Journal of the Australian Mathematical Society (Series A)

Research Article

Notes on uniform distribution modulo one

G. Myersona1 and A. D. Pollingtona2

a1 School of Mathematics, Physics, Computing and Electronics, Macquarie University Australia 2109

a2 Department of Mathematics Brigham Young University Provo, Utah 84602, U.S.A.

Abstract

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

1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision)

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