Sparse analytic systems

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

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Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Cody, Brent, Cox, Sean, Lee, Kayla
Format: Article
Language:English
Subjects:
Analytic functions
Entire functions
Equivalence
Mathematical analysis
Systems analysis
Online Access:Get full text
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Summary: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 \(\big\{ f(x) \ : \ f \in \mathcal{F} \big\}\) 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}.
ISSN:2331-8422