Arithmetic progressions in polynomial orbits

2024/03/07 by Sadek, Mohammad, Wafik, Mohamed, Yesin, Tuğba
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.04397

Abstract

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)),⋯\, 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.

Related

• Covering shrinking polynomials by quasi progressions
• Integer Dynamics
• Parameterizing degree n polynomials by multipliers of periodic orbits
• Narrow progressions in the primes
• The quantitative behaviour of polynomial orbits on nilmanifolds