Comply subtraction games avoiding arithmetic progressions

TL;DR

This paper introduces comply subtraction games that generate winning positions avoiding arithmetic progressions, linking combinatorial game theory with number theory and providing new classes of heap games with specific structural properties.

Contribution

It defines a new comply rule for subtraction games that produces sets avoiding arithmetic progressions, connecting combinatorial game theory with number-theoretic sets.

Findings

Winning positions characterized by base-3 digit restrictions.

Generalization to multi-dimensional games avoiding arithmetic progressions.

Connections to classical games like Nim and Wythoff Nim.

Abstract

Impartial subtraction games on the nonnegative integers have been studied by many and discussed in detail in for example the remarkable work Winning Ways by Conway, Berlekamp and Guy. We describe how comply variations of these games, similar to those introduced by Holshouser, Reiter, Smith, St\u{a}nic\u{a}, can be defined as having its sets of winning positions identical to well-known sets avoiding arithmetic progressions such as $x+z=2y$, studied by Szerkeres, Erd\H os and Tur\'an, and many others, thus exploring a new territory combining ideas from combinatorial games and combinatorial number theory. The sets we have in mind are greedy, that is, for our example: recursively a new nonnegative integer is included to the set if and only if it does not form a three term arithmetic progression with the smaller entries. It is known that the set thus obtained is equivalent to the following log-linear time closed expression: each winning position contains exclusively the digits 0 and 1 in base 3 expansion. In fact this set is impossible as a set of winning positions for a classical subtraction game, in a sense introduced recently by Duch\^ene and Rigo. Therefore our comply-rule generalization of the subtraction games can be seen to resolve new classes of sets as winning positions for heap games. In this context the $\star$-operator for invariant subtraction games was introduced by Larsson, Hegarty and Fraenkel. We define a similar operator for our game. Our comply games generalize into several dimensions. In two dimensions the winning positions can be represented by certain greedy permutations avoiding arithmetic progressions, one of which was recently introduced by Hegarty; while others generalize classical combinatorial games such as Nim and Wythoff Nim.