论文标题
特征$ p $中的动力二磷抗抗剂近似指数
Dynamical Diophantine Approximation Exponents in Characteristic $p$
论文作者
论文摘要
令$ ϕ（z）$在单变量中成为非等分的有理函数，系数为$ \ overline {\ mathbb {f}} _ p（t）$，并假设$ c}^in \ mathbb {p}^1（\ mathbb {p}^1（\ mathbb {\ mathbb {\ mathbb {cription） $ ϕ $。然后，我们证明了$ ϕ^{ - m}（γ）$的元素的二苯胺近似指数最终以$ \ lceil d^m/2 \ rceil+1 $限制。为此，我们将特征性$ p $的Diophantine技术与Berkovich空间中小点的Adelic等分组合在一起。作为应用程序，我们在此环境中推断出了Silverman著名限制定理的一种形式。也就是说，如果我们采用任何流浪点$ a \ in \ mathbb {p}^1（\ edimelline {\ mathbb {f}} _ p（t））$并写入$ ϕ^n（a）= a_n/b_n $ $ a_n，b_n \ in \ overline {\ mathbb {f}} _ p [t] $，然后我们证明\ [\ frac {1} {2} {2} \ leq \ liminf_ { \ frac {\ text {deg}（a_n）} {\ text {deg}（b_n）} \ leq \ limsup_ {n \ rightarrow \ rightarrow \ infty} \ frac {\ frac {\ frac {\ text {deg} $ \ infty $都是$ ϕ $的临界点。在特征性$ p $中，thue-siegel-dyson-roth定理是错误的，因此我们的证明需要与Silverman使用的技术不同。
Let $ϕ(z)$ be a non-isotrivial rational function in one-variable with coefficients in $\overline{\mathbb{F}}_p(t)$ and assume that $γ\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ is not a post-critical point for $ϕ$. Then we prove that the diophantine approximation exponent of elements of $ϕ^{-m}(γ)$ are eventually bounded above by $\lceil d^m/2\rceil+1$. To do this, we mix diophantine techniques in characteristic $p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we take any wandering point $a\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ and write $ϕ^n(a)=a_n/b_n$ for some coprime polynomials $a_n,b_n\in\overline{\mathbb{F}}_p[t]$, then we prove that \[ \frac{1}{2}\leq \liminf_{n\rightarrow\infty} \frac{\text{deg}(a_n)}{\text{deg}(b_n)} \leq\limsup_{n\rightarrow\infty} \frac{\text{deg}(a_n)}{\text{deg}(b_n)}\leq2,\] whenever $0$ and $\infty$ are both not post-critical points for $ϕ$. In characteristic $p$, the Thue-Siegel-Dyson-Roth theorem is false, and so our proof requires different techniques than those used by Silverman.
