Uncanny Subsequence Selections That Generate Normal Numbers

Abstract

Abstract Given a real number 0.a1a2a3 . . . that is normal to base b, we examine increasing sequences ni so that the number 0.an1an2an3 . . . are normal to base b. Classically, it is known that if the ni form an arithmetic progression, then this will work. We give several more constructions including ni that are recursively defined based on the digits ai. Of particular interest, we show that if a number is normal to base b, then removing all the digits from its expansion which equal (b−1) leaves a base-(b−1) expansion that is normal to base (b − 1)