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.

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic.Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial p(n) with at least one irrational coefficient (except for the constant one) and integer m ≥ 2, the sequence ⌊p(n)⌋ mod m is never automatic.We also prove that the conjecture is equivalent to the claim that the set of powers of an integer k ≥ 2 is not given by a generalised polynomial.

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We study periodic points for endomorphisms σ of abelian varieties A over algebraically closed fields of positive characteristic p. We show that the dynamical zeta function ζσ of σ is either rational or transcendental, the first case happening precisely when σ n − 1 is a separable isogeny for all n. We call this condition very inseparability and show it is equivalent to the action of σ on the local p-torsion group scheme being nilpotent.The "false" zeta function Dσ, in which the number of fixed points of σ n is replaced by the degree of σ n −1, is always a rational function. Let 1/Λ denote its largest real pole and assume no other pole or zero has the same absolute value. Then, using a general dichotomy result for power series proven by Royals and Ward in the appendix, we find that ζσ(z) has a natural boundary at |z| = 1/Λ when σ is not very inseparable.We introduce and study tame dynamics, ignoring orbits whose order is divisible by p. We construct a tame zeta function ζ * σ that is always algebraic, and such that ζσ factors into an infinite product of tame zeta functions. We briefly discuss functional equations.Finally, we study the length distribution of orbits and tame orbits. Orbits of very inseparable endomorphisms distribute like those of Axiom A systems with entropy log Λ, but the orbit length distribution of not very inseparable endomorphisms is more erratic and similar to S-integer dynamical systems. We provide an expression for the prime orbit counting function in which the error term displays a power saving depending on the largest real part of a zero of Dσ(Λ −s ).