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「254B, Notes 3: Quasirandom groups, expansion, and Selberg’s 3/16 theorem」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

In the previous set of notes we saw how a representation-theoretic property of groups, namely Kazhdan’s property (T) , could be used to demonstrate expansion in Cayley graphs. In this set of notes we discuss a different representation-theoretic property of groups, namely quasirandomness , which is also useful for demonstrating expansion in Cayley graphs, though in a somewhat different way to property (T). For instance, whereas property (T), being qualitative in nature, is onl

The definition of quasirandomness is easy enough to state:

已知结果和反例

Definition 1 (Quasirandom groups) Let be a finite group, and let . We say that is -quasirandom if all non-trivial unitary representations of have dimension at least . (Recall a representation is trivial if is the identity for all .)

Exercise 1 Let be a finite group, and let . A unitary representation is said to be irreducible if has no -invariant subspaces other than and . Show that is -quasirandom if and only if every non-trivial irreducible representation of has dimension at least .

证明或构造的主线

Remark 1 The terminology “quasirandom group” was introduced explicitly (though with slightly different notational conventions) by Gowers in 2008 in his detailed study of the concept ; the name arises because dense Cayley graphs in quasirandom groups are quasirandom graphs in the sense of Chung, Graham, and Wilson , as we shall see below. This property had already been used implicitly to construct expander graphs by Sarnak and Xue in 1991, and more recently by Gamburd in 2002

Quasirandomness behaves fairly well with respect to quotients and short exact sequences:

阅读时建议盯住的点

Exercise 2 Let be a short exact sequence of finite groups .

Informally, 下面会 call quasirandom if it is -quasirandom for some “large” , though the precise meaning of “large” will depend on context. For applications to expansion in Cayley graphs, “large” will mean “ for some constant independent of the size of “, but other regimes of are certainly of interest.

值得单独记下的条目

• (i) If is -quasirandom, show that is -quasirandom also. (Equivalently: any quotient of a -quasirandom finite group is again a -quasirandom finite group.)
• (ii) Conversely, if and are both -quasirandom, show that is -quasirandom also. (In particular, the direct or semidirect product of two -quasirandom finite groups is again a -quasirandom finite group.)
• (i) Show that if is non-trivial, then . ( Hint: use the mean zero component of the regular representation .) In particular, non-trivial finite groups cannot be infinitely quasirandom.
• (ii) Show that any proper subgroup of has index . ( Hint: use the mean zero component of the quasiregular representation .)
• (i) If is abelian and non-trivial, show that is not -quasirandom. ( Hint: use Fourier analysis or the classification of finite abelian groups .)
• (ii) Show that is -quasirandom if and only if it is perfect , i.e. the commutator group is equal to . (Equivalently, is -quasirandom if and only if it has no non-trivial abelian quotients.)
• (i) If , are subsets of , show that (compare with the expander mixing lemma, Exercise 9 from Notes 1 ).
• (ii) Show that is a two-sided -expander whenever

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