Erd\H{ o}s \cite{ MR168482} proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal{ F}$ of (real or complex) analytic functions, such that $\big\{ f(x) \ : \ f \in \mathcal{ F} \big\}$ is countable for every $x$. We strengthen Erd\H{ o}s' result by proving that CH is equivalent to the existence of what we call \emph{ sparse analytic systems} of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim$ on $\mathbb{ R}$ such that any "analytic-anonymous" attempt to predict the map $x \mapsto [x]_\sim$ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman \cite{ MR3552748}.
Categories: math.LO math.CV