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「Sum-difference exponents for boundedly many slopes, and rational complexity」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 “ Sum-difference exponents for boundedly many slopes, and rational complexity “. This is the second spinoff of my previous project with Bogdan Georgiev, Javier Gómez–Serrano, and Adam Zsolt Wagner that I recently posted about . One of the many problems we experimented using the AlphaEvolve tool with was that of computing sum-difference constants. While AlphaEvolve did modest improve one of the known lower bounds on sum-difference constant

In the original formulation of the sum-difference problem, one is given a finite subset of with some control on projections, such as

已知结果和反例

Because is clearly determined by and , one can trivially get an upper bound of on (1) . In 1999, Bourgain utilized what was then the very recent “Balog–Szemerédi–Gowers lemma” to improve this bound to , which gave a new lower bound of on the (Minkowski) dimension of Kakeya sets in , which improved upon the previous bounds of Tom Wolff in high dimensions. (A side note: Bourgain challenged Tom to also obtain a result of this form, but when they compared notes, Tom obtained the

One can get better bounds by adding more projections. For instance, if one also assumes

证明或构造的主线

It was observed by Ruzsa that all of these questions can be equivalently formulated in terms of Shannon entropy. For instance, the upper bound of (1) turns out to be equivalent to the entropy inequality

As part of the AlphaEvolve experiments, we directed this tool to obtain lower bounds for for various rational numbers , defined as the best constant in the inequality

阅读时建议盯住的点

The first main result of this paper is to confirm that this is indeed the case, in that

The arguments also apply to settings where there are more projections under control than just the projections. If one also controls projections for various rationals and denotes the set of slopes of the projections under control, then it turns out that the associated sum-difference constant still decays to , but now the key parameter is not the height of , but rather what I call the rational complexity of with respect to , defined as the smallest integer for which one can wri

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