Ergodic averages with deterministic weights

i.e., there exists a constant C such that SN(θ, u) ≤ CN . We define δ(θ, u) to be the infimum of the δ satisfying H1 for θ and u. About H1, in the case where θ takes its values in U (the set of complex numbers of modulus 1), it is clear that for all sequences θ and u, δ(θ, u) is smaller than or equal to 1 and it is well-known (see [Ka] for example) that it is greater than or equal to 1/2. Few explicit sequences θ are known to have δ(θ, u) strictly smaller than 1. When uk = k, for all k ∈ IN, we know [Ru, Sh] that for the Rudin-Shapiro sequence (and its generalizations [AL, MT]) we have δ(θ, u) = 1/2. For the Thue-Morse sequence δ(θ, u) = (log 3)/(log 4) [G]. When θ is a q-multiplicative sequence we will give a way to construct sequences fulfilling H1. When u is a subsequence of IN we will also give some examples of sequences θ satisfying H1. More attention will be payed to the special case uk = k + vk, k ∈ IN, where v = (vk; k ∈ IN) is non-decreasing with vk = O(k), ε < 1.