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Balanced rectangles over Sturmian words and minimal discrepancy intervals

Ingrid Vukusic

Abstract

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

Balanced rectangles over Sturmian words and minimal discrepancy intervals

Abstract

We consider rectangular matrices formed from Sturmian words with slope and we fully characterise their balance properties in terms of the Ostrowski representations of an with respect to . This generalises recent results by Anselmo et al., as well as by Shallit and Vukusic, where only quadratic irrationals were considered. In contrast to the two mentioned papers, our approach is based on the distribution of .
Paper Structure (17 sections, 34 theorems, 100 equations)

This paper contains 17 sections, 34 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Full characterization via Ostrowski representations
  3. Correspondence between balancedness and low discrepancy intervals
  4. Balanced intervals and certain bijective maps
  5. Properties of convergents and Ostrowski representations
  6. Basic properties of convergents
  7. Basic properties of Ostrowski representations
  8. One-sided approximations
  9. Minimising distances in certain ranges of integers
  10. Ostrowski representations of $n$ and $q_T - n$
  11. Proof of the full characterisation
  12. The case $n = x^{*}$
  13. Switching beween $n$ and $q_T - n$
  14. Technical lemmas for the case $x^{*} < n$
  15. The cases where $x^{*} \neq n$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\alpha\in (0,1)$ be irrational and $2 \leq m \leq n$. Then the $m\times n$ rectangles of the Sturmian words with slope $\alpha$ are balanced if and only if the Ostrowski representations of $m,n$ with respect to $\alpha$ are of at least one of the following four shapes: They have "split represen The smaller number $m$ is the denominator of a (semi-)convergent, and we have certain parity restri

Theorems & Definitions (78)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 2.1
  • Example 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Definition 4.1
  • ...and 68 more