Predictive Sets

A set $P\subset \mathbb{N}$ is called predictive if for any zero entropy finite-valued stationary process $(X_i{)}_{i\in \mathbb{Z}}$, $X_0$ is measurable with respect to $(X_i{)}_{i\in P}$. We know that $\mathbb{N}$ is a predictive set. In this paper we give sufficient conditions and necessary ones for a set to be predictive. We also discuss linear predictivity, predictivity among Gaussian processes and relate these to Riesz sets which arise in harmonic analysis.