Invariant games and non-homogeneous Beatty sequences

TL;DR

This paper characterizes pairs of non-homogeneous Beatty sequences for which an invariant game exists with those sequences as its P-positions, providing a decision procedure based on algebraic and combinatorial tools.

Contribution

It offers a complete characterization of certain Beatty sequence pairs related to invariant games, using symbolic dynamics and algebraic tests.

Findings

Characterization of all such sequence pairs for invariant games.

A decision procedure based on algebraic and combinatorial criteria.

Ability to determine the existence of an invariant game from four real numbers.

Abstract

We characterize all the pairs of complementary non-homogenous Beatty sequences $(A_n)_{n\ge 0}$ and $(B_n)_{n\ge 0}$ for which there exists an invariant game having exactly $\{(A_n,B_n)\mid n\ge 0\}\cup \{(B_n,A_n)\mid n\ge 0\}$ as set of $\mathcal{P}$-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can therefore decide whether or not such an invariant game exists.