Simultaneous Diophantine approximation to points on the Veronese curve

TL;DR

This work determines the Hausdorff dimension of the set of points on the Veronese curve that are simultaneously well approximable by rationals, across a significant range of the Diophantine exponent $\lambda$. The authors develop a dual-approximation framework, decompose the curve into tractable pieces, and leverage polynomial-discriminant control via Möbius-equivalence classes to bound counts of relevant polynomials. For cubic polynomials ($n=3$) they obtain explicit dimension bounds, including sharp results on $\lambda$-ranges and improved bounds in the mid-range, with a general bound for arbitrary $n$ obtained through resultant methods. A key auxiliary result is a near-optimal bound on the number of cubic polynomials with bounded height and discriminant, which underpins the dimension estimates and highlights a deep connection between Diophantine approximation, algebraic properties, and polynomial height growth.

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 $c(ε) H^{2/3 + ε} D^{5/6}$.

Theorems & Definitions (21)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 1
• Lemma 1
• Lemma 2
• Lemma 3