Simultaneous Diophantine approximation to points on the Veronese curve

Abstract

We compute the Hausdorff dimension of the set of simultaneously $q^{-\hat{I}»}$-well approximable points on the Veronese curve in ${\mathbb{R}}^n$ for $\hat{I}»$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given for a wider range of $\hat{I}»$ between $\frac{1}{3}$ and $\frac{1}{2}$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $\hat{I}»\le \frac{2}{n}$. 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 $c(\hat{I}\mu )H^{2/3+\hat{I}\mu }{D}^{5/6}$.

31 pages. Better bounds on lambda and better upper bound on the Hausdorff dimension for n=3 are provided in the updated version of the paper