  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 non-zero 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
polynomails. Finally, we show that any sufficiently sparse $\{0,1\}$-valued
sequence is given by a generalised polynomial.
  (This paper is essentially the first half of our earlier submission
arXiv:1610.03900 [math.NT]. Because the material in arXiv:1610.03900 [math.NT]
touches upon many different subjects, we believe it is preferable to split it
into two independent papers.)
