Sparse analytic systems

Abstract

Erdős \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 $\{f(x):f\in \mathcal{F}\}$ is countable for every $x$. We strengthen Erdős' 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}.

to appear in Forum of Mathematics, Sigma