Note on the Number of Balanced Independent Sets in the Hamming Cube

Let $Q_d$ be the $d$-dimensional Hamming cube and $N=|V(Q_d)|=2^d$. An independent set $I$ in is called balanced if contains same number of even odd vertices. We show that logarithm sets is
 \[(1-\Theta(1/\sqrt d))N/2.\]
 The key ingredient proof an improved version "Sapozhenko's graph container lemma".