Arithmetic progressions in polynomial orbits

Official URL: https://dx.doi.org/10.1142/S1793042124500970

Let f be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit Orbf (t) = {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