On the topology of sums in powers of an algebraic number

TL;DR

This paper investigates the conditions under which the set of sums of powers of an algebraic number, with coefficients in {-1,0,1}, is dense in the real numbers, especially focusing on roots of polynomials with specific properties.

Contribution

It provides new sufficient conditions for the denseness of these sums when the algebraic number is a root of a polynomial with coefficients 0, ±1, extending previous knowledge.

Findings

If q is not a root of a polynomial with coefficients 0, ±1, then Λ(q) is dense in ℝ.

If q is not a Perron number, Λ(q) is dense.

If q has a conjugate α with q|α|<1, then Λ(q) is dense.

Abstract

Let $1<q<2$ and \[ \Lambda(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $\Lambda(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $\Lambda(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $\alpha$ such that $q|\alpha|<1$, then $\Lambda(q)$ is dense in $\mathbb{R}$.