On finite $\beta$-expansions for the set of natural numbers

TL;DR

This paper investigates the conditions under which natural numbers have finite $eta$-expansions for certain Pisot numbers, revealing complex behaviors related to polynomial residue classes and providing criteria for finiteness.

Contribution

It identifies new classes of Pisot numbers with unpredictable finiteness properties and offers a concise criterion for finiteness of $eta$-expansions.

Findings

Some Pisot numbers satisfy $F_1$, others do not, depending on polynomial residue classes.

When $F_1$ fails, the set of numbers with infinite $eta$-expansions is infinite.

A sufficient condition for $F_1$ is the finiteness of the expansion of $(eta-loor{eta})^2$.

Abstract

We present a study of the problem of finiteness of the $\beta$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $\beta$-expansion of 1 is not a non-decreasing sequence. We show a class of simple $\beta$-numbers which display a puzzling behaviour, in the sense that an infinite subset of such $\beta$ satisfy $F_1$, whereas a complementary infinite subset does not. This puzzle is organized by the residue class modulo 3 of an integer coefficient of the main family of polynomials discussed. We prove that, when $F_1$ does not hold, the complement of ${{\rm Fin}(\beta)}$ is infinite. Finally, we give a concise sufficient condition for $F_1$, which is the finitude of the $\beta$-expansion of $(\beta-\lfloor \beta\rfloor)^2$.