Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength

A well-known open problem of Meir and Moser asks if the squares of sidelength 1/n for n ≥ 2 can be packed perfectly into a rectangle of area ∑n=2∞n-2 = π2/6 - 1. In this paper we show that for any 1/2 < t < 1, and any n0 that is sufficiently large depending on t, the squares of sidelength n-t for n ≥ n0 can be packed perfectly into a square of area ∑n=n0∞n-2t. This was previously known (if one packs a rectangle instead of a square) for 1/2 < t ≤ 2/3 (in which case one can take n0 = 1).