Description
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. By a theorem of
Bergelson-Leibman, generalised polynomials can be completely described
in terms of dynamics on nilmanifolds. Our main result is that the set of
integers where a sparse generalised polynomial takes non-zero value
cannot be combinatorially rich (specifically, cannot contain a translate
of an IP set). We 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. We
apply these results to a question on automatic sequences. This is joint
work with Jakub Konieczny (Jerusalem).