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「Dyadic models」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

One of the oldest and most fundamental concepts in mathematics is the line . Depending on exactly what mathematical structures we want to study (algebraic, geometric, topological, order-theoretic, etc.), we model lines nowadays by a variety of standard mathematical objects, such as the real line , the complex line , the projective line , the extended real line , the affine line , the continuum , the long line , etc. We also have discrete versions of the line, such as the natu

Broadly speaking, the line has three major families of structures on it:

已知结果和反例

Of course, these structures are inter-related, and it is an important phenomenon that a mathematical concept which appears to be native to one structure, can often be equivalently defined in terms of other structures. For instance, the absolute value of an integer can be defined geometrically as the distance from 0 to , algebraically as the index of the subgroup of the integers generated by n, or one-dimensionally as the number of integers between 0 and (including 0, but excl

What I want to talk about today is an important toy model for the line (in any of its incarnations), in which the geometric and algebraic structures are enhanced (and become neatly nested and recursive), at the expense of the one-dimensional structure (which is largely destroyed). This model has many different names, depending on what field of mathematics one is working in and which structures one is interested in. In harmonic analysis it is called the dyadic model, the Walsh

证明或构造的主线

Very broadly speaking, one of the key advantages that dyadic models offer over non-dyadic models is that they do not have any “spillover” from one scale to the next. This spillover is introduced to us all the way back in primary school, when we learn about the algorithms for decimal notation arithmetic: long addition , long subtraction , long multiplication , and long division . In decimal notation, the notion of scale is given to us by powers of ten (with higher powers corre

It is thus natural to look for models of arithmetic in which this spillover is not present. One is first exposed to such models in high school, when the arithmetic of polynomials in one unknown is introduced (i.e. one works with rings such as or rather than or ). For instance, to quotient one polynomial by another, one uses the polynomial long division (or synthetic division ) algorithm, which is formally identical to long division for integers in decimal notation but without

阅读时建议盯住的点

Now, polynomial rings such as or are a bit “too big” to serve as models for or (unless one adjoins some infinitesimals, but that’s another story), as they have one more dimension. One can get a more accurate model by considering the decimal representation again, which identifies natural numbers as polynomials over the space of digits . This space is not closed under addition (which is what causes spillover in the first place); but we can remedy this by replacing this space of

There is also a base 10 dyadic model for the real numbers, in which we allow infinitely many negative powers of t but only finitely many positive powers in t; in other words, the model is , the ring of formal Laurent series in . This ring again differs slightly from the reals; for instance, is now no longer equal to (in fact, they differ by ). So the decimal notation maps onto the positive real axis , but there is a small amount of non-injectivity caused by this map.

值得单独记下的条目

• Geometric structures , such as a metric or a measure, completeness , scales (coarse and fine), rigid motions (translations and reflection), similarities (dilation, affine maps), and differential structures (tangent bundle, etc.);
• Algebraic structures , such group, ring, or field structures, and everything else that comes from those categories (e.g. subgroups, homomorphisms, involutions, etc.); and

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