论文标题
稀疏分析系统
Sparse analytic systems
论文作者
论文摘要
erdős\ cite {MR168482}证明，连续假设（CH）等于（（真实或复杂）分析函数的一个无数的家庭$ \ MATHCAL {F} $的存在，以便每个分析功能$ x $。我们通过证明CH等效于我们所谓的函数\ emph {稀疏分析系统}的存在来增强ERD的结果。我们使用此类系统来构建CH，假设$ \ Mathbb {r} $上的等价关系$ \ sim $，以便任何“分析性匿名”试图预测地图$ x \ mapsto [x] _ \ sim $几乎在任何地方都必须失败。这为Bajpai-Velleman \ Cite {MR3552748}的问题提供了一个始终如一的负面答案。
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}.
