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 Orb(f)(t) = {t,f(t),f(f(t)),& mldr;}, 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 Orb(f)(t) using k such arithmetic progressions unless 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 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.