I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.


E.g.


* "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.
* "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.


No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.


In technical terms, I'm looking for a $\Pi^1\_1$ sentence that we don't know how to reduce to an arithmetical sentence.


I'd also like it to be easy to state and obviously $\Pi^1\_1$, so that it can be included in a logic paper without requiring much explanation.



Open problem in analysis with just one quantifier?
The
[Littlewood conjecture](https://en.wikipedia.org/wiki/Littlewood_conjecture) is an example that meets all my requirements.


It is easy to state and obviously $\Pi^1\_1$. Furthermore [this comment by Christian Reiher](https://gowers.wordpress.com/2009/11/17/problems-related-to-littlewoods-conjecture-2/#comment-4395) gives me confidence that it has no known reduction to an arithmetical sentence. (Hopefully it even lacks a known reduction to a $\Sigma^1\_1$ sentence.)


