On the Asymptotic Behavior of Guessing Sequences
Tom Benhamou, Sean LeClair
TL;DR
Problem addressed: characterize the asymptotic behavior and topological size of the set of reals guessed by diamond-style sequences tied to ultrafilters on $\\omega$. Approach: establish a probabilistic threshold via the Borel-Cantelli lemmas and realize explicit guessing sequences using random walks (and Polya's recurrence) to drive $G(\\vec{\\mathcal{A}})$ to probability one, linking to Tukey-top behavior through $\\diamondsuit^-(\\mathcal{U})$. Key contributions: a sharp threshold given by $\\sum_{n=0}^{\\infty} \\frac{\\pi(n)}{2^{n}}=\\infty$ guaranteeing probability one for $G(\\vec{\\mathcal{A}})$, concrete constructions for growth like $\\binom{n}{\\frac{n}{2}}$ and $\\binom{\\frac{n}{2}}{\\frac{n}{4}}^{2}$ (via random walks), and results on meager vs non-meager guessed sets under different forcing. Significance: clarifies the size and regularity properties of guessed reals and informs the understanding of ultrafilter guessing principles within the Tukey framework.
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 $π$ which infinitely guesses a probability one set is equivalent to the divergence of $\sum_{n=0}^{\infty}\frac{π(n)}{2^n}$. 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}.