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arxiv: 2604.11387 · v2 · submitted 2026-04-13 · 🧮 math.DS

Frequency of patterns in smooth sequences over the alphabet {1, 3}

Damien Jamet (LORIA) , Ir\`ene Marcovici (LMRS) , L\'eo Poirier (I2M) , Thierry de la Rue (LMRS) This is my paper

Pith reviewed 2026-05-10 15:54 UTC · model grok-4.3

classification 🧮 math.DS
keywords smooth sequencessymbolic dynamicsunique ergodicitysubstitutive systemspattern frequencytype sequencesodd alphabetergodic theory
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The pith

Smooth sequences over the alphabet {1,3} always have well-defined asymptotic pattern frequencies determined by their type sequences.

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

The paper develops an ergodic theory framework for smooth sequences on the alphabet {1,3}. It partitions the space into subshifts according to the sequence of types that record local structure and the types of successive derivatives. Establishing that these subshifts admit a substitutive structure yields unique ergodicity for each of them. This directly implies that the asymptotic frequency of every finite pattern exists and is fixed once the type sequence is known.

Core claim

Fixing the sequence of types of the successive derivatives produces smaller subshifts of smooth sequences that possess a substitutive structure. These subshifts are therefore uniquely ergodic, so the asymptotic frequency of any finite pattern is well-defined and depends only on the type sequence.

What carries the argument

The substitutive structure of the subshifts obtained by fixing the type sequence of successive derivatives, which guarantees unique ergodicity and hence the existence of pattern frequencies.

Load-bearing premise

The substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which unique ergodicity follows.

What would settle it

A smooth sequence over {1,3} in which the limiting frequency of some finite pattern fails to exist or fails to be determined solely by the type sequence.

Figures

Figures reproduced from arXiv: 2604.11387 by Damien Jamet (LORIA), Ir\`ene Marcovici (LMRS), L\'eo Poirier (I2M), Thierry de la Rue (LMRS).

Figure 1
Figure 1. Figure 1: The Oldenburger-Kolakoski sequence K and its run-length encoding ∆(K) = K. However, this specific sequence has been introduced and also extensively investigated much earlier by Oldenburger in 1939 [14, 3]. The Oldenburger￾Kolakoski sequence, hereafter referred to as such and denoted by K, is the unique sequence over the alphabet {1, 2} starting with 1 that is a fixed point of the run-length encoding operat… view at source ↗
Figure 2
Figure 2. Figure 2: The map Φ provides a one-to-one correspondence between the set of smooth sequences and the set of all infinite sequences over Σ [5]. them remain entirely open: Let w ∈ Σ ∗ be a finite word of length |w|. An occurrence of w in a sequence x ∈ Σ ∗∪Σ N∪Σ Z is an index i such that xi+k = wk for all 0 ≤ k < |w|. We denote by |x0 · · · xn−1|w the number of occurrences of w in the prefix of length n of x. The freq… view at source ↗
Figure 3
Figure 3. Figure 3: Successive derivatives of a bi-infinite smooth sequence [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reconstruction (integration) of a smooth bi-infinite sequence [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: First 4 steps in the construction of the array, for [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Homographies h0 and h1 acting on the square K = [0, 1 2 ] × [0, 1 2 ] 3.1.2 Contracting homographies The purpose of this section is to establish nice contracting properties for the homographies h0 and h1 defined in Lemmas 3.3 and 3.4. We first observe that the quantities a L and b L are always in [0, 1 2 ], therefore it is natural to consider the action of h0 and h1 on the square K := [0, 1 2 ] × [0, 1 2 ]… view at source ↗
Figure 7
Figure 7. Figure 7: Decomposing a recoding y in terms of images of letters of one of its derivative ∆L y The larger the degree L of derivation, the less likely occurrences of w in y will be to straddle two adjacent SL. So, as it will be justified in the following lemma, we can approximate |w 0 k |w by P S∈{A,B,C,D} |w L k |S|SL|w. Dividing this approximation by the length nk of w L k , we get X S∈{A,B,C,D} |w L k |S nk |SL|w … view at source ↗
Figure 8
Figure 8. Figure 8: An infinitely well-aligned smooth sequence [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: First iterations of the homographies h0 and h1 on the square K. It is straightforward to check that, as indicated on [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Action of h0 and h1 on the sides of the square 1. a(τ ), b(τ )  ∈ D if and only if τ = 1∞, which is the only case where ντ ([D]) = 0. 2. a(τ ), b(τ )  ∈ C if and only if τ = 01∞, which is the only case where ντ ([C]) = 0. 3. a(τ ), b(τ )  ∈ A if and only if τ = 101∞, which is the only case where ντ ([A]) = 0. 4. a(τ ), b(τ )  ∈ B if and only if τ = 001∞, which is the only case where ντ ([B]) = 0. This… view at source ↗
Figure 11
Figure 11. Figure 11: Getting w by substitution from w ′ with |w ′ | ≥ |w| of w ′ the only possibility is that |w ′ | = |w| (every letter in w ′ corresponds via 41 [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Successive approximations of F obtained by computing the images of the square K by sequences of iterations of the two homographies h0 and h1. When both α and β are odd integers, the properties of Section 2 still hold, with the two substitutions becoming: φ0 :    A 7→ A(BA) p B 7→ D(CD) p C 7→ A(BA) q D 7→ D(CD) q φ1 :    A 7→ B(AB) p B 7→ C(DC) p C 7→ B(AB) q D 7→ C(DC) q . with p = α−1 … view at source ↗
Figure 13
Figure 13. Figure 13: Successive approximations of F (from bottom up) derived from the approximations of F using (25), up to rank 7. h1(a, b) := 1 β − (β − α)(a + b)  β + 1 4 − a · β − α 2 , β − 1 4 − a · β − α 2  . However, if we want to use the same arguments as in Section 3 to prove the unique ergodicity, we need to ensure that these homographies are con￾tracting on a suitable domain. But this does not seem to be straight… view at source ↗
Figure 14
Figure 14. Figure 14: Images of the square K by sequences of length 13 of iterations of the two homographies h0 and h1. (a) {α, β} = {3, 7} (b) {α, β} = {9, 11} [PITH_FULL_IMAGE:figures/full_fig_p046_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Images of the square K by sequences of length up to 6 of iterations of the two homographies h0 and h1. 5.3 Measure-preserving systems The analysis conducted in the paper provides a continuum of measure￾preserving systems: (Xτ , µτ , S)τ∈{0,1} N . All these systems are ergodic, and have zero Kolmogorov-Sinai entropy: this latter fact follows from the vari￾ational principle, since the subshift X of smooth s… view at source ↗
read the original abstract

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1, 3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts, and propose some perspectives.

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

1 major / 0 minor

Summary. The paper develops an ergodic theory framework for studying statistical properties of smooth sequences over the odd alphabet {1, 3}. It partitions the subshift of smooth sequences according to local structure to define a notion of type, describes the substitutive structure on the smaller subshifts obtained by fixing the type sequence of successive derivatives, and deduces unique ergodicity of these subshifts from that structure. A direct consequence is that the asymptotic frequency of any finite pattern is always well-defined and depends only on the type sequence. The paper also characterizes minimality of the subshifts and outlines perspectives.

Significance. If the central claims hold, the work supplies a uniform method for establishing existence and type-dependence of pattern frequencies in this arithmetic class of sequences, connecting derivative-based type sequences to unique ergodicity via substitution dynamics. This is a concrete contribution to symbolic dynamics and ergodic theory on substitutive systems, with potential for computing frequencies explicitly in applications. The framework itself, once the unique ergodicity step is secured, is a strength.

major comments (1)
  1. The deduction of unique ergodicity for every subshift X_τ (fixed type sequence τ of successive derivatives) from the described substitutive structure is load-bearing for the main claim that frequencies are always well-defined. Standard results require that each substitution (or a power) be primitive and that the subshift coincide with the orbit closure of a fixed point. The manuscript must explicitly verify these conditions uniformly across arbitrary type sequences τ; without such verification, some X_τ may admit multiple ergodic measures, rendering frequencies undefined or sequence-dependent within the same type class. This directly affects the abstract's assertion that frequencies depend only on the type sequence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for a more explicit verification of unique ergodicity. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: The deduction of unique ergodicity for every subshift X_τ (fixed type sequence τ of successive derivatives) from the described substitutive structure is load-bearing for the main claim that frequencies are always well-defined. Standard results require that each substitution (or a power) be primitive and that the subshift coincide with the orbit closure of a fixed point. The manuscript must explicitly verify these conditions uniformly across arbitrary type sequences τ; without such verification, some X_τ may admit multiple ergodic measures, rendering frequencies undefined or sequence-dependent within the same type class. This directly affects the abstract's assertion that frequencies depend only on the type sequence.

    Authors: We agree that the unique ergodicity of each X_τ is central to the claim that pattern frequencies are well-defined and depend only on τ. The manuscript derives a substitutive structure for each fixed τ and deduces unique ergodicity from it, but we acknowledge that the primitivity of the substitution (or a power) and the identification of X_τ as the orbit closure of a fixed point are not verified explicitly and uniformly for arbitrary τ. In the revised manuscript we will insert a new lemma establishing these two properties for every infinite type sequence τ: primitivity follows from the recursive definition of derivatives on the alphabet {1,3}, which forces every letter to appear in sufficiently long iterates independently of the particular τ; the orbit-closure property follows from the construction of the infinite fixed point by iterating the substitution according to τ. With these two facts in place, the standard theorems on unique ergodicity of primitive substitutive systems apply directly and uniformly, confirming that frequencies exist and depend only on τ. This addition does not change any of the stated results but makes the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: unique ergodicity follows from described substitutive structure via standard ergodic theory

full rationale

The derivation begins by partitioning smooth sequences into type sequences τ, constructs the corresponding subshifts X_τ via the substitutive structure obtained from successive derivatives, and concludes unique ergodicity for each such X_τ. This is not self-definitional because the frequencies are not fitted to data nor renamed as predictions; they are consequences of the unique invariant measure whose existence is asserted to follow from the substitution rules on X_τ. No self-citation is invoked as load-bearing justification, no ansatz is smuggled, and no uniqueness theorem is imported from prior author work. The central claim therefore remains independent of its inputs and does not reduce by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract invokes standard ergodic theory and subshift concepts without introducing free parameters, new entities, or ad-hoc axioms beyond the domain assumptions of symbolic dynamics.

axioms (1)
  • standard math Standard properties of subshifts and invariant measures in topological dynamics
    The framework relies on unique ergodicity implying constant pattern frequencies for almost every sequence.

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

  1. The complexity of finite smooth words over binary alphabets

    cs.FL 2026-03 unverdicted novelty 6.0

    f-smooth words are precisely the factors of smooth words, with complexity growing as Θ(n^{log(a+b)/log((a+b)/2)}) proven for even alphabets, lower bound for all binary, and tighter upper bound for odd alphabets.

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