Normality of the Ehrenfeucht-Mycielski Sequence

TL;DR

This paper investigates the balance and normality properties of the Ehrenfeucht-Mycielski sequence, providing detailed proofs of weaker forms of the balance conjecture and extending the analysis to simple normality for binary strings of length 2.

Contribution

It offers detailed proofs of weaker balance conjectures and extends the analysis to simple normality for length-2 binary strings, advancing understanding of the sequence's randomness properties.

Findings

Proved weaker forms of the balance conjecture.

Extended balance analysis to simple normality for length-2 words.

Provided detailed proofs for previously stated lemmas.

Abstract

We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that 1 and 0 occur equally often in it. Our contribution is twofold. First, we study weaker forms of the conjecture proved in the past and lay out detailed proofs for many lemmas which were stated without proofs. Secondly, we extend the claim of balance to that of normality and prove a weaker form of simple normality to word length 2.