On Modifying Normal Numbers

A real number x is called a normal number (in a base β ≥ 2) if all possible blocks of digits appear with the same asymptotic frequency in the β-ary expansion of x. This notion was introduced by E. Borel in 1909, who proved that almost all real numbers (in the sense of Lebesgue measure) are normal numbers. There exist many constructions of normal numbers, most of which are based on assembling appropriate function values to obtain the digital expansion of x. Volkmann showed that normal numbers constructed in this way can be slightly modified without losing the normality property. In the present note, we generalize the result of Volkmann. Let C(β) denote the set of reals which have a β-ary representation with asymptotic frequency of non-zero digits equal to 0. We prove that if x is normal in base β, then x + qy is also normal in the same base, for any y ∈ C(β) and q ∈ Q. Likewise, let S denote a set of positive integers of asymptotic density 0. We prove that a normal number x retains the normality property, if an arbitrary digit is inserted between positions j and j + 1 for each j ∈ S.