Simultaneous Diophantine approximation on the three-dimensional Veronese curve: the complete Hausdorff dimension story

Abstract

We determine the Hausdorff dimension of the intersection of the set $\mathcal W_3(λ)$ of simultaneously $λ$-well approximable points with the Veronese curve $\mathcal V_3 \subset \mathbb R^3$ for $λ\ge3/5$, thus completing the full range of $λ$ values. Precisely, we show that for $λ\ge 1/3$, $$\dim\bigl(\mathcal W_3(λ)\cap\mathcal V_3\bigr)= \max\left\{\frac{2-2λ}{1+λ}, \frac{2}{3(1+λ)}\right\}.$$ To the best of the authors' knowledge, this makes $\mathcal V_3$ the first nondegenerate, non-planar curve with a completely determined Hausdorff dimension theory.