论文标题
高度差距猜想，$ d $ finitesions和弱动力莫德尔·兰格（Mordell-Lang）
Height Gap Conjectures, $D$-Finiteness, and Weak Dynamical Mordell-Lang
论文作者
论文摘要
在先前的工作中，第一作者吉奥卡（Ghioca）和第三作者引入了广泛的动力框架，从而从数字理论和代数组合学中产生了许多经典序列。具体而言，这些是$ f（φ^n（x））$的序列，其中$φ\ colon x \ to x $和$ f \ colon x \ to \ mathbb {p}^1 $是$ \ overline {\ mathb {q}} $ and x（$ x \ intlline and x（$ x \ intline）定义的有理图向前的轨道避免了$φ$和$ f $的不确定性位置的观点。他们推测，如果序列是无限的，则$ \ limsup \ frac {h（f（φ^n（x）））} {\ log n}> 0 $。他们还为$ \ liminf $做出了相应的猜想，并表明这意味着动态的mordell-lang猜想。在本文中，我们证明了$ \ limsup $猜想以及$ \ liminf $ supenture从一组密度$ 0 $的猜测之外。作为应用程序，我们证明了有关$ d $ finite功率系列的增长率以及动态的mordell-lang猜想的增长率，直到一组密度$ 0 $。
In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f(Φ^n(x))$, where $Φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $Φ$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(Φ^n(x)))}{\log n} > 0$. They also made a corresponding conjecture for $\liminf$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\limsup$ conjecture as well as the $\liminf$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.
