Sadek, Mohammad and Wafik, Mohamed and Yesin Elsheikh, Emine Tuğba (2024) Arithmetic progressions in polynomial orbits. International Journal of Number Theory, 20 (8). pp. 20092025. ISSN 17930421 (Print) 17937310 (Online)
This is the latest version of this item.
Official URL: https://dx.doi.org/10.1142/S1793042124500970
Abstract
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
Item Type:  Article 

Uncontrolled Keywords:  Arithmetic dynamics; covering systems; intersection of orbits; polynomial orbits; primitive divisors 
Divisions:  Faculty of Engineering and Natural Sciences 
Depositing User:  Mohammad Sadek 
Date Deposited:  23 Sep 2024 12:41 
Last Modified:  23 Sep 2024 12:41 
URI:  https://research.sabanciuniv.edu/id/eprint/50035 
Available Versions of this Item

Arithmetic progressions in polynomial orbits. (deposited 12 Jun 2024 15:19)
 Arithmetic progressions in polynomial orbits. (deposited 23 Sep 2024 12:41) [Currently Displayed]