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「Open question: best bounds for cap sets」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

Earlier this month, in the previous incarnation of this page , I posed a question which 一个常见想法是 was unsolved, and obtained the answer (in fact, it was solved 25 years ago ) within a week. Now that this new version of the page has better feedback capability, I am now tempted to try again, since I have a large number of such questions which I would like to publicise. (Actually, I even have a secret web page full of these somewhere near my home page , though it will take a non-t

Perhaps my favourite open question is the problem on the maximal size of a cap set – a subset of ( being the finite field of three elements) which contains no lines, or equivalently no non-trivial arithmetic progressions of length three. As an upper bound, one can easily modify the proof of Roth’s theorem to show that cap sets must have size (see e.g. this paper of Meshulam ). This of course is better than the trivial bound of once n is large. In the converse direction, the t

已知结果和反例

One reason why I find this question important is that it serves as an excellent model for the analogous question of finding large sets without progressions of length three in the interval . Here, the best upper bound of is due to Bourgain (he also has a recent, not yet published, improvement to , while the best lower bound of is an ancient result of Behrend . Using the finite field heuristic that “behaves like” , we see that the Bourgain bound should be improvable to , wherea

The Roth bound of appears to be the natural limit of the purely Fourier-analytic approach of Roth, and so any breakthrough would be extremely interesting, as it almost certainly would need a radically new idea. The lower bound might be improvable by some sort of algebraic geometry construction, though it is not clear at all how to achieve this.

证明或构造的主线

( Update, Feb 25: After some feedback and advice, and moving the entire blog to another site, I have finally gotten the math formulae to work out nicely. Thanks for all the help!)

( Update , Feb 27: As pointed out in the comments, one can interpret this problem in terms of the wonderful game Set , in which case the problem is to find the largest number of cards one can put on the table for which nobody has a valid move. As far as I know, the best bounds on the cap set problem in small dimensions are the ones cited in the Edel paper mentioned above .)

阅读时建议盯住的点

( Update , Mar 5: After discussions with Jordan Ellenberg, we realised that there is a variant formulation of the problem which may be a little bit more tractable. Given any , the fewest number of lines in a set of of density at least is known to be for some ; this is essentially a result of Croot . The reformulated question is then to get as strong a bound on ) as one can. For instance, the counterexample shows that , while the Roth-Meshulam argument gives .)

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