Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength

Abstract

A well-known open problem of Meir and Moser asks if the squares of sidelength 1/n for $n\ge 2$ can be packed perfectly into a rectangle of area ${\sum }_{n=2}^{\infty }{n}^{-2}={\pi }^2/6-1$. In this paper we show that for any $1/2<t<1$, and any $n_0$ that is sufficiently large depending on t, the squares of sidelength $n^{-t}$ for $n\ge n_0$ can be packed perfectly into a square of area ${\sum }_{n=n_0}^{\infty }{n}^{-2t}$. This was previously known (if one packs a rectangle instead of a square) for $1/2<t\le 2/3$ (in which case one can take $n_0=1$).

Keywords: Harmonic series; Meir–Moser problem; Square packing.