p进Wirsing问题
On the p-adic Wirsing problem
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中文总结 AI 辅助
本文建立了Poëls关于实超越数的Wirsing问题结果的p进对应，证明了p进情形下超越数的逼近指数下界，改进了此前的相关已知下界。
中文摘要 AI 辅助
对于实超越数ξ，设ωₙ^*(ξ)为满足以下条件的所有ω的上确界：存在无限多个次数≤n的实代数数α，使得|ξ-α|≤H(α)^(-ω-1)，其中H(α)是α的极小多项式的朴素高度。Wirsing的著名结果给出了统一下界ωₙ^*(ξ)≥(n+1)/2，Poëls近期的工作将其大幅改进为n/(2-log2)。本文建立了Poëls结果的p进对应：设p为素数，ξ∈Qₚ为超越数，ωₙ,ₚ^*(ξ)为满足以下条件的所有实数ω的上确界：存在无限多个次数≤n的Qₚ中代数数α，使得|ξ-α|ₚ≤H(α)^(-ω-1)。本文证明ωₙ,ₚ^*(ξ)≥n/(2-log2)-1，该结果改进了Morrison和Teulié给出的p进情形下Wirsing定理的类似物等已知下界。
英文摘要
For a real transcendental number $ξ$, let $ω_n^*(ξ)$ denote the supremum of all $ω$ for which there exist infinitely many real algebraic numbers $α$ of degree $\leq n$ satisfying $|ξ-α|\leq H(α)^{-ω-1}$, where $H(α)$ is the naive height of the minimal polynomial of $α$. A celebrated result of Wirsing gives the uniform lower bound $ω_n^*(ξ)\geq\frac{n+1}{2}$, which was improved significantly in a recent work of Poëls to $\frac{n}{2-\log 2}$. In this paper, we establish a $p$-adic counterpart of Poëls's result. Let $p$ be a prime and $ξ\in\Qp$ be transcendental. Let $ω_{n,p}^*(ξ)$ be the supremum of all real numbers $ω$ for which there exist infinitely many algebraic numbers $α\in \Qp$ of degree $\leq n$ such that $|ξ-α|_p\leq H(α)^{-ω-1}$. We show that $ω^*_{n,p}(ξ)\geq\frac{n}{2-\log 2}-1$. This improves the known lower bounds in the $p$-adic setting, namely the analogue of Wirsing's theorem, due to Morrison and Teulié.
