良性逼近与不可加速性
Benign approximations and non-speedability
AI总结：
该研究通过构造两类特殊左可计算数，厘清恢复可逼近数与近可计算数的不可比较且非平凡相交的关系，并借此否定回答了Merkle和Titov关于左可计算数中Martin-Löf随机与不可加速性等价的开放问题。
摘要：
一个左可计算数$x$被称为恢复可逼近的，若存在一个收敛于$x$的可计算递增有理数列$(x_n)_n$，使得对无穷多个$n \in \mathbb{N}$，有$x - x_n < 2^{-n}$；若存在这样的$(x_n)_n$，使得对每个可计算递增函数$s \colon \mathbb{N} \to \mathbb{N}$，序列$(x_{s(n+1)} - x_{s(n)})_n$可计算地收敛于0，则称$x$为近可计算的。本文研究了这两个概念之间的关系：一方面构造了一个同时是恢复可逼近和近可计算的非可计算数，另一方面构造了一个是近可计算但非恢复可逼近的左可计算数；由此可直接推出，这两个概念不可比较且存在非平凡的交集。厘清这一关系后，我们便掌握了回答Merkle和Titov的一个开放问题的关键：他们研究了可加速数，即其逼近在某种意义下可被加速的左可计算数，并提出疑问：在左可计算数中，Martin-Löf随机是否等价于不可加速。由于我们证明了在近可计算数中，可加速数与恢复可逼近数的概念是等价的，我们的第二个构造给出了否定的答案。
英文摘要：
A left-computable number $x$ is called regainingly approximable if there is a computable increasing sequence $(x_n)_n$ of rational numbers converging to $x$ such that $x - x_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$; and it is called nearly computable if there is such an $(x_n)_n$ such that for every computable increasing function $s \colon \mathbb{N} \to \mathbb{N}$ the sequence ${(x_{s(n+1)} - x_{s(n)})_n}$ converges computably to 0. In this article we study the relationship between both concepts by constructing on the one hand a non-computable number that is both regainingly approximable and nearly computable, and on the other hand a left-computable number that is nearly computable but not regainingly approximable; it then easily follows that the two notions are incomparable with non-trivial intersection. With this relationship clarified, we then hold the keys to answering an open question of Merkle and Titov: they studied speedable numbers, that is, left-computable numbers whose approximations can be sped up in a certain sense, and asked whether, among the left-computable numbers, being Martin-Löf random is equivalent to being non-speedable. As we show that the concepts of speedable and regainingly approximable numbers are equivalent within the nearly computable numbers, our second construction provides a negative answer.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
