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

We characterize the class of non‐decreasing functions f such that for any increasing sequence { n k } of integerslim N → ∞Σ k ⩽ N cos 2 π n k ωN 1 / 2f ( N )= 0a.e. Combined with an inequality of Koksma our results prove the existence of an increasing sequence { n k } of integers such that the discrepancy D N (ω) of the sequence { n k ω} mod 1 satisfieslim sup N → ∞( N / log N ) 1 / 2D 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 { n k } of integers withn k + 1n k⩾ 1 + ρ kfork ⩾ 1 ,where {ρ k } is a given non‐increasing sequence of real numbers, for which the discrepancy D N (ω) of { n k ω} mod 1 fails the upper half of the law of the iterated logarithm by a factor of log log (1/ ρ N ).