Consider the infinite sum $S^2=\sum 1/(a_nb_n)$ with $a_n$ and $b_n$ positive and monotonically increasing, is it always true that we can cover a square of sidelength S with rectangles of sides $1/a_n$ and $1/b_n$ ?
Disregarding cases with trivial obstructions eg. $(1/a_1>S)$.
I am looking for cases where such a tiling is possible or impossible and the tiling or obstruction is nontrivial. Or a result that says its always a trivial tiling (ie a method to construct the tiling) or trivial obstruction