  Consider irrational affine subspace $ A\subset \mathbb{R}^d$ of dimension
$a$. We prove that the set $$ \{\xi =(\xi_1,...,\xi_d) \in {A}:\,\,\, \
q^{1/a}\cdot \max_{1\le i \le d} ||q\xi_i|| \to \infty,\,\,\,\, q\to \infty\}
$$ is an $\alpha$-winning set for every $\alpha \in (0,1/2]$
