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Local obstructions in sequences revisited

Matthieu Rosenfeld, Alexander Shen

TL;DR

This work introduces a unifying combinatorial game based on β-weighted trees to streamline existence proofs for objects like lacunary Diophantine approximations and nonrepetitive sequences. The core method yields explicit, improved constants and, crucially, computable constructions that avoid reliance on non-constructive probabilistic lemmas. By applying the game to Miller’s forbidden-factor framework and to denominators in irrational approximations, the authors obtain new or sharpened bounds on regularity and square-free/avoidance properties, including computable versions for 4-list square-freeness and related phenomena. The approach not only recovers classical results (e.g., Thue-type square-free words, Beck-type separation) but also connects to broader themes like left-handed LLL and entropy compression, offering a versatile, algorithm-friendly perspective on combinatorics on words and Diophantine approximation.

Abstract

In this article, we consider some simple combinatorial game and a winning strategy in this game. This game is then used to prove several known results about non-repetitive sequences and approximations with denominators from a lacunary sequence. In this way we simplify the proofs, improve the bounds and get for free the computable versions that required a separate treatment.

Local obstructions in sequences revisited

TL;DR

This work introduces a unifying combinatorial game based on β-weighted trees to streamline existence proofs for objects like lacunary Diophantine approximations and nonrepetitive sequences. The core method yields explicit, improved constants and, crucially, computable constructions that avoid reliance on non-constructive probabilistic lemmas. By applying the game to Miller’s forbidden-factor framework and to denominators in irrational approximations, the authors obtain new or sharpened bounds on regularity and square-free/avoidance properties, including computable versions for 4-list square-freeness and related phenomena. The approach not only recovers classical results (e.g., Thue-type square-free words, Beck-type separation) but also connects to broader themes like left-handed LLL and entropy compression, offering a versatile, algorithm-friendly perspective on combinatorics on words and Diophantine approximation.

Abstract

In this article, we consider some simple combinatorial game and a winning strategy in this game. This game is then used to prove several known results about non-repetitive sequences and approximations with denominators from a lacunary sequence. In this way we simplify the proofs, improve the bounds and get for free the computable versions that required a separate treatment.

Paper Structure

This paper contains 11 sections, 11 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. The game
  3. Miller's argument
  4. Nonrepetitive words
  5. Separated occurrences
  6. Making adjacent blocks very different
  7. Application to denominators
  8. Optimizing the choice of parameters
  9. Historical remarks
  10. Erdős' question
  11. Forbidden factors

Key Result

Theorem 1

Assume that for some integer $k>1$, some $\beta$ in $(1,k)$ and some $\omega>0$ the inequality holds. Then Bob has a winning strategy in the $(\beta,\omega,k)$-game.

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Remark 1
  • ...and 10 more