Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?:

For any $A\subseteq\left[0,1\right]^k$ with the measure of $A$ satisfied $\mu(A)\geq\gamma$, there exists integer $k_1\geq\sqrt{k}$ such that $A$ can be approximated by a product space (Cartesian product) formed by a subset in $\mathbb{R}^{k-k_1}$ and a hypersquare in $\mathbb{R}^{k_1}$?

i.e. there exist $K_1\subseteq[k],|K_1|=k_1$ and $A_0\subseteq\mathbb{R}^{[k]-K_1}$ and for any $i\in[K_1]$ there exist $U_i\subseteq[0,1]$, such that $A\triangle (A_0\cdot\prod U_i)=(A_0\cdot\prod U_i-A)\cup(A-A_0\cdot\prod U_i)\leq\varepsilon$ for some $\varepsilon\ll\gamma$.