A metric discrepancy result with given speed

It is known that the discrepancy D_N{ kx} of the sequence { kx} satisfies ND_N{ kx} = O((log N) (log log N) ^{1 + ε}) a.e. for all ε> 0 , but not for ε= 0. For n_k= θ^k, θ> 1 we have ND_N{ n_kx} ≦ (Σ _θ+ ε) (2 Nlog log N) ^{1 / 2} a.e. for some 0 < Σ _θ< ∞ and N≧ N₀ if ε> 0 , but not for ε< 0. In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed Ψ (N) between (log N) (log log N) ^{1 + ε} and (Nlog log N) ^{1 / 2} and for any Σ > 0 , there exists a sequence { n_k} of positive integers such that ND_N{ n_kx} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums.