Simultaneous Diophantine approximation on the three dimensional Veronese curve

We compute the Hausdorff dimension of the set of simultaneously $\lambda$-well approximable points on the Veronese curve in ${\text{RR}}^3$ for $1\mathrm{/}3\le \lambda \le 3\mathrm{/}5$. 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 ${\text{VVV}}_3$ the first nondegenerate curve in ${\text{RR}}^n$, $n\ge 3$, to confirm the lower bound part of this conjecture.