We study the convergence of the ergodic averages 1 N ∑ k = 0 N - 1 θ ( k ) f ∘ T u k where ( θ ( k ) ) k ∈ ℕ is a bounded sequence and ( u k ) k ∈ ℕ a strictly increasing sequence of integers such that Sup α ∈ ℝ | ∑ k = 0 N - 1 θ ( k ) exp ( 2 i π α u k ) | = O ( N δ ) for some δ < 1 . Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q -multiplicative sequence.