A. Baker's conjecture for approximation on the Veronese curve

Let $\psi:\mathbb{N}\to\mathbb{R}^+$ be decreasing, let $A_n(\psi):=\{\mathbf{x}\in\mathbb{R}^n:\exists^\infty\mathbf{q}\in\mathbb{Z}^n\text{ such that }\|\mathbf{q}\cdot\mathbf{x}\|<\psi(|\mathbf{q}|)\}$, and let $\mathcal{M}=\mathcal{V}_n$ with induced Lebesgue measure $|\cdot|_\mathcal{M}$. A. Baker's conjecture.

$$|A_n(\psi)\cap\mathcal{M}|_\mathcal{M}=\begin{cases}0&\text{if }\displaystyle\sum_{q=1}^{\infty}q^{n-1}\psi(q)<\infty,\\ |\mathcal{M}|_\mathcal{M}&\text{if }\displaystyle\sum_{q=1}^{\infty}q^{n-1}\psi(q)=\infty.\end{cases}$$

Here the full-measure alternative means that the complement in $\mathcal{M}$ has measure zero. The conjecture was subsequently proved for all non-degenerate submanifolds, so it is solved.