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Arithmetic progressions in polynomial orbits

Mohammad Sadek, Mohamed Wafik, Tuğba Yesin

Abstract

Let $f$ be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit $\operatorname{Orb}_f(t)=\{t,f(t),f(f(t)),\cdots\}$, where $t$ is an integer, using arithmetic progressions each of which contains $t$. Fixing an integer $k\ge 2$, we prove that it is impossible to cover $\operatorname{Orb}_f(t)$ using $k$ such arithmetic progressions unless $\operatorname{Orb}_f(t)$ is contained in one of these progressions. In fact, we show that the relative density of terms covered by $k$ such arithmetic progressions in $\operatorname{Orb}_f(t)$ 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.

Arithmetic progressions in polynomial orbits

Abstract

Let be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit , where is an integer, using arithmetic progressions each of which contains . Fixing an integer , we prove that it is impossible to cover using such arithmetic progressions unless is contained in one of these progressions. In fact, we show that the relative density of terms covered by such arithmetic progressions in is uniformly bounded from above by a bound that depends solely on . In addition, the latter relative density can be made as close as desired to by an appropriate choice of arithmetic progressions containing if is allowed to be large enough.
Paper Structure (6 sections, 18 theorems, 27 equations)

This paper contains 6 sections, 18 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Intersection of polynomial orbits with linear polynomial orbits
  3. Primitive divisors and intersections with linear orbits
  4. Relative density of orbits intersections
  5. Covering polynomial orbits using arithmetic progressions
  6. Densities of intersections of polynomial orbits and arithmetic progressions

Key Result

Proposition 2.1

Let $f(x)\in K[x]$ be of degree at least $2$. Let $g(x)=ax+b\in K[x]$ be such that $\operatorname{Orb}_f(s)\cap \operatorname{Orb}_g^\pm(t)$ is infinite for some fixed $s,t\in K$. Then either $a$ is a root of unity in $\mathcal{O}_K$ or $f^\phi$ is a power of a linear polynomial, where $\phi(x) = x+

Theorems & Definitions (28)

  • Proposition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 18 more