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 $R^n$ for λ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given for a wider range of λ between $\frac{1}{3}$ and $\frac{1}{2}$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $\lambda ⩽\frac{2}{n}$. 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 $c\left(ϵ\right)H^{2/3+ϵ}{D}^{5/6}$.