We give upper bounds for several highness properties in computability randomness theory. First, we prove that a certain discrete covering property (which requires to almost cover all sets in a certain class) does not imply the ability to compute a 1-random real, answering a question of Greenberg, Miller and Nies. This also implies that an infinite set of incompressible strings does not necessarily compute a 1-random real. Second, we prove that given a homogeneous binary tree that does not admit an infinite computable path, a sequence of bounded martingales whose initial capitals tend to zero (where a martingale is bounded if its range is a bounded set of reals), there exists a martingale $S$ majorizing infinitely many of them such that $S$ does not compute an infinite path of the tree. This implies that: (1) a certain highness notion (which degenerates non 1-random into noncomputably random) does not imply PA-completeness, answering a question of Miller; (2) the computably random reducibility is not equivalent to Turing reducibility, answering a question of Nies. The proof of the second result suggests that the coding power of the universal computably enumerable martingale lies in its infinite variance.

中文翻译：

---

对马丁格尔进行专业化的可计算分析

我们在可计算性随机性理论中给出了几个高级属性的上限。首先，我们证明了某个离散覆盖属性（它需要几乎覆盖某个类中的所有集合）并不意味着能够计算 1-随机实数，回答格林伯格、米勒和 Nies 的问题。这也意味着不可压缩字符串的无限集合不一定计算 1-随机实数。其次，我们证明给定一个不允许无限可计算路径的齐次二叉树，一个初始大写趋于零的有界鞅序列（如果鞅的范围是有界实数集，则它是有界的），存在鞅$秒$ 将它们中的无穷大化，使得 $秒$不计算树的无限路径。这意味着：（1）某种高级概念（将非 1-随机退化为不可计算的随机）并不意味着 PA 完备性，回答米勒的问题；(2) 可计算的随机可约性不等于图灵可约性，回答一个 Nies 的问题。第二个结果的证明表明，通用可计算可枚举鞅的编码能力在于它的无限方差。