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Journal of the London Mathematical Society 1994 50(3):454-464; doi:10.1112/jlms/50.3.454
© 1994 by London Mathematical Society
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© The London Mathematical Society

The Size of Trigonometric and Walsh Series and Uniform Distribution Mod 1

István Berkes and Walter Philipp

Mathematical Institute, Hungarian Academy of Sciences Reáltanoda u 13–15, H-1053 Budapest, Hungary
Department of Statistics, University of Illinois 725 S. Wright Street, Champaign, Illinois 61820, USA

Received 19 February 1993.

We characterize the class of non-decreasing functions f such that for any increasing sequence {nk} of integers

Formula
Combined with an inequality of Koksma our results prove the existence of an increasing sequence {nk} of integers such that the discrepancy DN({omega}) of the sequence {nk{omega}} mod 1 satisfies

Formula
This disproves conjectures of Erdos [9] and R. C. Baker [2]. We prove the analogue of the above result for the Walsh system, thereby solving a problem of Révész [18]. Finally we solve a problem raised in [16], by showing the existence of sequences {nk} of integers with

Formula
where {{rho}k} is a given non-increasing sequence of real numbers, for which the discrepancy DN({omega}) of {nk{omega}} mod 1 fails the upper half of the law of the iterated logarithm by a factor of log log (1/{rho}N).

Dedicated to Pál Révész on the occasion of his 60th birthday


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