The dimension of ergodic random sequences

TL;DR

This paper proves that for Martin-Löf random sequences with respect to a computable ergodic measure, both the effective dimension and the strong effective dimension equal the measure's entropy, confirming a long-standing conjecture.

Contribution

It establishes that the effective dimension of Martin-Löf random sequences also equals the entropy, extending previous results on strong effective dimension.

Findings

Effective dimension equals entropy for Martin-Löf random sequences.

Extended Birkhoff's ergodic theorem for Martin-Löf randomness.

Confirmed the conjecture linking effective dimension and entropy.

Abstract

Let \mu be a computable ergodic shift-invariant measure over the Cantor space. Providing a constructive proof of Shannon-McMillan-Breiman theorem, V'yugin proved that if a sequence x is Martin-L\"of random w.r.t. \mu then the strong effective dimension Dim(x) of x equals the entropy of \mu. Whether its effective dimension dim(x) also equals the entropy was left as an problem question. In this paper we settle this problem, providing a positive answer. A key step in the proof consists in extending recent results on Birkhoff's ergodic theorem for Martin-L\"of random sequences.