Question 55
The infinite grid of lines of the form $\mathbb{R} \times \{m\}$ and $\{m\} \times \mathbb{R}$, where $m$ runs through all integers, subdivide the Euclidean plane $\mathbb{R} \times \mathbb{R}$ into $1 \times 1$ cells. Let $S$ be the set-theoretic union of a finite number of such cells, and let $a$ be a positive real number less than or equal to $1/4$. Show that $S$ can be covered by a finite number of squares satisfying the following three conditions simultaneously:
(1) Each square in the cover is an array of $1 \times 1$ cells;
(2) The squares in the cover have pairwise disjoint interiors; and
(3) For each square $Q$ in the cover, the ratio of the area of $S \cap Q$ to the area of $Q$ is at least $a$ and at most $a\lfloor a^{-1/2} \rfloor^2$.
(1) Each square in the cover is an array of $1 \times 1$ cells;
(2) The squares in the cover have pairwise disjoint interiors; and
(3) For each square $Q$ in the cover, the ratio of the area of $S \cap Q$ to the area of $Q$ is at least $a$ and at most $a\lfloor a^{-1/2} \rfloor^2$.