陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Perfectly packing a square by squares of nearly harmonic sidelength」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

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 for into a single square or rectangle of area . (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 perfectl

Another direction in which partial progress has been made is to consider instead the problem of packing squares of sidelength , perfectly into a square or rectangle of total area , for some fixed constant (this lower bound is needed to make the total area finite), with the aim being to get as close to as possible. Prior to this paper, the most recent advance in this direction was by Januszewski and Zielonka last year, who achieved such a packing in the range .

已知结果和反例

In this paper we are able to get arbitrarily close to (which turns out to be a “critical” value of this parameter), but at the expense of deleting the first few tiles:

As in previous works, the general strategy is to execute a greedy algorithm, which can be described somewhat incompletely as follows.

证明或构造的主线

The main innovation of this paper is to perform Step 3 somewhat more efficiently than in previous papers.

The above algorithm can get stuck if one reaches a point where one has already packed squares of sidelength for , but that all remaining rectangles in have width less than , in which case there is no obvious way to fit in the next square. If we let and denote the width and height of these rectangles , then the total area of the rectangles must be

阅读时建议盯住的点

In comparison, the perimeter of the squares that one has already packed is equal to

By choosing the parameter suitably large (and taking sufficiently large depending on ), one can then prove the theorem. (In order to do some technical bookkeeping and to allow one to close an induction in the verification of the algorithm’s correctness, it is convenient to replace the perimeter by a slightly weighted variant for a small exponent , but this is a somewhat artificial device that somewhat obscures the main ideas.)

值得单独记下的条目

• Step 2: Amongst all the rectangles in , locate the rectangle of the largest width (defined as the shorter of the two sidelengths of ).
• Step 3: Pack (as efficiently as one can) squares of sidelength for into for some , and decompose the portion of not covered by this packing into rectangles .
• Step 4: Replace by , replace by , and return to Step 1.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源