Simultaneous Diophantine approximation to points on the Veronese curve

Abstract

We compute the Hausdorff dimension of the set of simultaneously $q^{-λ}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ 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 $\frac13$ and $\frac12$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $λ\le \frac{2}{n}$. 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 $c(ε) H^{2/3 + ε} D^{5/6}$.