Arithmetic progressions in polynomial orbits
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909131268423680 |
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| author | Sadek, Mohammad Wafik, Mohamed Yesin, Tuğba |
| author_facet | Sadek, Mohammad Wafik, Mohamed Yesin, Tuğba |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic progressions in polynomial orbits Sadek, Mohammad Wafik, Mohamed Yesin, Tuğba Number Theory 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. |
| title | Arithmetic progressions in polynomial orbits |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.04397 |