On the Asymptotic Behavior of Guessing Sequences

TL;DR

This paper investigates the properties of guessing sequences for reals, establishing conditions for their success, providing examples, and exploring ultrafilter constructions to understand their measure and topological characteristics.

Contribution

It characterizes the divergence condition for guessing sequences with certain growth, offers concrete examples using random walks, and compares ultrafilter and Cohen forcing constructions.

Findings

Guessing sequences with asymptotic growth  diverge in sum for probability one guessing.

Concrete examples of low-growth guessing sequences are constructed via random walks.

Ultrafilter constructions always produce guessed sets that are meager, while Cohen forcing can produce non-meager guessed sets.

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}.