Year: 2022
Type: article
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 ⩾ 2 $k \geqslant 2$ is an integer and g : N → R $g \colon \mathbb {N} \rightarrow \mathbb {R}$ is a generalised polynomial such that g ( k n ) = 0 $g(k^n) = 0$ for all n ⩾ 0 $n \geqslant 0$ then there exist infinitely many m ∈ N $m \in \mathbb {N}$ , not divisible by k $k$ , such that g ( m k n ) = 0 $g(mk^n) = 0$ for some n ⩾ 0 $n \geqslant 0$ . As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.
DOI: 10.1112/jlms.12509
Author: Jakub Konieczny
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2944276417