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Dimension of Besicovitch-Eggleston sets for non-autonomous systems with countable symbolic dynamics

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Pith's one-line read For non-autonomous countable affine systems, the Hausdorff dimension of each digit-frequency level set is exactly the larger of a system-wide convergence exponent and a fibre-entropy ratio.

desk verdict Theorem 1.3 is probably true, but Lemma 5.2 is false, so the beta_T lower bound is unsupported as written. read the letter →

arxiv 2506.01803 v1 pith:5GQ56VXG submitted 2025-06-02 math.DS

Jonny Imbierski , Charlene Kalle This is my paper
classification math.DS MSC 11K5537H9911A67
keywords Besicovitch\textendashEgglestonsetsHausdorffdimensionnon-autonomousiteratedfunctionsystemsgeneralisedL\"urothseriesdigitfrequenciescountablesymbolicdynamicsfibreexponentofconvergence
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

This paper establishes an exact Hausdorff-dimension formula for Besicovitch–Eggleston level sets of non-autonomous affine iterated function systems: systems where a switching sequence $\omega$ chooses, at each step, which countable GLS IFS acts, and the digit frequencies in the resulting expansion are prescribed by a vector $\alpha$. The formula says the dimension is the larger of two numbers: $\eta_{\mathcal{T}}$, the maximal exponent of convergence of the contraction ratios across the component systems, and $\beta_{\mathcal{T}}(\alpha)$, a fibre-entropy-to-Lyapunov ratio computed from $\alpha$ and the contraction ratios. A sympathetic reader should care because the answer is sharp and universal: the dimension is exact, the autonomous theorem for L\"uroth expansions becomes a special case, and the universal-lower-bound phenomenon, a dimension floor coming from infinitely many small contraction ratios, survives time dependence.

What carries the argument

The machinery is the non-autonomous Generalised L\"uroth Series (NGLS) framework: the maps are affine contractions $f_{s,b}(x)=a_{s,b}+(-1)^{\varepsilon}x/N_{s,b}$ with inverse contraction ratios $N_{s,b}\ge 1$, arranged so that the images tile $[0,1]$ in measure and each component system satisfies the open set condition. Two quantities drive the formula. $\eta_{\mathcal{T}}=\max_s\inf\{t:\sum_b N_{s,b}^{-t}<\infty\}$ is the maximal exponent of convergence, a measure of how many digits have very small contraction ratios. $\beta_{\mathcal{T}}(\alpha)$ is the fibre dimension, a limit inferior over truncated digit sets of fibre entropy divided by $\sum_{d\in\mathcal{D}_m}\alpha_d\log N_d$. The upper bound is a covering argument using level-$n$ fundamental intervals and Stirling estimates. The lower bound splits into two constructions: for $\eta_{\mathcal{T}}$, a special sequence is woven so that the switching to the extremal system is slowed down, and a measure supported on sequences whose selected digits lie in carefully chosen windows is analysed with a Billingsley-type lemma; for $\beta_{\mathcal{T}}(\alpha)$, the full system is approximated by finite IFSs $\mathcal{T}^{(m)}$, the finite-system formula is applied to Bernoulli-like measures, and their dimensions are shown to converge to $\beta_{\mathcal{T}}(\alpha)$.

What would settle it

Find one Borel set $B$, for instance $B=F_{\mathcal{T},\omega}(\alpha)$ itself for a L\"uroth-type system and a frequency vector $\alpha$ with infinitely many positive entries, for which $\lim_m \mu^{(m)}(B)\neq\mu(B)$, or compute $\dim_H F_{\mathcal{T},\omega}(\alpha)$ for such a system and check whether it drops below $\beta_{\mathcal{T}}(\alpha)$. The first would identify the gap in the proof of Proposition 5.3; the second would refute the formula.

Watch

Extended reading notes

Core claim

Theorem 1.3 is the paper's central claim. For a finite family $\mathcal{T}$ of GLS IFSs and a frequency vector $\alpha$ satisfying the non-degeneracy condition $(\dagger)$, for every switching sequence $\omega$ whose own symbol frequencies equal the aggregated frequencies $\alpha_s$, the Hausdorff dimension of the level set $F_{\mathcal{T},\omega}(\alpha)$ is exactly $\max\{\eta_{\mathcal{T}},\beta_{\mathcal{T}}(\alpha)\}$, provided $\lim_n \log n/\log N_{s,n}$ exists for every infinite component system. When the expected logarithmic contraction sum $\sum_{d\in\mathcal{D}} \alpha_d\log N_d$ diverges, the dimension is exactly $\eta_{\mathcal{T}}$. The theorem also characterises the set of admissible switching sequences. The first term $\eta_{\mathcal{T}}$ is the maximal exponent of convergence of the systems' contraction ratios; the second term $\beta_{\mathcal{T}}(\alpha)$ is the limit inferior of the fibre entropy divided by the corresponding logarithmic contraction sum. The result recovers the autonomous theorem for L\"uroth expansions as the special case of a single stationary system.

Load-bearing premise

The proof that $\beta_{\mathcal{T}}(\alpha)$ is a lower bound assumes that the approximating measures $\mu^{(m)}$ converge to $\mu$ on every Borel set, in the strong sense of setwise convergence; the text verifies equality only on cylinders, and the subsequent claim $\mu^{(m)}(F_{\mathcal{T},\omega}(\alpha))>1-\delta$ depends on that missing setwise step.

Editorial extensions

If this is right

  • For every NGLS system satisfying condition (1.2), the dimension of each Besicovitch\textendash Eggleston level set is known exactly, not merely bounded.
  • The autonomous L\"uroth result and its universal $1/2$ lower bound are recovered by taking $\mathcal{T}=\{\mathcal{T}_L\}$; the same phenomenon persists in the non-autonomous setting as the maximum over systems of their convergence exponents.
  • When all component systems have finite digit sets, $\eta_{\mathcal{T}}=0$ and the formula reduces to an explicit entropy-over-Lyapunov ratio, extending earlier finite-GLS results to a wider class of switching sequences.
  • The formula determines the Hausdorff dimension of vertical slices of self-affine carpets of Lalley\textendash Gatzouras type built from a finite horizontal GLS IFS and the non-autonomous family $\mathcal{T}$.
  • When $\sum_d \alpha_d\log N_d=+\infty$, the dimension is $\eta_{\mathcal{T}}$ independently of $\alpha$, so the universal floor dominates the fibre term completely.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the setwise-convergence step in Lemma 5.2 is not valid, the $\beta_{\mathcal{T}}(\alpha)$ lower bound may only be proved for sets of full measure rather than full Hausdorff dimension; the formula itself could still hold via a different argument, but the current proof of the second half of Theorem 1.3 would need repair.
  • The formula suggests a general additivity principle for countably branched non-autonomous conformal systems: dimension should be the maximum of a transverse exponent of convergence and a fibre entropy ratio whenever the growth of the largest contraction ratios is regular.
  • Viewing $\omega$ as a random switching sequence, the theorem is a quenched statement; one could test whether the same formula holds almost surely for annealed averages over $\omega$, or whether quenched fluctuations produce exceptional $\omega$ with lower dimension.
  • The inverse contraction ratios are ordered increasingly and condition (1.2) is essentially regular variation; systems whose largest contraction ratios oscillate without a limit might have a dimension formula governed by limsup behaviour, possibly with an extra term.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper studies Besicovitch-Eggleston level sets F_{T,ω}(α) for non-autonomous systems built from finite families of countable affine iterated function systems (GLS systems). The main result, Theorem 1.3, asserts that under condition (1.2) and non-degeneracy (†) the Hausdorff dimension equals max{η_T, β_T(α)}, where η_T is the maximal exponent of convergence and β_T(α) is a fibre-dimension term. The proof has three parts: an upper bound via cylinder counting (Proposition 3.1), a lower bound by the exponent of convergence using a non-autonomous Billingsley lemma (Proposition 4.1), and a lower bound by the fibre dimension using finite IFS approximations (Proposition 5.3).

Significance. If correct, the formula would be a genuine extension of the autonomous countable-IFS result of Fan et al. (2010) to time-dependent systems, unifying several earlier results and giving a clean two-term dimension formula. The upper-bound proof and the η_T lower-bound proof are carefully written and appear sound, and they provide useful techniques for non-autonomous countable IFSs. The paper contains no fitted parameters or circular reasoning; the main quantities are defined directly from the IFS data. However, the lower-bound proof for the β_T(α) term is built on a lemma that is both unproved and false, so the central theorem is not established in the submitted form.

major comments (2)
  1. [Section 5.2, Lemma 5.2] The proof of Lemma 5.2 asserts setwise convergence μ_{ω,α}^{(m)}(B) → μ_{ω,α}(B) for every Borel set B. The only argument offered is that it suffices to check equality on cylinders, because the cylinders generate the Borel σ-algebra. This is a logical error: pointwise convergence on a generating π-system does not imply setwise convergence on the full σ-algebra. The proof establishes, at most, that for every cylinder C there is m0(C) such that μ^{(m)}(C)=μ(C) for all m≥m0(C), which is a strictly weaker statement. The assertion is in fact false. For example, for the Luroth system with α concentrated on powers of two, α_{2^k}=2^{-(k+1)} and α_d=0 otherwise, the measure μ=μ_{ω,α} gives full mass to the level set F=F_{T,ω}(α) by the strong law of large numbers. For each m, the finite approximation T^{(m)} lumps all digits >m into one map with positive tail weight, so μ^{(m)} gives positive mass to intervals I_d for infinitely many d>m with α_d=0; consequently the frequencies of those digits under the finite-system typical orbit are positive, and μ^{(m)}(F)=0 for every m. Thus Lemma 5.2 fails as stated.
  2. [Section 5, Proposition 5.3] The proof of the β_T(α) lower bound depends on the line 'Since μ_{ω,α}(F_{T,ω}(α))=1, Lemma 5.2 implies that there is M(δ) such that μ^{(m)}_{ω,α}(F_{T,ω}(α))>1−δ for all m≥M(δ).' Because Lemma 5.2 is false, this premise is unsupported; in the Luroth counterexample above, μ^{(m)}(F)=0 for every m, so the inequality fails completely. No alternative argument is given to show that μ^{(m)}(F_{T,ω}(α)) is close to 1. Proposition 5.3 is the only proof of the β_T(α) lower bound in Theorem 1.3, so that half of the main theorem is not established.
minor comments (2)
  1. [Section 4, proof of Lemma 4.2] In the second displayed inequality of Lemma 4.2, the constant C is introduced as C = ∑_{1≤ℓ≤N_a} log N_{ωℓ,aℓ}, but the bound (C + c_a) n log n is stated with C and c_a depending on ε; this is harmless but the notation could be clarified.
  2. [Section 2.3] The definition of the exceptional set X_ω and the statement that [0,1]\X_ω is countable are correct, but the splitting into three cases would benefit from a reference or a brief justification that the set of points with two expansions is countable; as written, the reader must infer this from the OSC and affineness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dimension formula is derived from independent covering and measure arguments; self-citations are auxiliary and not load-bearing.

full rationale

The central formula (1.3) is not equivalent to its inputs by construction. The quantities eta_T and beta_T(alpha) are defined directly from the IFS contraction ratios and the frequency vector alpha; no parameter is fitted to any dimension value. The upper bound (Proposition 3.1) is a direct covering estimate using fibre fundamental intervals; the eta_T lower bound (Proposition 4.1) constructs a set E_a with a measure mu_a via the Ionescu-Tulcea theorem and a Billingsley-type lemma, yielding a dimension bound independent of beta_T(alpha); the beta_T(alpha) lower bound (Proposition 5.3) approximates by finite GLS systems and computes pointwise dimensions of the approximating measures, then lets m tend to infinity. In each case the claimed dimension only appears as the conclusion, not as an assumption. Citations to [FLMW10], [IKM24], and [BK25] are used for external, independently checkable lemmas about the existence of digit-frequency sequences and the autonomous dimension theorem; they are not used to import the non-autonomous formula. The statement 'Theorem 1.3 generalises this result' about [IKM24, Section 3.3] is an honest comparison, not a renaming. I note one non-circularity issue: Lemma 5.2 in Section 5.2 asserts setwise convergence mu_(omega,alpha)(B) = lim_m mu^(m)_(omega,alpha)(B) for every Borel B, but its proof only establishes agreement on cylinders; agreement on cylinders does not imply setwise convergence, and the later use of mu^(m)(F_(T,omega)(alpha)) > 1 - delta in Proposition 5.3 is therefore not fully justified. This is a proof-gap or correctness risk, not a circularity, and it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or unexplained objects; NGLS is a notational framework built from existing concepts. All parameters in the formula are derived from the given IFS data and the frequency vector. The main weakness is a proof step, not an extra axiom.

assumptions (5)
  • domain assumption Each T_s is a GLS IFS: countable index set B_s, affine contractions, open set condition, and sum_{b in B_s} N_{s,b}^{-1} = 1.
    Stated in Section 2.3; ensures the fundamental intervals partition [0,1] up to a countable set, giving unique expansions outside X_omega.
  • domain assumption The frequency vector alpha satisfies the non-degeneracy condition (dagger): for each s in S there exists b in B_s with alpha_{(s,b)} > 0.
    Stated after Theorem 1.2; used in Theorem 1.2 and Lemma 3.2 to guarantee that approximating frequency vectors exist.
  • domain assumption Condition (1.2): the limit lim_n log n / log N_{s,n} exists for every s in S_N.
    Assumed in Theorem 1.3 and Section 4; used to identify eta(T_s) via (4.3) and to obtain the growth bound (4.4).
  • domain assumption The index set B_s is ordered so that N_{s,b} is nondecreasing in b.
    Used in Section 4, for example N_{zeta,1} <= ... <= N_{zeta,b} and N_{s,b} >= b; this ordering is part of the GLS setup.
  • standard math Standard external results: Ionescu-Tulcea extension, Caratheodory extension, Young's pointwise dimension theorem, Billingsley dimension comparison, Stirling's formula, and Polya-Szego's exponent-of-convergence identity.
    Invoked in Sections 2, 4 and 5; each is cited to a reference and is standard, not proved afresh in this paper.

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Pith. "Pith review of Dimension of Besicovitch-Eggleston sets for non-autonomous systems with countable symbolic dynamics." pith.science (2026). https://pith.science/paper/5GQ56VXG

@misc{pith2026250601803,
  author       = {Pith},
  title        = {Pith review of: Dimension of Besicovitch-Eggleston sets for non-autonomous systems with countable symbolic dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GQ56VXG}},
  note         = {Machine review of arXiv:2506.01803}
}
read the original abstract

In this article we derive a formula for the Hausdorff dimension of Besicovitch-Eggleston level sets associated with non-autonomous dynamics constructed from families of countable affine iterated function systems. The formula obtained shows that the universal-lower-bound phenomenon present in the autonomous case studied by Fan et al. (2010) persists in this non-autonomous setting.

Figures

Figures reproduced from arXiv: 2506.01803 by the authors.

Figure 1
Figure 1. Examples of finite and infinite IFSs generating GLS expansions. Bi￾nary expansions (a) and L¨uroth expansions (c) are particular examples of GLS expansions. Now consider the IFS TL = {fk : [0, 1] → [0, 1]}k∈N with fk(x) = x+k k(k+1) , see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Three examples of GLS IFSs Ts with their first three approximations T (m) s for m = 1, 2, 3. In (a)-(c), we see the first three approximations of the L¨uroth system TL shown in (d). In (e)-(g), we see the first three approximations of the GLS IFS TF with finite set BF shown in (h). In (i)-(k), we see the first three approximations of the GLS IFS TI with infinite set BI shown in (l). Let πω,m : B N ω,m → [0, 1], (bn)… view at source ↗

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