Geodesic
Sparse analytic systems
Forum of Mathematics, Sigma, Tome 11 (2023)

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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]. 
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     title = {Sparse analytic systems},
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Brent Cody; Sean Cox; Kayla Lee. Sparse analytic systems. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.54

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