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 square of area ${\sum }_{n=2}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}-1$. 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 ${\sum }_{n=n_0}^{\infty }\frac{1}{n^{2t}}$. 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 $n_0=1$).

arXiv comment11 pages, 1 figure. Several minor corrections