arXiv 2403.043972024-03-07math.NT

多项式轨道中的等差数列

Arithmetic progressions in polynomial orbits

Mohammad Sadek, Mohamed Wafik, Tuğba Yesin

AI总结：

该研究针对次数≥2的整系数多项式轨道的等差数列覆盖问题，证明k个含初始点的等差数列无法覆盖轨道（除非轨道含于其中一个），其覆盖相对密度有仅依赖k的一致上界，且k足够大时可使该密度任意接近1。

摘要：

设$f$是次数至少为2的整系数多项式。我们研究用均包含整数$t$的等差数列覆盖轨道$\text{Orb}_f(t)=\u200bt,f(t),f(f(t)),\u2026\u200b$的问题。固定整数$k\uf0652$，我们证明：除非$\text{Orb}_f(t)$包含于其中一个等差数列，否则无法用$k$个此类等差数列覆盖$\text{Orb}_f(t)$。事实上，我们证明$k$个此类等差数列覆盖的项在$\text{Orb}_f(t)$中的相对密度，一致上界仅依赖于$k$。此外，若允许$k$足够大，通过适当选取$k$个包含$t$的等差数列，可使该相对密度任意接近1。

英文摘要：

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.

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