The Generalized Bergman Game

TL;DR

The paper introduces the Generalized Bergman Game, a two-player game that decomposes integers into base-$eta$ expansions, and analyzes its termination time and complexity.

Contribution

It generalizes previous Zeckendorf decomposition games to a broad class of base-$eta$ expansions and provides bounds on game length and complexity.

Findings

Longest game terminates in $ heta(n^2)$ time

Shortest game terminates between $ ext{o}(n)$ and $ ext{O}(n^2)$ time

Linear bound on maximum length of the tuple during the game

Abstract

Every positive integer may be written uniquely as a base-$\beta$ decomposition--that is a legal sum of powers of $\beta$--where $\beta$ is the dominating root of a non-increasing positive linear recurrence sequence. Guided by earlier work on a two-player game which produces the Zeckendorf Decomposition of an integer (see [Bai+19]), we define a broad class of two-player games played on an infinite tuple of non-negative integers which decompose a positive integer into its base-$\beta$ expansion. We call this game the Generalized Bergman Game. We prove that the longest possible Generalized Bergman game on an initial state $S$ with $n$ summands terminates in $\Theta(n^2)$ time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between $\Omega(n)$ and $O(n^2)$ time. We also show a linear bound on the maximum length of the tuple used throughout the game.