[21] Let = 12 ... be any infinite binary sequence. Define = 12 ... by i = i + i+1, i 1.

[21] Let ω = ω1ω2 ... be any infinite binary sequence. Define

ζ = ζ1ζ2 ... by ζi = ωi + ωi+1, i ≥ 1. This gives a sequence over the alphabet {0, 1, 2}. Show that ζ is not random in the sense of Martin-L¨of under the uniform measure (extend the definition from binary to ternary sequences).

Comments. Hint: The blocks 02 and 20 do not occur in ζ. Source: [R.

von Mises, Probability, Statistics and Truth, Dover, 1981].