Dimension 1 sequences are close to randoms

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-Löf random sequence. More generally, a sequence has effective dimension s if and only if it is coarsely similar to a weakly s-random sequence. Further, for any s ă t, every sequence of effective dimension s can be changed on density at most H ́1ptq ́H ́1psq of its bits to produce a sequence of effective dimension t, and this bound is optimal. The theory of algorithmic randomness defines an individual object in a probability space to be random if it looks plausible as an output of a corresponding random process. The first and the most studied definition was given by Martin-Löf [ML66]: a random object is an object that satisfies all “effective” probability laws, i.e., does not belong to any effectively null set. (See [DH10, UVS13, She15] for details; we consider only the case of uniform Bernoulli measure on binary sequences, which corresponds to independent tossings of a fair coin.) It was shown by Schnorr and Levin (see [Sch72, Sch73, Lev73]) that an equivalent definition can be given in terms of description complexity: a bit sequence X P 2 is Martin-Löf (ML) random if and only if the prefix-free complexity of its n-bit prefix X æn is at least n ́Op1q. (See [LV93, UVS13, She15] for the definition of prefix-free complexity and for the proof of this equivalence; one may use also monotone or a priori complexity.) This robust class also has an equivalent characterization based on martingales that goes back to Schnorr [Sch71]. The notion of randomness is in another way quite fragile: if we take a random sequence and change to zero, say, its 10th, 100th, 1000th, etc. bits, the resulting sequence is not random, and for a good reason: a cheater that cheats once in a while is still a cheater. To consider such sequences as “approximately random”, one option is to relax the Levin-Schnorr definition by replacing the Op1q term in the complexity characterisation of randomness by a bigger opnq term, thus requiring that limnÑ8KpX ænq{n “ 1. Such sequences coincide with the sequences of effective Hausdorff dimension 1. (Effective Hausdorff dimension was first explicitly introduced by Lutz [Lut00]. It can be defined in several equivalent ways via complexity, via natural generalizations of effective null sets, and via natural generalizations of martingales; again, see [DH10, UVS13, She15] for more information.) Another approach follows the above example more closely: we could say that a sequence is approximately random if it differs from a random sequence on a set The first and fourth authors were supported by a Rutherford Discovery Fellowship from the Royal Society of NZ. The second author was partially supported by grant #358043 from the Simons Foundation. The third author was partially supported by RaCAF ANR-15-CE40-0016-01 grant. 1 2 N. GREENBERG, J.S. MILLER, A. SHEN, AND L.B. WESTRICK of density 0. Our starting point is that this also characterizes the sequences of effective Hausdorff dimension 1. To set notation, for n ě 1, we let d be the normalised Hamming distance on t0, 1u, the set of binary strings of length n: dpσ, τq “ # tk : σpkq ‰ τpkqu n ; and we also denote by d the Besicovitch distance on Cantor space 2 (the space of infinite binary sequences), defined by dpX,Y q “ lim sup nÑ8 dpX æn, Y ænq, where Z æn stands for the n-bit prefix of Z. If dpX,Y q “ 0, then we say that X and Y are coarsely equivalent.1 Theorem 1.7. A sequence has effective Hausdorff dimension 1 if and only if it is coarsely equivalent to a ML-random sequence. In Section 2, we generalize this result to sequences of effective dimension s in various ways. Because a sequence X having effective dimension s implies that the prefix-free complexity of its n-bit prefix X æn is at least sn ́ opnq, it is natural to consider the weakly s-randoms, those sequences X such that KpX ænq ě sn ́Op1q. Theorem 2.5. Every sequence of effective Hausdorff dimension s is coarsely equivalent to a weakly s-random. Along the way to proving this, we pass through the question of how to raise the effective dimension of a given sequence while keeping density of changes at a minimum. If dpX,Y q “ 0, then dimpXq “ dimpY q; so sequences of effective Hausdorff dimension s ă 1 cannot be coarsely equivalent to a ML random sequence. It is natural then to ask, what is the minimal distance required between any sequence and a random? By Theorem 2.5, it is equivalent to ask about distances between sequences of dimension s and dimension 1; and naturally generalising, to ask, for any 0 ď s ă t ď 1, about distances between sequences of dimension s and dimension t. We start with a naive bound. For any X,Y P 2, | dimpY q ́ dimpXq| ď HpdpX,Y qq. This is our Proposition 3.1. Here Hppq “ ́pp log p`p1 ́pq logp1 ́pqq is the binary entropy function defined on r0, 1s. The binary entropy function is used to measure the size of Hamming balls. If V pn, rq “ ř