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Journal of the London Mathematical Society 1999 59(1):50-64; doi:10.1112/S0024610798006954
© 1999 by London Mathematical Society
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© The London Mathematical Society

On Irregularities of Distribution II

R. C. Baker

Department of Mathematics, Brigham Young University Provo, UT 84602, USA

Received 17 December 1996. Revision received 20 October 1997.

Let a=(a1, a2, a3, ...) be an arbitrary infinite sequence in U=[0, 1). Let


Formula

Van der Corput [5] conjectured that d(a, n) (n=1, 2, ...) is unbounded, and this was proved in 1945 by van Aardenne-Ehrenfest [1]. Later she refined this [2], obtaining


Formula

for infinitely many n. Here and later c1, c2, ... denote positive absolute constants.

In 1954, Roth [8] showed that the quantity


Formula

is closely related to the discrepancy of a suitable point set in U2.


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