Foliations and Diffeomorphism Groups — Question 22 (Freedman)

Definitions and mathematical statement

The supplied source asks whether, for sufficiently small real $\epsilon>0$, \[ g_{\mathscr{E}(p)}(\epsilon)\geq \frac{c\lvert\mathscr{E}(p)\rvert}{\epsilon^2+O(\epsilon^3)} \] holds for some constant $c$, with $c=2\pi$ suggested as a possibility.

The supplied record does not define the parameter $p$, the object $\mathscr{E}(p)$, the meaning of $\lvert\mathscr{E}(p)\rvert$, or the function $g_{\mathscr{E}(p)}$. Their domains and the quantification over $p$ are therefore unresolved. It also does not specify whether $c$ must be positive, which parameters it may depend on, or whether the smallness threshold for $\epsilon$ is uniform in $p$. The denominator is reproduced in the source's asymptotic notation: no particular remainder function or uniformity of its $O(\epsilon^3)$ bound is specified. These missing definitions and qualifications prevent a fully specified mathematical proposition from being extracted from this record alone.

Short English statement

Question 22, attributed in the supplied context to Freedman, asks for the displayed lower bound for small $\epsilon>0$, possibly with $c=2\pi$. The objects involved and several quantifiers remain unspecified. The registry labels the question open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $800081$, exact source problem code catalogue entry. The supplied record lists Nariman and Yazdi as proposers, gives the year 2025, and labels the problem open.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 22. Supplied source URL: https://arxiv.org/abs/2503.18274. The supplied background attributes the question or its context to Freedman.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no source visit or independent verification was performed for this draft.