Sparse generalised polynomials

TL;DR

This paper explores sparse generalized polynomials, showing their limitations in containing certain structured sets, and demonstrates their ability to represent characteristic functions of specific number sequences like Fibonacci and Tribonacci.

Contribution

It proves that sparse generalized polynomials cannot contain translated IP sets and constructs examples representing Fibonacci and Tribonacci numbers.

Findings

Sparse generalized polynomials cannot contain translated IP sets.

Characteristic functions of Fibonacci and Tribonacci numbers are generalized polynomials.

Any sufficiently sparse 0-1 sequence can be represented by a generalized polynomial.

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 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.)