How robustly can you predict the future?

Abstract

Hardin and Taylor \cite{MR2384262} proved that any function on the reals -- even a nowhere continuous one -- can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed in \cite{MR3100500} that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman \cite{MR3552748}, who provided upper and lower frontiers (in the subgroup lattice of $\text{Homeo}^+(\mathbb{R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of H\"older's Theorem (that every Archimedean group is abelian).