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「A counterexample to the periodic tiling conjecture」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

Rachel Greenfeld and I have just uploaded to the arXiv our announcement “ A counterexample to the periodic tiling conjecture “. This is an announcement of a longer paper that we are currently in the process of writing up (and hope to release in a few weeks), in which we disprove the periodic tiling conjecture of Grünbaum-Shephard and Lagarias-Wang . This conjecture can be formulated in both discrete and continuous settings:

The discrete periodic tiling conjecture can be easily established for by the pigeonhole principle (as first observed by Newman ), and was proven for by Bhattacharya (with a new proof given by Greenfeld and myself ). The continuous periodic tiling conjecture was established for by Lagarias and Wang . By an old observation of Hao Wang, one of the consequences of the (discrete) periodic tiling conjecture is that the problem of determining whether a given finite set tiles by tran

已知结果和反例

On the other hand, once one allows tilings by more than one tile, it is well known that aperiodic tile sets exist, even in dimension two – finite collections of discrete or continuous tiles that can tile the given domain by translations, but not periodically. Perhaps the most famous examples of such aperiodic tilings are the Penrose tilings , but there are many other constructions; for instance, there is a construction of Ammann, Grümbaum, and Shephard of eight tiles in which

This suggests that the techniques used to prove the discrete periodic conjecture in are already close to the limit of their applicability, as they cannot handle even virtually two-dimensional discrete abelian groups such as . The main difficulty is in constructing the counterexample in the setting.

证明或构造的主线

The approach starts by adapting some of the methods of a previous paper of Rachel and myself . The first step is make the problem easier to solve by disproving a “multiple periodic tiling conjecture” instead of the traditional periodic tiling conjecture. At present, Theorem 3 asserts the existence of a “tiling equation” (where one should think of and as given, and the tiling set is known), which admits solutions, all of which are non-periodic. It turns out that it is enough t

It is convenient to replace sets by functions, so that this tiling language can be translated to a more familiar language, namely the language of (certain types of) functional equations. The key point here is that the tiling equation

阅读时建议盯住的点

The non-periodic behaviour that we ended up trying to capture was that of a certain “ -adically structured function” associated to some fixed and sufficiently large prime (in fact for our arguments any prime larger than , e.g., , would suffice), defined by the formula

A feature of this function is that when one restricts to a row or diagonal of such a function, the resulting one-dimensional function exhibits “ -adic structure” in the sense that it behaves like a rescaled version of ; see the announcement for a precise version of this statement. It turns out that the converse is essentially true: after excluding some degenerate solutions in which the function is constant along one or more of the columns, all two-dimensional functions which

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。

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