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 ${\mathcal{W}}_3(\lambda )$ of simultaneously $\lambda$-well approximable points with the Veronese curve ${\mathcal{V}}_3\subset {\mathbb{R}}^3$ for $\lambda \ge 3/5$, thus completing the full range of $\lambda$ values. Precisely, we show that for $\lambda \ge 1/3$, $$\dim ({\mathcal{W}}_3(\lambda )\cap {\mathcal{V}}_3)=\max \left\{\frac{2-2\lambda }{1+\lambda },\frac{2}{3(1+\lambda )}\right\}.$$ 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.

27 pages