Perfectly packing a square by squares of nearly harmonic sidelength

TL;DR

This paper proves that for any exponent t between 1/2 and 1, sufficiently large sets of squares with side lengths decreasing as n^{-t} can be perfectly packed into a larger square, extending previous results.

Contribution

It establishes new packing results for squares with side lengths n^{-t} for t in (1/2, 1), including the case where n_0 is large, advancing the understanding of geometric packing problems.

Findings

Perfect packing for t in (1/2, 1) with large n_0

Extension of previous packing results to squares

Improved understanding of square packing constraints

Abstract

A well known open problem of Meir and Moser asks if the squares of sidelength $1/n$ for $n \geq 2$ 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 $n \geq n_0$ 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 $1/2 < t \leq 2/3$ (in which case one can take $n_0=1$).