Simultaneous Diophantine approximation on the three dimensional Veronese curve

TL;DR

This paper determines the Hausdorff dimension of points on the Veronese curve in three dimensions that are well approximable within a specific range, confirming a conjecture for this nondegenerate curve.

Contribution

It computes the Hausdorff dimension for the Veronese curve in  dimensions within a predicted range, confirming the lower bound part of a conjecture.

Findings

Confirmed the lower bound of the conjecture for the Hausdorff dimension

First nondegenerate curve in  in  to verify this conjecture

Established dimension results for -dimensional Veronese curve

Abstract

We compute the Hausdorff dimension of the set of simultaneously $\lambda$-well approximable points on the Veronese curve in $\RR^3$ for $1/3\le \lambda\le 3/5$. This range for $\lambda$ was predicted in the conjecture of Beresnevich and Yang from~\cite{ber_yan_2023}. To the best of the author's knowledge, this makes $\VVV_3$ the first nondegenerate curve in $\RR^n$, $n\ge 3$, to confirm the lower bound part of this conjecture.