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

The Kucera-Gacs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Lof 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. Kucera implicitly achieved redundancy nlogn while Gacs used a more elaborate coding procedure which achieves redundancy root nlogn. A similar bound is implicit in the later proof by Merkle and Mihailovic. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Lof random oracles. We show that any nondecreasing computable function g such that Sigma(n) 2(-g(n)) = infinity is not a general upper bound on the redundancy in computations from Martin-Lof random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Lof random oracle satisfies Sigma(n) 2(-g(n)) = infinity . Moreover, the class of such reals is comeager and includes a Delta(0)(2) real as well as all weakly 2-generic reals. On the other hand, it has been recently shown that any real is computable from a Martin-Lof random oracle with redundancy g, provided that g is a computable nondecreasing function such that Sigma(n) 2(-g(n)) = infinity. Hence our lower bound is optimal, and excludes many slow growing functions such as toga from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop. (C) 2016 Elsevier Inc. All rights reserved.