Note on normal numbers

Abstract

Introduction* Let a be a real number with fractional part .a when written to base r. Let Y n denote the block of the first n digits in this representation and let N(d, Y n ) denote the number of occurrences of the digit in B . The number a is said to be simply normal to base r if lim for each of the r distinct choices of d. is said to be normal to base r if each of the numbers a, roc, r 2 a, are simply normal to each of the bases r, r 2 , r 3 , . These definitions, due to Emile Borel [1], were introduced in 1909. In 1940 S. S. Pillai The purpose of the present note is to show that a is normal to base r if and only if there exists a set of positive integers m 1 < m % < ra 3 < such that a is simply normal to base r m * for each il>l, and also to show that no finite set of m's will suffice.