Arithmetic progressions in polynomial orbits

Abstract

Let $f$ be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit $\operatorname{Orb}_f(t)=\{t,f(t),f(f(t)),\cdots\}$, where $t$ is an integer, using arithmetic progressions each of which contains $t$. Fixing an integer $k\ge 2$, we prove that it is impossible to cover $\operatorname{Orb}_f(t)$ using $k$ such arithmetic progressions unless $\operatorname{Orb}_f(t)$ is contained in one of these progressions. In fact, we show that the relative density of terms covered by $k$ such arithmetic progressions in $\operatorname{Orb}_f(t)$ is uniformly bounded from above by a bound that depends solely on $k$. In addition, the latter relative density can be made as close as desired to $1$ by an appropriate choice of $k$ arithmetic progressions containing $t$ if $k$ is allowed to be large enough.