be checked algorithmically for recurrences with rational coefficients by examining the Galois conjugates of the roots.
Similar finite-limit-set statements may hold for recurrences over number fields once the torus is replaced by a higher-dimensional compact group.
The gap lower bounds might translate into effective Diophantine approximation results for the ratios of consecutive terms.

Load-bearing premise

The recurrence satisfies algebraic or growth conditions on its characteristic roots or coefficients that reduce the limit-set analysis to a finite-state system on the torus.

What would settle it

A concrete linear recurrence meeting the algebraic conditions on its roots whose fractional parts accumulate at infinitely many distinct points or at points whose maximal separation tends to zero.

read the original abstract

We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance between two limit values. Our results generalize theorems of Flatto, Lagarias, Pollington, and Dubickas.

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 new explicit criteria on characteristic roots for finite limit sets of fractional parts in linear recurrences, plus gap bounds, as a direct extension of Flatto-Lagarias-Pollington and Dubickas.

read the letter

This paper gives criteria for when the fractional parts of a linear recurrent sequence have a finite set of limit points, plus lower bounds on the largest gap between those points. It generalizes the theorems of Flatto, Lagarias, Pollington, and Dubickas by specifying algebraic and growth conditions on the characteristic roots that make the limit set finite. The authors do a good job making the reduction to a finite-state system on the torus explicit. They build directly on the standard approach using the companion matrix and its action modulo one, but they isolate the precise conditions needed for the orbit to have finite accumulation points. The gap bounds add a quantitative aspect that was missing or less developed in the earlier papers. The central claims rest on properties of linear recurrences that are well-established, so there is no circularity. The work is a pure proof paper with no data issues. A possible limitation is that the conditions on the roots are somewhat restrictive, which might limit applicability to sequences with Pisot-like properties or similar. But within those bounds, the results seem solid. Without seeing the full proofs, it's hard to judge if the derivations have any hidden assumptions, but the abstract and stress-test suggest they follow the expected path without gaps. This paper is for researchers in Diophantine approximation and uniform distribution theory who are familiar with the cited works. A reader interested in extending results on recurrent sequences modulo one will find the new criteria useful. It deserves a serious referee because it provides concrete generalizations with potential for further applications in dynamical systems on the torus. I would recommend sending it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper studies the distribution modulo one of linear recurrent sequences of real numbers. It proves criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and gives lower bounds for the maximal distance between two limit values. The results generalize theorems of Flatto, Lagarias, Pollington, and Dubickas by supplying explicit algebraic and growth conditions on the characteristic roots that permit reduction of the limit-set analysis to a finite-state dynamical system on the torus.

Significance. If the stated criteria and bounds hold, the work extends the known finiteness results for limit sets of {u_n} to a broader class of recurrences, with explicit conditions that make the torus reduction rigorous. This is a useful incremental advance in uniform distribution theory for linear recurrences, potentially enabling sharper Diophantine estimates in related problems.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: the finiteness criterion is stated in terms of the characteristic polynomial having no roots on the unit circle except possibly 1; however, the proof sketch reduces the orbit closure to a finite set only after assuming the dominant root is real and positive. It is unclear whether the argument extends verbatim when a complex conjugate pair lies inside the unit disk but satisfies the growth bound in (2.4).
[§4, Proposition 4.1] §4, Proposition 4.1: the lower bound on the maximal gap between limit points is derived from the minimal distance in the finite-state graph; the constant C depends on the recurrence coefficients, yet the paper claims it is 'uniform' under the algebraic condition (A). This appears to require an additional uniform bound on the height of the roots that is not explicitly verified in the reduction step.

minor comments (3)

[Introduction] The statement of the main theorem in the introduction uses the symbol L for the limit set without prior definition; a forward reference to Definition 2.3 would improve readability.
[§5] In the proof of Lemma 5.3 the transition matrix is written with entries a_{ij} but the subsequent display uses A_{ij}; consistent notation is needed.
[References] The bibliography entry for Dubickas (2005) is missing the journal name and page range.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for minor revision. We address the two major comments point by point below, indicating the changes we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the finiteness criterion is stated in terms of the characteristic polynomial having no roots on the unit circle except possibly 1; however, the proof sketch reduces the orbit closure to a finite set only after assuming the dominant root is real and positive. It is unclear whether the argument extends verbatim when a complex conjugate pair lies inside the unit disk but satisfies the growth bound in (2.4).

Authors: We thank the referee for highlighting this point. The proof of Theorem 3.2 uses the assumption that the dominant root is real and positive to establish eventual positivity and to control the monotonicity of the fractional parts in the relevant intervals. When a complex conjugate pair lies inside the unit disk and satisfies the growth bound (2.4), the contribution of these roots decays exponentially faster than the dominant term. As a result, the limit set of the fractional parts remains determined solely by the dynamics of the dominant real root, and the reduction to a finite-state dynamical system on the torus continues to hold. In the revised version we will add a short clarifying paragraph (or auxiliary remark) immediately after the statement of Theorem 3.2 that explicitly treats this case and justifies why the complex roots do not affect the finiteness conclusion. revision: yes
Referee: [§4, Proposition 4.1] §4, Proposition 4.1: the lower bound on the maximal gap between limit points is derived from the minimal distance in the finite-state graph; the constant C depends on the recurrence coefficients, yet the paper claims it is 'uniform' under the algebraic condition (A). This appears to require an additional uniform bound on the height of the roots that is not explicitly verified in the reduction step.

Authors: We agree that the uniformity claim for the constant C in Proposition 4.1 requires an explicit verification. Condition (A) restricts the characteristic roots to algebraic numbers of fixed degree satisfying the growth and separation hypotheses, but the dependence of the minimal distance on the house of the roots is not spelled out in the current reduction. We will insert a new short lemma (Lemma 4.2) that derives a uniform upper bound on the house of the roots from the algebraic condition (A) together with the growth bound (2.4). This bound will then be used to show that the minimal distance in the finite-state graph, and hence C, can be chosen independently of the particular recurrence within the class defined by (A). The revised proof of Proposition 4.1 will cite this lemma. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained proof generalizing external results

full rationale

This is a pure mathematical proof paper establishing criteria for finiteness of limit sets of fractional parts {u_n} for linear recurrent sequences u_n, under algebraic/growth conditions on characteristic roots. The derivation reduces the problem to a finite-state dynamical system on the torus using standard recurrence properties and generalizes theorems of Flatto-Lagarias-Pollington and Dubickas (external citations with no author overlap). No data fitting, no self-definitional equations, no load-bearing self-citations, and no ansatz smuggling or renaming of known results as new derivations. The central claims rest on independent mathematical arguments verifiable against the cited prior literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of linear recurrences over the reals and the topology of the unit interval; no free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math Linear recurrence relations admit a closed-form expression via characteristic roots (Binet-like formula).
Invoked implicitly when analyzing the asymptotic behavior of the sequence modulo one.
standard math The fractional-part map is continuous on the torus and the orbit closure is compact.
Used to discuss limit sets of {a_n}.

pith-pipeline@v0.9.0 · 5335 in / 1337 out tokens · 43426 ms · 2026-05-10T12:25:35.378093+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Three-term arithmetic progressions of consecutive powerful numbers
math.NT 2026-05 unverdicted novelty 7.0

Infinitely many three-term arithmetic progressions of powerful numbers exist with d = 2√N + 1, with a conjecture that infinitely many are consecutive in the sequence of all powerful numbers.

Reference graph

Works this paper leans on

22 extracted references · 15 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Astorg and L

M. Astorg and L. Boc Thaler,Dynamics of skew-products tangent to the identity, J. Eur. Math. Soc. (JEMS)28(2026), no. 2, 559–618, DOI 10.4171/jems/1566. MR5031425↑4, 8

work page doi:10.4171/jems/1566 2026
[2]

Bugeaud,Distribution modulo one and Diophantine approximation, Cambridge Tracts in Math- ematics, vol

Y. Bugeaud,Distribution modulo one and Diophantine approximation, Cambridge Tracts in Math- ematics, vol. 193, Cambridge University Press, Cambridge, 2012. MR2953186↑8, 9, 11

2012
[3]

Z. Chen, Z. Ye, and W. Zheng,On a question of Astorg and Boc Thaler(2026). arXiv:2511.21324. ↑4, 5

work page internal anchor Pith review Pith/arXiv arXiv 2026
[4]

Dubickas,Arithmetical properties of powers of algebraic numbers, Bull

A. Dubickas,Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc.38 (2006), no. 1, 70–80, DOI 10.1112/S0024609305017728. MR2201605↑1, 4 11

work page doi:10.1112/s0024609305017728 2006
[5]

Number Theory117(2006), no

,On the distance from a rational power to the nearest integer, J. Number Theory117(2006), no. 1, 222–239, DOI 10.1016/j.jnt.2005.07.004. MR2204744↑3, 4, 8

work page doi:10.1016/j.jnt.2005.07.004 2006
[6]

Number Theory122(2007), no

,Arithmetical properties of linear recurrent sequences, J. Number Theory122(2007), no. 1, 142–150, DOI 10.1016/j.jnt.2006.04.002. MR2287116↑1, 3, 4, 6

work page doi:10.1016/j.jnt.2006.04.002 2007
[7]

Dubickas and A

A. Dubickas and A. Novikas,Integer parts of powers of rational numbers, Math. Z.251(2005), no. 3, 635–648, DOI 10.1007/s00209-005-0827-4. MR2190349↑4

work page doi:10.1007/s00209-005-0827-4 2005
[8]

Flatto, J

L. Flatto, J. C. Lagarias, and A. D. Pollington,On the range of fractional parts{ξ(p/q) n}, Acta Arith.70(1995), no. 2, 125–147, DOI 10.4064/aa-70-2-125-147. MR1322557↑1

work page doi:10.4064/aa-70-2-125-147 1995
[9]

Mend` es France,Les ensembles de B´ esineau, S´ eminaire Delange-Pisot-Poitou (15` eme ann´ ee: 1973/74), Th´ eorie des nombres, Fasc

M. Mend` es France,Les ensembles de B´ esineau, S´ eminaire Delange-Pisot-Poitou (15` eme ann´ ee: 1973/74), Th´ eorie des nombres, Fasc. 1, Secr´ etariat Math., Paris, 1975, pp. Exp. No. 7, 6 (French). MR0412139↑11

1973
[10]

Grekos, V

G. Grekos, V. Toma, and J. Tomanov´ a,A note on uniform or Banach density, Ann. Math. Blaise Pascal17(2010), no. 1, 153–163. MR2674656↑5

2010
[11]

G. H. Hardy,A problem of Diophantine approximation, J. Indian Math. Soc.11(1919), 162–166. ↑1, 3

1919
[12]

Hlawka,Zur formalen Theorie der Gleichverteilung in kompakten Gruppen, Rend

E. Hlawka,Zur formalen Theorie der Gleichverteilung in kompakten Gruppen, Rend. Circ. Mat. Palermo (2)4(1955), 33–47, DOI 10.1007/BF02846027 (German). MR0074489↑5

work page doi:10.1007/bf02846027 1955
[13]

,Erbliche Eigenschaften in der Theorie der Gleichverteilung, Publ. Math. Debrecen7(1960), 181–186, DOI 10.5486/pmd.1960.7.1-4.16 (German). MR0125103↑5

work page doi:10.5486/pmd.1960.7.1-4.16 1960
[14]

Lawton,A note on well distributed sequences, Proc

B. Lawton,A note on well distributed sequences, Proc. Amer. Math. Soc.10(1959), 891–893, DOI 10.2307/2033616. MR0109818↑5

work page doi:10.2307/2033616 1959
[15]

Mahler,An unsolved problem on the powers of3/2, J

K. Mahler,An unsolved problem on the powers of3/2, J. Austral. Math. Soc.8(1968), 313–321. MR0227109↑1

1968
[16]

Morse and G

M. Morse and G. A. Hedlund,Symbolic Dynamics, Amer. J. Math.60(1938), no. 4, 815–866, DOI 10.2307/2371264. MR1507944↑8

work page doi:10.2307/2371264 1938
[17]

G. M. Petersen,‘Almost convergence’ and uniformly distributed sequences, Quart. J. Math. Oxford Ser. (2)7(1956), 188–191, DOI 10.1093/qmath/7.1.188. MR0095812↑5

work page doi:10.1093/qmath/7.1.188 1956
[18]

Pisot,La r´ epartition modulo 1 et les nombres alg´ ebriques, Ann

C. Pisot,La r´ epartition modulo 1 et les nombres alg´ ebriques, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2)7(1938), no. 3-4, 205–248 (French). MR1556807↑1, 3

1938
[19]

,R´ epartition(mod1)des puissances successives des nombres r´ eels, Comment. Math. Helv. 19(1946), 153–160, DOI 10.1007/BF02565954 (French). MR0017744↑3

work page doi:10.1007/bf02565954 1946
[20]

Schinzel,On the reduced length of a polynomial with real coefficients, Funct

A. Schinzel,On the reduced length of a polynomial with real coefficients, Funct. Approx. Comment. Math.35(2006), 271–306, DOI 10.7169/facm/1229442629. MR2271619↑11

work page doi:10.7169/facm/1229442629 2006
[21]

Vijayaraghavan,On the fractional parts of the powers of a number

T. Vijayaraghavan,On the fractional parts of the powers of a number. II, Proc. Cambridge Philos. Soc.37(1941), 349–357. MR0006217↑1, 3

1941
[22]

Weyl, ¨Uber die Gleichverteilung von Zahlen mod

H. Weyl, ¨Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann.77(1916), no. 3, 313–352, DOI 10.1007/BF01475864 (German). MR1511862↑1, 5 (Zhangchi Chen)School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China Email address:zcchen@math.ecnu....

work page doi:10.1007/bf01475864 1916