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「New Nikodym set constructions over finite fields」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 have uploaded to the arXiv my paper “ New Nikodym set constructions over finite fields “. This is a spinoff of my previous project with Bogdan Georgiev, Javier Gómez–Serrano, and Adam Zsolt Wagner that I recently posted about . In that project we experimented with using AlphaEvolve (and other tools, such as DeepThink and AlphaProof) to explore various mathematical problems which were connected somehow to an optimization problem. For one of these — the finite field Nikodym s

Let be a finite field of some order (which must be a prime or a power of a prime), and let be a fixed dimension. A Nikodym set in is a subset of with the property that for every point , there exists a line passing through such that all points of other than lie in . Such sets are close cousins of Kakeya sets (which contain a line in every direction); indeed, roughly speaking, applying a random projective transformation to a Nikodym set will yield (most of) a Kakeya set. As a c

已知结果和反例

For Kakeya sets, Bukh and Chao showed this bound to be sharp up to the lower order error ; but for Nikodym sets it is conjectured that in fact such sets should asymptotically have full density, in the sense that

In our experiments we focused on the opposite problem of constructing Nikodym sets of size as small as possible. In the plane , constructions of size

证明或构造的主线

We set AlphaEvolve to try to optimize the three dimensional problem with a variable field size (which we took to be prime for simplicity), with the intent to get this tool to come up with a construction that worked asymptotically for large , rather than just for any fixed value of . After some rounds of evolution, it arrived at a construction which empirically had size about . Inspecting the code, it turned out that AlphaEvolve had constructed a Nikodym set by (mostly) removi

The arguments can be sketched here as follows. Let be a random surface of degree , and let be a point in which does not lie in . A random line through then meets in a number of points, which is basically the set of zeroes in of a random polynomial of degree . The (function field analogue of the) Chebotarev density theorem predicts that the probability that this polynomial has no roots in is about , where

阅读时建议盯住的点

DeepThink took the degrees to be large, so that the derangement probabilities were close to . This led it to predict that could be taken to be as large as , leading to the claimed bound (2) . However, on inspecting this argument we realized that these moderately high degree surfaces were effectively acting as random sets, so one could dramatically simplify DeepThink’s argument by simply taking to be a completely random set of the desired cardinality (2) , in which case the ve

On the other hand, the derangement probabilities oscillate around , and in fact are as large as when . This suggested that one could do better than the purely random construction if one only removed quadratic surfaces instead of higher degree surfaces, and heuristically predicted the improvement

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