Ergodic averages with deterministic weights

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 $\theta = (\theta (k) ; k\in \NN)$isaboundedsequenceand$u = (u_k ; k\in \NN)$ a strictly increasing sequence of integers such that for some $\delta <1$ $$S_N(\theta ,u):=\underset{\alpha \in \text{pRR}}{\sup }\mathrm{\mid }\sum _{k=0}^{N-1}\theta (k)\exp (2i\pi \alpha u_k)\mathrm{\mid }=O(N^{\delta })\text{leqno}({\mathcal{H}}_1)$$ i.e., there exists a constant $C$ such that $S_N(\theta ,u)\le C{N}^{\delta }$. We define $\delta (\theta, u)$tobetheinfimumofthe$\delta $satisfying$\H_1$for$\theta $and$u$.