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We state a norm sensitive Diophantine approximation problem arising form the work of the first-named author and Nick Wadleigh. The work there dealt with an approximation problem in terms of a supremum norm. Here we adapt the setting to an $\ell^2$ norm. Specifically, for a given approximation function $\psi$ we establish a zero-one law on the set of $\alpha \in \mathbb{R}$ for which the inequality $\left(\frac{\alpha q -p}{\psi(t)}\right)^2 + \left(\frac{q}{t}\right)^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions for all large enough $t$. Several related questions and higher-dimensional generalizations are also discussed.

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm