Normal numbers and selection rules

Abstract

Given anormal number x=0,x ₁ x ₂ ··· to base 2 and aselection rule S ⊂{0, 1}*=∪ ^/t8_n=0 {0, 1}ⁿ, we define a subsequencex,=0,$\chi _{t_1 } \chi _{t_2 }$·· where {t ₁<t ₂<···}={i; x ₁ x ₂···x _i−1 εS}.x _s is called aproper subsequence ofx if lim_i/∞ ti/</t8. A selection ruleS is said topreserve normality if for any normal numberx such thatx _s is a proper subsequence ofx, x _s 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 ∼ _s η 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]).