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arXiv · 2608.25335

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

Dmitry Badziahin·Nikita Shulga
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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 $\lambda \ge3/5$, thus completing the full range of $\lambda$ values. Precisely, we show that for $\lambda\ge 1/3$, $$ \dim\bigl(\mathcal W_3(\lambda)\cap\mathcal V_3\bigr)= \max\left\{\frac{2-2\lambda}{1+\lambda}, \frac{2}{3(1+\lambda)}\right\}. $$ 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.

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Dmitry Badziahin, Nikita Shulga. 2026-08-26. Simultaneous Diophantine approximation on the three-dimensional Veronese curve: the complete Hausdorff dimension story. https://arxiv.org/abs/2608.25335

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