公式渲染测试
I’ve just uploaded to the arXiv my preprint “Perfectly packing a square by squares of nearly harmonic sidelength“. This paper concerns a variant of an old problem of Meir and Moser, who asks whether it is possible to perfectly pack squares of sidelength $1/n$ into a single square or rectangle of area ${\sum_{n=2}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} - 1}$. (The following variant problem, also posed by Meir and Moser and discussed for instance in this MathOverflow post, is perhaps even more well known: is it possible to perfectly pack rectangles of dimensions ${1/n \times 1/(n+1)}$ for ${n \geq 1}$ into a single square of area ${\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1}$?) For the purposes of this paper, rectangles and squares are understood to have sides parallel to the axes, and a packing is perfect if it partitions the region being packed up to sets of measure zero. As one partial result towards these problems, it was shown by Paulhus that squares of sidelength ${1/n}$ for ${n \geq 2}$ can be packed (not quite perfectly) into a single rectangle of area ${\frac{\pi^2}{6} - 1 + \frac{1}{1244918662}}$, and rectangles of dimensions ${1/n \times 1/n+1}$ for ${n \geq 1}$ can be packed (again not quite perfectly) into a single square of area ${1 + \frac{1}{10^9+1}}$. (Paulhus’s paper had some gaps in it, but these were subsequently repaired by Grzegorek and Januszewski.)
$1 + \frac{1}{10^9+1}$ In this paper we are able to get $t$ arbitrarily close to $1$ (which turns out to be a “critical” value of this parameter), but at the expense of deleting the first few tiles:
Theorem 1 If ${1/2 < t < 1}$, and ${n_0}$ is sufficiently large depending on $t$, then one can pack squares of sidelength ${n^{-t}}$, ${n \geq n_0}$ perfectly into a square of area ${\sum_{n=n_0}^{\infty} \frac{1}{n^{2t}}}$.
As in previous works, the general strategy is to execute a greedy algorithm, which can be described somewhat incompletely as follows.
原文来自陶哲轩的博客
ml的看着好多了，但是代码太长了