  The purpose of this paper is to study ergodic averages with deterministic
weights. More precisely we study the convergence of the ergodic averages of the
type $\frac{1}{N} \sum_{k=0}^{N-1} \theta (k) f \circ T^{u_k}$ where $\theta =
(\theta (k) ; k\in \NN)$ is a bounded sequence and $u = (u_k ; k\in \NN)$ a
strictly increasing sequence of integers such that for some $\delta<1$ $$ S_N
(\theta, u) := \sup_{\alpha \in \pRR} | \sum_{k=0}^{N-1} \theta (k) \exp (2i\pi
\alpha u_k) | = O (N^{\delta}) \leqno{({\cal H}_1)} $$ i.e., there exists a
constant $C$ such that $S_N (\theta, u) \leq C N^{\delta} $. We define $\delta
(\theta, u)$ to be the infimum of the $\delta $ satisfying $\H_1$ for $\theta $
and $u$.
