On to my sixth attempt.  How about the 
[Littlewood conjecture](https://en.wikipedia.org/wiki/Littlewood_conjecture)?

Does this have a known reduction to an arithmetical sentence?

Encouragingly, [this comment](https://gowers.wordpress.com/2009/11/17/problems-related-to-littlewoods-conjecture-2/#comment-4395) would seem to suggest that it doesn't, or at least didn't in 2009.