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 [Formula: see text], where t is an integer, using arithmetic progressions each of which contains t. Fixing an integer [Formula: see text], we prove that it is impossible to cover [Formula: see text] using k such arithmetic progressions unless [Formula: see text] is contained in one of these progressions. In fact, we show that the relative density of terms covered by k such arithmetic progressions in [Formula: see text] 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.