A Normality Conjecture on Rational Base Number Systems
Pith reviewed 2026-05-18 09:34 UTC · model grok-4.3
The pith
Every minimal and maximal word in a rational base number system is normal over an appropriate subalphabet.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, we present extensive numerical experiments that examine the richness threshold and the deviation from normality of these words. We also discuss the implications that the validity of our conjecture would have for several long-standing open problems, including the existence of Z-numbers and Z_{p/q}-numbers, the existence of triple expansions in rational base p/q, and the Collatz-inspired 4/3 problem.
What carries the argument
Minimal and maximal words generated by the rational base number system, which serve as the infinite sequences whose normality over a subalphabet is conjectured.
If this is right
- The conjecture would imply the existence of Z-numbers in the sense of Mahler.
- It would imply the existence of Z_{p/q}-numbers in the sense of Flatto.
- It would settle the existence or non-existence of triple expansions in rational base p/q.
- It would resolve aspects of the 4/3 problem studied by Dubickas and Mossinghoff.
Where Pith is reading between the lines
- If the conjecture is true, one could test normality of these words by checking only a finite but sufficiently large prefix once the richness threshold is passed.
- The result might extend to other generalized number systems that produce analogous minimal and maximal sequences, such as those with non-constant digit sets.
- Confirmation could provide a new route to constructing normal sequences with explicit combinatorial or arithmetic definitions.
Load-bearing premise
Finite-prefix numerical checks on richness threshold and deviation from normality are indicative of the limiting behavior of the infinite words.
What would settle it
A concrete minimal or maximal word whose digit frequencies deviate from uniformity by a fixed positive amount that does not tend to zero as the length of the examined prefix grows without bound.
read the original abstract
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, we present extensive numerical experiments that examine the richness threshold and the deviation from normality of these words. We also discuss the implications that the validity of our conjecture would have for several long-standing open problems, including the existence of $Z$-numbers (Mahler, 1968) and $Z_{p/q}$-numbers (Flatto, 1992), the existence of triple expansions in rational base $p/q$ (Akiyama, 2008), and the Collatz-inspired `4/3 problem' (Dubickas and Mossinghoff, 2009).
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper conjectures that every minimal and maximal infinite word arising in rational base number systems is normal with respect to an appropriate subalphabet. Support is provided via numerical experiments that compute richness thresholds and empirical deviations from normality on finite prefixes; the authors also outline implications for open problems including the existence of Z-numbers, Z_{p/q}-numbers, triple expansions in base p/q, and the 4/3 problem.
Significance. If the conjecture is true, it would unify several long-standing questions in Diophantine approximation and symbolic dynamics under a single normality statement, potentially resolving the existence of Z-numbers (Mahler) and the 4/3 problem. The manuscript supplies extensive computational evidence on prefix statistics, which constitutes a genuine strength for a conjecture paper; however, the asymptotic character of normality means that the evidential value hinges on whether finite checks can be extrapolated.
major comments (2)
- [Section 4] Section 4 (Numerical Experiments): the reported checks on richness thresholds and block-frequency deviations are performed only on finite prefixes; no discrepancy bound, rate-of-convergence estimate, or uniform control in block length is given that would guarantee the limiting frequencies lim (1/n) |{i : w[i..i+k-1]=u}| = μ(u) required for normality.
- [Introduction and Section 5] Introduction and Section 5 (Implications): the conjecture is invoked to imply resolutions of the Z-number and 4/3 problems, yet the manuscript does not address whether the observed trends on finite prefixes could reverse or accumulate at scales beyond the tested lengths, leaving the logical bridge from computation to the claimed consequences unestablished.
minor comments (2)
- Notation for the subalphabet in the normality statement is introduced without a dedicated definition paragraph; a short clarifying sentence or table would improve readability.
- Figure captions for the deviation plots do not explicitly state the prefix lengths or the precise subalphabet used in each panel.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments, which highlight key limitations in the evidential basis of our conjecture. We address each major point below and have made revisions to clarify the scope of the numerical support and the conditional nature of the implications.
read point-by-point responses
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Referee: [Section 4] Section 4 (Numerical Experiments): the reported checks on richness thresholds and block-frequency deviations are performed only on finite prefixes; no discrepancy bound, rate-of-convergence estimate, or uniform control in block length is given that would guarantee the limiting frequencies lim (1/n) |{i : w[i..i+k-1]=u}| = μ(u) required for normality.
Authors: We agree that the experiments in Section 4 examine only finite prefixes and provide no rigorous discrepancy bounds or convergence rates. Proving the existence of the limiting frequencies for all block lengths would constitute a proof of the normality conjecture itself, which remains open. The computations instead offer empirical support by documenting consistent trends across increasingly long prefixes and multiple rational bases. In the revised version we have added an explicit paragraph in Section 4 stating these limitations and clarifying that the observed approach to uniform frequencies is suggestive rather than conclusive. revision: partial
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Referee: [Introduction and Section 5] Introduction and Section 5 (Implications): the conjecture is invoked to imply resolutions of the Z-number and 4/3 problems, yet the manuscript does not address whether the observed trends on finite prefixes could reverse or accumulate at scales beyond the tested lengths, leaving the logical bridge from computation to the claimed consequences unestablished.
Authors: The implications for Z-numbers, Z_{p/q}-numbers, triple expansions, and the 4/3 problem are presented strictly as consequences that would follow if the conjecture holds. We acknowledge that finite-prefix statistics cannot preclude reversals or slow accumulations of deviations at larger scales. The revised manuscript now states more explicitly in both the introduction and Section 5 that the numerical evidence is supportive but does not establish the asymptotic results, and that the connections to the open problems remain conditional on the conjecture being true. revision: yes
Circularity Check
No circularity: explicit conjecture with independent numerical support
full rationale
The paper states an explicit conjecture that every minimal and maximal word is normal over an appropriate subalphabet and supports it with numerical experiments on richness thresholds and finite-prefix deviations from normality. No derivation chain, first-principles result, or prediction is claimed that could reduce to the inputs by construction. The numerical checks are presented as external evidence rather than a fitted parameter renamed as a prediction, and no self-citation load-bearing step or ansatz is invoked to justify the central claim. The work is therefore self-contained as a conjecture plus supporting data.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Standard definitions of rational base p/q number systems and the associated minimal and maximal infinite words (Akiyama-Frougny-Sakarovitch 2008).
discussion (0)
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