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

Abstract

We characterize the class of non-decreasing functions f such that for any increasing sequence {nk} of integers lim N → ∞ Σ k ⩽ N cos 2 π n k ω N 1 / 2 f ( N ) = 0 a.e. Combined with an inequality of Koksma our results prove the existence of an increasing sequence {nk} of integers such that the discrepancy DN(ω) of the sequence {nkω} mod 1 satisfies lim sup N → ∞ ( N / log N ) 1 / 2 D N ( ω ) = ∞ a.e. This disproves conjectures of Erdős [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 n k + 1 n k ⩾ 1 + ρ k for k ⩾ 1 , where {ρk} is a given non-increasing sequence of real numbers, for which the discrepancy DN(ω) of {nkω} mod 1 fails the upper half of the law of the iterated logarithm by a factor of log log (1/ρN).