Sparse analytic systems

Erdős [7] 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 eq...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2023-07, Vol.11, Article e58
Main Authors: Cody, Brent, Cox, Sean, Lee, Kayla
Format: Article
Language:English
Subjects:
03E25
03E50
26E05
30D20
Analytic functions
Equivalence
Foundations
Mathematical analysis
Systems analysis
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Summary:Erdős [7] 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 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 [2].
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2023.54