Simultaneous Diophantine approximation to points on the Veronese curve

We compute the Hausdorff dimension of the set of simultaneously $q^{-\lambda }$-well approximable points on the Veronese curve in ${\mathbb{R}}^n$ for $\lambda$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given for a wider range of $\lambda$ between $\frac{1}{3}$ and $\frac{1}{2}$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $\lambda \le \frac{2}{n}$. In the course of the proof we establish that the number of cubic polynomials of height at most $H$ and non-zero discriminant at most $D$ is bounded from above by $c(\epsilon )H^{2\mathrm{/}3+\epsilon }{D}^{5\mathrm{/}6}$.