Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

Abstract: The Ku\v{c}era-G\'acs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-L\"of random real. If the computation of the first $n$ bits of a sequence requires $n+h(n)$ bits of the random oracle, then $h$ is the redundancy of the computation. Ku\v{c}era implicitly achieved redundancy $n\log n$ while G\'acs used a more elaborate coding procedure which achieves redundancy $\sqrt{n}\log n$. A similar upper bound is implicit in the later proof by Merkle and Mihailovi\'c. In this paper we obtain strict optimal lower bounds on the redundancy in computations from Martin-L\"of random oracles. We show that any nondecreasing computable function $g$ such that $\sum_n 2^{-g(n)}=\infty$ is not a general upper bound on the redundancy in computations from Martin-L\"of random oracles. In fact, there exists a real $X$ such that the redundancy $g$ of any computation of $X$ from a Martin-L\"of random oracle satisfies $\sum_n 2^{-g(n)}<\infty$. Moreover, the class of such reals is comeager and includes a $\Delta^0_2$ real as well as all weakly 2-generic reals. This excludes many slow growing functions such as $\log n$ from bounding the redundancy in computations from random oracles for a large class of reals. On the other hand it was recently shown that if $\sum_n 2^{-g(n)}<\infty$ then $g$ is a general upper bound for the redundancy in computations of any real from some Martin-L\"of random oracle. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.