Normality of the Ehrenfeucht-Mycielski Sequence

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. 1 ar X iv :1 71 0. 01 32 5v 1 [ cs .D M ] 3 O ct 2 01 7