Badly approximable vectors in affine subspaces: Jarn\'ık-type result

Consider irrational affine subspace $A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $\{ξ=(ξ_1,...,ξ_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||qξ_i|| \to \infty,\,\,\,\, q\to \infty\}$ is an $α$-winning set for every $α\in (0,1/2]$

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