Consider the set of simultaneously $\lambda$-well approximable points in $\mathbb{R}^n$, i.e. these are the points $\matbf{x}$ such that $||\mathbf{x} - \mathbf{p}/q|| < q^{-1-\lambda}$ for infinitely many rational vectors $\mathbf{p}/q$. Measuring the set of such points on manifolds is one of the most intricate problems in metric theory of Diophantine approximation. Unlike the dual case of well approximable linear forms, the results here are known to depend on a manifold. For example, for large enough $\lambda$ some of the manifolds do not contain simultaneously $\lambda$-well approximable points at all, while for the others the set of such points always has positive Hausdorff dimension. In his landmark work, Beresnevich provided a lower bound on the Hausdorff dimension of the set of simultaneously $\lambda$-well approximable points on non-degenerate curves as soon as $1/n \le \lambda\le 3/(2n-1)$. I will talk about the recent result which shows that Beresnevich's bound is sharp for the three-dimensional Veronese curve $\{x, x^2, x^3\}$ and for all $\lambda$ between 1/3 and 3/5:
