November 18
Brent Cody, Virginia Commonwealth University
Sparse analytic systems

Given a set $S$, an $S$-predictor $\mathcal{P}$ is a function that takes as inputs functions of the form $f:(-\infty,t)\rightarrow S$, where $t\in\mathbb{R}$, and outputs a guess $\mathcal{P}(f)$ for what $f(t)$ 'should be.' An $S$-predictor is good if for all total functions $F:\mathbb{R}\to S$ the set of $t\in\mathbb{R}$ for which the guess $\mathcal{P}(F\upharpoonright(-\infty,t))$ is not equal to $F(t)$ has measure zero. Hardin and Taylor proved that every set $S$ has a good $S$-predictor and they raised various questions asking about the extent to which the prediction $\mathcal{P}(f)$ made by a good predictor might be invariant after precomposing $f$ with various well-behaved functions - this leads to the notion of 'anonymity' of good predictors under various classes of functions. Bajpai and Velleman answered several of Hardin and Taylor's questions and asked: Does there exist, for every set $S$, a good $S$-predictor that is anonymous with respect to the strictly increasing analytic homeomorphisms of $\mathbb{R}$? We provide a consistently negative answer to this question by strengthening a result of Erdős, which states that the Continuum Hypothesis is equivalent to the existence of an uncountable family $F$ of (real or complex) analytic functions, such that $\{f(x):f\in F\}$ 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. This is joint work with Sean Cox and Kayla Lee.