Sparse generalised polynomials

Abstract

We investigate generalised polynomials (i.e., polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a sparse generalised polynomial takes nonzero value cannot contain a translate of an IP set. We also study some explicit constructions and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomials. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial.