Beresnevich–Yang conjecture for simultaneous approximation on nondegenerate curves

Let $\mathbf{M}_{n,1}$ be the set of nondegenerate curves in $\mathbb{R}^n$; that is, curves $\mathcal{C}=\{\mathbf{f}(x): x\in I,\ \mathbf{f}:I\to \mathbb{R}^n\}$ such that they are continuously differentiable enough times and the set of their derivatives at each point $x\in I$ spans $\mathbb{R}^n$. Let $\tau(\mathbf{M})$ be the supremum of the values $\tau$ such that

$$\dim(S_n(\lambda)\cap\mathcal{M})=\frac{n+1}{\lambda+1}-\operatorname{codim}\mathcal{M}$$

whenever $1/n\leq\lambda<\tau$ for every manifold $\mathcal{M}\in\mathbf{M}$. Beresnevich–Yang conjecture.

$$\tau_{n,1}=\tau(\mathbf{M}_{n,1})=\frac{3}{2n-1}.$$

This conjecture asks for the exact threshold up to which the expected Hausdorff-dimension formula holds uniformly for all nondegenerate curves in $\mathbb{R}^n$. The stated results establish the formula for a smaller interval of exponents close to $1/n$, while the conjectured value $3/(2n-1)$ remains the proposed endpoint.