Local obstructions in sequences revisited

2025-03-26

AI Summary

This work addresses the local obstruction problem in Diophantine approximation with non-repeating sequences and sparse denominators. We introduce a novel combinatorial game-theoretic approach: modeling obstruction structures as finite two-player games, where unavoidable obstructions are characterized by winning strategies. This framework unifies and simplifies proofs of several classical theorems—including Thue-type and Mahler-type results—while significantly improving asymptotic bounds. Moreover, it naturally yields a computable decision procedure for obstruction detection: an explicit, implementable algorithm is obtained directly from the game formulation, without requiring ad hoc constructions. The key innovation lies in the first systematic application of combinatorial game theory to the analysis of sequence-based obstructions, thereby integrating proof simplification, bound optimization, and computational decidability within a single coherent framework.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

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.

Problem

• Study a combinatorial game and its winning strategy
• Simplify proofs on non-repetitive sequence results
• Improve bounds for lacunary sequence approximations

Innovation

• Combinatorial game for non-repetitive sequences
• Simplified proofs with improved bounds
• Computable versions without separate treatment