Sparse analytic systems

TL;DR

This paper strengthens Erdős's result by showing the Continuum Hypothesis is equivalent to the existence of sparse analytic systems, and uses these to construct a complex equivalence relation with properties that challenge analytic prediction.

Contribution

It introduces the concept of sparse analytic systems and demonstrates their equivalence to CH, providing new tools for analyzing analytic functions and equivalence relations.

Findings

CH is equivalent to the existence of sparse analytic systems

Constructed an equivalence relation resistant to analytic prediction

Provided a negative answer to a question on analytic-anonymous prediction

Abstract

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}.