On the Asymptotic Behavior of Guessing Sequences

Abstract

We continue the study of probabilistic and topological properties of the set of reals that are being guessed by a diamond sequence from \cite{Benhamou_Wu}. We show that the existence of sequence of a asymptotic growth $\pi$ which infinitely guesses a probability one set is equivalent to the divergence of $\sum_{n=0}^{\infty}\frac{\pi(n)}{2^n}$. We then provide concrete examples for guessing sequences of certain low asymptotic growth using random walks. Finally, we show that the ultrafilter construction from \cite{Benhamou_Wu} always yield an ultrafilters and a sequence which guesses a meager set, while a simple construction using Cohen forcing gives a non-meager set of guessed reals. These results answer \cite[Question 6.13]{Benhamou_Wu} and partially addresses \cite[Question 6.8]{Benhamou_Wu}.