Given \(\gamma>0\) and \(a=(a_1,a_2, \ldots, a_d) \in \mathbb{R}^d_{>0}\). A measurable set \(S \subset \mathbb{R}^d\) is called \((\gamma, a)\)-thick if for every \(x\in \mathbb{R}^d\)
\begin{equation*} |S \cap (x+[0,a_1] \times \cdots \times [0,a_d])| \ge \gamma \prod_{i=1}^{d} a_i. \end{equation*}The thick set are colored black. A \(\gamma\)-portion of the the red rectangle is covered \(S\).