Thanks for pointing it out. I have no solution to either case. @Joe Bebel

Thanks. In your example, do you mean $\epsilon'=\sqrt{\epsilon}$? That is good for the purpose. Here being dimension-independent means that $\epsilon'=f(\epsilon)$, which only depends on $\epsilon$. @Anthony Quas

Thanks a lot. I am more interesting in the relation between the number of rectangles and closeness of $S$ and a rectangle, which is more or less described by factor $r$.

Sorry that I didn't make the definition of rectangles clear. You pick several rows, maybe not contiguous, and several columns, maybe also not contiguous, then all the squares in the crossing forms a rectangle. So all the black squares in the chessboard can be covered by only two rectangles