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 (Formula presented.) is an integer and (Formula presented.) is a generalised polynomial such that (Formula presented.) for all (Formula presented.) then there exist infinitely many (Formula presented.), not divisible by (Formula presented.), such that (Formula presented.) for some (Formula presented.). As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.