Badly approximable vectors in affine subspaces: Jarn\'{\i}k-type result

Consider irrational affine subspace $A\subset {\mathbb{R}}^d$ of dimension $a$. We prove that the set $$\{\xi =({\xi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\xi }_d)\in A:q^{1\mathrm{/}a}\cdot \underset{1\le i\le d}{\max }\mathrm{\mid }\mathrm{\mid }q{\xi }_i\mathrm{\mid }\mathrm{\mid }\to \mathrm{\infty },q\to \mathrm{\infty }\}$$ is an $\alpha$-winning set for every $\alpha \in (0,1\mathrm{/}2]$