Arithmetic progressions in polynomial orbits

Let f be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit Orbf(t)=,f(t),f(f(t)),⋯\, where t is an integer, using arithmetic progressions each of which contains t. Fixing an integer k≥ 2, we prove that it is impossible to cover Orbf(t) using k such arithmetic progressions unless Orbf(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 Orbf(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.