Slack-Pack algorithm for Meir-Moser packing problem

TL;DR

This paper introduces the Slack-Pack algorithm, a novel tiling method for the Meir-Moser problem, which manages gaps and controls empty space ratios, extending Tao's recent results on tiling with scaled rectangles.

Contribution

The paper presents the Slack-Pack algorithm, enabling controlled tiling with gaps, and provides arguments supporting the extension of Tao's tiling results to the case t=1.

Findings

The Slack-Pack algorithm effectively manages gaps in tiling.

It controls the ratio of empty space to total area.

The method supports extending Tao's results to t=1.

Abstract

The well-known problem stated by A. Meir and L. Moser consists in tiling the unit square with rectangles (details), whose side lengths equal $1/n\times 1/(n+1)$, where indices~$n$ range from 1 to infinity. Recently, Terence Tao has proved that it is possible to tile with $1/n^t\times 1/(n+1)^t$ rectangles (squares with the side length of $1/n^t$), $1/2<t<1$, the square, whose area equals the sum of areas of these details, provided that only those details, whose indices exceed certain~$n_0$, are taken into consideration. We adduce arguments in favor of the assumption that the result obtained by T. Tao is also valid for $t=1$. We use a new tiling method (the Slack-Pack algorithm), which initially admits gaps between stacks of details. The algorithm uses a pre-fixed parameter $\gamma$, $\sqrt{3/2}<\gamma<3/2$, connected with the gap value. The new algorithm allows one to control the ratio of the area of the large rectangular part, which is free of details, to the whole area of the remaining empty space. This ratio (under certain natural assumptions) always exceeds $1-1/\gamma-\delta$, where $\delta$ tends to zero as $n_0$ increases.