Fourier dimension for exactly approximable matrices

We determine the Fourier dimension of exactly approximation set for all dimensions. More precisely, let $m,n\ge 1$, $\mathit{\theta }\in {\mathbb{R}}^m$, and $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$ be a non-increasing approximate function. Denote by ${\text{Exact}}_{m,n}(\psi ,\mathit{\theta })$ the set of $m\times n$ matrices which are inhomogeneously $\psi$-approximable with shift $\mathit{\theta }$, but not $c\psi$-approximable for any fixed $0<c<1$. We prove that $${\dim }_{\text{F}}{\text{Exact}}_{m,n}(\psi ,\mathit{\theta })=\frac{2n}{1+{\tau }_{\psi }},\text{ whenever }{\tau }_{\psi }>\frac{n}{m},$$ where ${\tau }_{\psi }$ is the lower order of $\psi$ at infinity.