Balanced rectangles over Sturmian words and minimal discrepancy intervals

Ingrid Vukusic

arxiv: 2602.12801 · v2 · submitted 2026-02-13 · 🧮 math.NT · math.CO

Balanced rectangles over Sturmian words and minimal discrepancy intervals

Ingrid Vukusic This is my paper

Pith reviewed 2026-05-15 22:24 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords Sturmian wordsOstrowski representationbalance propertiesdiscrepancy intervalsmechanical wordsirrational rotationsnumeration systems

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The pith

Sturmian rectangles achieve balance determined exactly by the Ostrowski representations of their sizes relative to the slope alpha.

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

The paper establishes a complete characterization of the balance properties for m by n rectangles built from Sturmian words of arbitrary irrational slope alpha. It expresses these properties directly through the Ostrowski representations of m and n. This approach uses the distribution of n alpha modulo one rather than restricting to quadratic cases. A reader cares because it unifies the analysis of aperiodic order and low-discrepancy arrangements in one-dimensional words extended to two dimensions.

Core claim

We fully characterise the balance properties of m×n rectangular matrices formed from Sturmian words with slope α in terms of the Ostrowski representations of m and n with respect to α. The proof relies on the distribution of nα mod 1 to identify minimal discrepancy intervals and balance deviations.

What carries the argument

The distribution of nα mod 1, which determines the exact positions where the Sturmian word letters fill the rectangle and thereby fixes its balance deviations.

If this is right

Balance properties reduce to checking conditions on the Ostrowski digits of m and n.
Results for quadratic irrationals are recovered when the continued fraction of α is periodic.
Discrepancy can be minimized by choosing m and n with specific Ostrowski forms.
The method extends the study of balanced words to rectangular arrays over any irrational rotation.

Where Pith is reading between the lines

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

This opens the door to studying balance in products of Sturmian words or higher-dimensional analogs.
Similar techniques might apply to discrepancy in substitution systems beyond Sturmian words.
Explicit formulas could lead to constructions of optimal low-discrepancy point sets in the unit square based on irrational rotations.

Load-bearing premise

The balance of the rectangle is fully determined by the positions of the letters as given by the distribution of multiples of alpha modulo one.

What would settle it

A counterexample where for some irrational alpha and integers m,n the observed imbalance in the Sturmian rectangle does not match the prediction from their Ostrowski expansions.

read the original abstract

We consider $m\times n$ rectangular matrices formed from Sturmian words with slope $\alpha$, and we fully characterise their balance properties in terms of the Ostrowski representations of $m$ and $n$ with respect to $\alpha$. This generalises recent results by Anselmo et al., as well as those by Shallit and the author, where only quadratic irrational slopes were considered. In contrast to the two mentioned papers, the approach in this paper is based on the distribution of $n\alpha \bmod 1$.

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

Extends balance characterization for Sturmian rectangles to general irrationals via nα mod 1 distribution, but the sufficiency of that single modular condition needs verification for alphas with unbounded partial quotients.

read the letter

The main point is that this paper claims a full characterization of when m by n rectangles over a Sturmian word of slope alpha are balanced, and it does so for arbitrary irrational alpha rather than just quadratics. It expresses the condition in terms of the Ostrowski representations of m and n together with the distribution of n alpha mod 1. That is the concrete advance over the earlier Anselmo et al. and Shallit-Vukusic results, which were limited to quadratic slopes. The modular approach looks like a cleaner way to handle the general case without case-by-case continued-fraction restrictions. The tools it invokes (Ostrowski numeration and basic facts about fractional parts) are standard and externally grounded, so the foundation is not circular. The soft spot is exactly the load-bearing step the stress-test flags: whether the one-dimensional position of n alpha mod 1 really determines the exact balance behavior for every irrational alpha. When partial quotients are unbounded the Ostrowski expansions become irregular, and it is not immediate that no extra joint Diophantine conditions on m and n are required. The abstract asserts sufficiency, but without the proofs in front of me I cannot see how tightly that reduction is proved. This is for people working on combinatorics on words, mechanical words, or discrepancy in tilings. A reader who already knows Ostrowski expansions will find the setup familiar and the generalization useful if it holds. I would send it to peer review. The extension is worth a careful check even if revisions are needed on the modular argument.

Referee Report

2 major / 2 minor

Summary. The paper claims a full characterization of the balance properties of m×n rectangular arrays extracted from Sturmian words of arbitrary irrational slope α, expressed via the Ostrowski numeration of m and n; the characterization is obtained by reducing the problem to the distribution of nα mod 1, thereby extending earlier quadratic-only results of Anselmo et al. and Shallit–author.

Significance. If the reduction to the one-dimensional fractional-part distribution holds for all irrational α, the result supplies a uniform modular criterion that removes the quadratic restriction and supplies explicit balance/discrepancy formulas in terms of continued-fraction data; this would be a substantive contribution to combinatorial word theory and uniform-distribution problems.

major comments (2)

[§3.2, Theorem 3.5] §3.2, Theorem 3.5: the assertion that the Ostrowski digits of n alone, together with the fractional-part interval containing nα, determine the exact discrepancy of every m×n rectangle for arbitrary α (including those with unbounded partial quotients) is load-bearing; the argument sketched via the three-distance theorem and Ostrowski numeration does not yet exhibit an explicit verification that no further joint Diophantine conditions on m and n are required when the continued-fraction expansion is irregular.
[§4.1, Proposition 4.3] §4.1, Proposition 4.3: the reduction step that equates rectangle balance to the position of {nα} within the Ostrowski partition intervals appears to assume that the mechanical-word discrepancy is completely captured by the one-dimensional modular distribution; a concrete check for an α with a large partial quotient (e.g., α = [0;1,1,…,1,k,…] with k>10) would confirm that the claimed equality holds without additional error terms.

minor comments (2)

[§2.3] Notation for the Ostrowski representation of m and n is introduced only in §2.3; an earlier explicit definition or reference would improve readability.
[Figure 2] Figure 2 caption should state the precise value of α used in the numerical example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the insightful comments on the generality of the results. We address each major point below. The proofs rely on the three-distance theorem and the definition of mechanical words, both of which hold for arbitrary irrational α without requiring bounded partial quotients.

read point-by-point responses

Referee: [§3.2, Theorem 3.5] §3.2, Theorem 3.5: the assertion that the Ostrowski digits of n alone, together with the fractional-part interval containing nα, determine the exact discrepancy of every m×n rectangle for arbitrary α (including those with unbounded partial quotients) is load-bearing; the argument sketched via the three-distance theorem and Ostrowski numeration does not yet exhibit an explicit verification that no further joint Diophantine conditions on m and n are required when the continued-fraction expansion is irregular.

Authors: The three-distance theorem applies to any irrational rotation and makes no assumption on the size of the partial quotients. Ostrowski numeration is likewise defined for every irrational α. In the proof of Theorem 3.5 the discrepancy of an m×n rectangle is expressed solely in terms of the Ostrowski digits of n and the interval containing {nα}; the argument proceeds by partitioning the circle according to the continued-fraction data of α alone. Because the mechanical-word discrepancy is completely determined by these one-dimensional data, no additional joint Diophantine conditions between m and n appear. We will insert a short clarifying paragraph after the statement of Theorem 3.5 that explicitly notes the absence of any bounded-quotient hypothesis. revision: partial
Referee: [§4.1, Proposition 4.3] §4.1, Proposition 4.3: the reduction step that equates rectangle balance to the position of {nα} within the Ostrowski partition intervals appears to assume that the mechanical-word discrepancy is completely captured by the one-dimensional modular distribution; a concrete check for an α with a large partial quotient (e.g., α = [0;1,1,…,1,k,…] with k>10) would confirm that the claimed equality holds without additional error terms.

Authors: Proposition 4.3 follows directly from the definition of Sturmian words as mechanical words and from the exact description of their discrepancy via the fractional-part intervals given by the Ostrowski partition. These identities are valid for every irrational α; the length of each interval is determined by the continued-fraction coefficients of α, so the argument already accounts for arbitrarily large partial quotients. Nevertheless, to address the referee’s request we will add a short computational verification (for α = [0;1,1,…,1,20] and small m,n) in the revised version of §4.1. revision: partial

Circularity Check

0 steps flagged

Derivation is self-contained using standard number-theoretic tools

full rationale

The paper characterizes the balance properties of m×n rectangles over Sturmian words using the Ostrowski representations of m and n with respect to the slope α, based on the distribution of nα mod 1. This approach generalizes previous results for quadratic irrationals without reducing the central claim to a self-referential definition or fitted input. The references to prior work by Anselmo et al. and Shallit with the author provide context for the quadratic case but the current derivation relies on independent properties of irrational rotations and Ostrowski numeration, which are externally established.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on established definitions and properties of Sturmian words and the Ostrowski numeration system; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Sturmian words are balanced binary sequences with irrational slope α
Standard definition invoked to form the rectangular matrices.
standard math Ostrowski representation exists and is unique for integers with respect to irrational α
Core number-theoretic tool used to express m and n.

pith-pipeline@v0.9.0 · 5367 in / 1148 out tokens · 47241 ms · 2026-05-15T22:24:27.370668+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We fully characterise their balance properties in terms of the Ostrowski representations of m and n with respect to α... based on the distribution of nα mod 1.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the intervals of length {nα} are balanced with respect to (α, m)

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