Normal numbers and selection rules

AbstractGiven anormal number x=0,x1x2 ··· to base 2 and aselection rule S ⊂{0, 1}*=∪n=0/t8{0, 1}n, we define a subsequencex,=0, $$\chi _{t_1 } \chi _{t_2 } $$ ·· where {t1<t2<···}={i; x1x2···xi−1 εS}.xs is called aproper subsequence ofx if limi/∞ti/</t8. A selection ruleS is said topreserve normality if for any normal numberx such thatxs is a proper subsequence ofx, xs is also a normal number. We prove that ifS/∼s is a finite set, where ∼s is an equivalence relation on {0, 1}* such that ξ ∼sη if and only if {ζ; ξζ εS}={ζ; ηζ εS}, thenS preserves normality. This is a generalization of the known result in finite automata case, where {0, 1}*/∼s is a finite set (Agafonov [1]).