J Austral Math Soc Ser A 49 pp264--272, 1990.

Notes on Uniform Distribution Modulo One

G. Myerson and A. D. Pollington

Abstract

We exhibit a sequence (u_n) which is not uniformly distributed modulo one even though for each fixed integer k ³ 2 the sequence (ku_n) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit s sequence (u_n) which is u.d. (mod 1) even though no subsequence of the form (u_kn+j) is u.d. (mod 1) for any k ³ 2. We prove that, if the subsequences (u_kn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set P, then (u_n) is u.d. (mod 1) if the sum of the reciprocals of the primes in P diverges. We show that this reuslt is the best possible of its type.

1980 AMS Subject Classification (1985 Revision): 11K06

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Authors

G. Myerson

School of Mathematics, Physics, Computing and Electronics, Macquarie University, Australia 2109.

A. D. Pollington

Department of Mathematics, Brigham Young University, Provo, Utah 84602, U.S.A.