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Let $\alpha,\beta \in \mathbb{R}$. Is it true that \[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\] where $\|x\|$ is the distance from $x$ to the nearest integer?
The infamous Littlewood conjecture.
Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there integers $x,y,z$ such that \[\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?\]
Originally a conjecture due to Oppenheim. Davenport and Heilbronn [DaHe46] solve the analogous problem for quadratic forms in 5 variables.

This is true, and was proved by Margulis [Ma89].