Generalised polynomials and integer powers

Abstract

We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k\ge 2$ is and integer and $g:\mathbb{N}\to \mathbb{R}$ is a generalised polynomial such that $g(k^n)=0$ for all $n\ge 0$ then there exists infinitely many $m\in \mathbb{N}$, not divisible by $k$, such that $g(m{k}^n)=0$ for some $n\ge 0$. As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.

51 pages

References in corpus (1)

• Automatic sequences and generalised polynomials