Digit expansions in rational and algebraic basis

TL;DR

This paper introduces a novel method for expanding complex numbers in arbitrary algebraic bases using digit sets, generalizing rational base systems and connecting to tilings and $p$-adic analysis.

Contribution

It develops a new framework for $eta$-expansions in complex bases, extending previous rational systems, and explores their properties, algorithms, and connections to tilings and $p$-adic fields.

Findings

Defined $eta$-expansions for complex numbers with unique almost everywhere.

Developed algorithms for Gaussian integer expansions.

Linked expansions to complex plane tilings and $p$-adic completions.

Abstract

Consider $\alpha \in \Q(i)$ satisfying $|\alpha| >1$. Let $\D = \{0,1,\ldots,|a_0|-1\}$, where $a_0$ is the independent coefficient of the minimal primitive polynomial of $\alpha$. We introduce a way of expanding complex numbers in base $\alpha$ with digits in $\D$ that we call $\alpha$-expansions, which generalize rational base number systems introduced by Akiyama, Frougny and Sakarovitch, and are related to rational self-affine tiles introduced by Steiner and Thuswaldner. We define an algorithm to obtain the expansions for certain Gaussian integers and show results on the language. We then extend the expansions to all $x \in \C$ (or $x \in \R$ when $\alpha = \ab \in \Q$, the rational case will be our starting point) and show that they are unique almost everywhere. We relate them to tilings of the complex plane. We characterize $\alpha$-expansions in terms of $p$-adic completions of $\Q(i)$ with respect to Gaussian primes.