MathDB

Divide the plane into

1

1

squares formed by the lattice points. 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: 1) Each square in the cover is an array of

1

1

cells 2) The squares in the cover have pairwise disjoint interios and 3)For each square

Q

in the cover the ratio of the area

S \cap Q

to the area of Q is at least

a

and at most

a {(\lfloor a^{-1/2} \rfloor)} ^2

Problem Statement