On Gromov’s theorem on groups of polynomial growth

This article documents my presentation of Gromov’s theorem on groups of polynomial growth at the MIT combinatorics reading group. The presentation is based on Gromov’s 1981 paper, Groups of polynomial growth and expanding maps, Kleiner’s 2007 paper, A new proof of Gromov’s theorem on groups of polynomial growth, and Tao’s 2009 blog post, A finitary version of Gromov’s polynomial growth theorem.


Throughout, we fix a finitely generated group G and a finite symmetric generating set S (that is \forall x \in S. x^{-1}\in S). For every group x \in G, the word length \lVert x \rVert of x is the shortest length n of a word s_1s_2 \dots s_n in S that expresses x.

Gromov’s theorem connects a group property of G with the growth of the cardinality of the ball B(r) := \left\{x \in G : \lVert x \rVert \le r\right\} of radius r. The ball B(r) can be seen as the set of vertices within distance r from the identity element of G in the Cayley graph of (G, S).

Definition (nilpotent and virtually nilpotent). A group H is nilpotent of class n if there is a lower central series H = H_0 \rhd H_1 \rhd \dots \rhd H_n = \{e\}, where H_{i+1} = [H_i, H]. A group G is virtually nilpotent if there is a finite index subgroup H of G that is nilpotent.

Example 1. When G is abelian, G is nilpotent and \lvert B(r) \rvert = r^{\mathrm{rank}(G)}.

Example 2. When G is the discrete Heisenberg group, that is G = \left\{ \begin{pmatrix} 1 & a & b \\ & 1 & c \\ & & 1 \end{pmatrix} : a, b, c \in \mathbb{Z} \right\}, we have the lower central series G > \langle \begin{pmatrix} 1 & & 1 \\ & 1 & \\ & & 1\end{pmatrix} \rangle > \{e\}, and the growth of \lvert B(r) \rvert is bounded by r^4.

Theorem (Gromov 1981). For every group G generated by a finite symmetric set S, \lvert B(r)\rvert is at most polynomial in r if and only if G is virtually nilpotent.

Gromov’s proof uses the deep Montgomery–Zippin–Yamabe structure theory of locally compact groups on Hilbert’s fifth problem. Colding and Minicozzi, solving a conjecture of Yau, showed that the space of harmonic functions with polynomial growth on an open manifold with non-negative Ricci curvature. To state (a weak version of) the discrete analog of their result, we need the notion of Lipschitz harmonic on the group G with the finite symmetric generating set S.

Definition (Lipschitz and harmonic). A function f\colon G\to\mathbb{R} is Lipschitz if \sup_{g \in G, s \in S}\lvert f(gs) - f(g) \rvert is finite, and is harmonic if f(g) = \frac{1}{\lvert S \rvert} \sum_{s \in S} f(gs) for all g \in G.

Theorem (Colding and Minicozzi 1997, Kleiner 2010). If \vert B(r)\lvert is at most polynomial in r, then the linear space of Lipschitz harmonic functions on G is finite dimensional.

Kleiner provided a new proof of Gromov’s theorem by establishing the Colding–Minicozzi theorem from scratch. Later Shalom and Tao pushed Kleiner’s methods to obtain the following quantitative version of Gromov’s theorem.

Theorem (Shalom and Tao 2010). There exists an absolute constant c such that if \lvert B(r) \rvert \le r^d for some r > \exp(\exp(cd^c)) then G contains a finite index subgraph H which is nilpotent of class \le c^d.

In the rest of the article, we will prove the Colding–Minicozzi theorem through the Poincaré inequality and the reverse Poincaré inequality.

Poincaré inequality

Lemma (Poincaré inequality). For every function f \colon G \to \mathbb{R}, if f has mean 0 on B(r), the l^2-norm of f on B(r) is bounded by the fluctuation of f on B(2r): \sum_{x\in B(r)}f(x)^2 \le \frac{\lvert B(2r) \rvert}{\lvert B(r) \rvert}\cdot 2r^2 \sum_{x,y\in B(2r), x\sim y}(f(x) - f(y))^2.

Proof. One can check that the left hand side is equal to \frac{1}{2\lvert B(r)\rvert}\sum_{x,y\in B(r)}(f(x)-f(y))^2.

For each z \in B(2r), we fix a shortest path e = z_0, z_1, \dots, z_{\lVert z \rVert} = z from e to z in the Cayley graph of (G, S). Given x,y\in B(r), let z = x^{-1}y \in B(2r) and get f(x) - f(y) = \sum_{i=1}^{\lVert z \rVert}f(xz_{i-1})-f(xz_i) \\ \implies (f(x)-f(y))^2 \le \lVert z \rVert \sum_{i=1}^{\lVert z \rVert}(f(xz_{i-1})-f(xz_i))^2.

When summing over the last inequality over all x,y\in B(r), we can regroup the summands by z and i as follows: \sum_{z \in B(2r)}\lVert z\rVert\sum_{i=1}^{\lVert z \rVert} \left(\sum_{x\in B(r) : xz \in B(r)}(f(xz_{i-1})-f(xz_i))^2\right).

Fix z and i for a moment. One can show that both xz_{i-1} and xz_i are in B(2r), and moreover the directed edges (xz_{i-1}, xz_{i}) are distinct when x varies in B(r). Thus \sum_{x\in B(r) : xz \in B(r)}(f(xz_{i-1})-f(xz_i))^2 \le \sum_{x,y\in B(2r), x\sim y}(f(x)-f(y))^2.

We obtain the Poincaré inequality by putting everything together and noticing that \lVert z \rVert \le 2r.

Reverse Poincaré inequality

Lemma (Reverse Poincaré inequality). For every harmonic function f\colon G \to \mathbb{R}, the fluctuation of f on B(r) is bounded by the l^2-norm of f on B(2r): \sum_{x,y\in B(r), x\sim y}(f(x) - f(y))^2 \le \lvert S \vert \cdot 4r^{-2}\sum_{x\in B(2r)}f(x)^2.

To facilitate the proof, we introduce the following notations. Given a function f\colon G\to \mathbb{R} and s\in S, write f_s(x) := f(xs) and \partial_s f := f_s - f. It is easy to see that

  1. \sum_{s\in S}\partial_{s^{-1}}\partial_s f = 0 when f is harmonic, and
  2. \sum_{x\in G}f(x)\partial_s g(x) = \sum_{x\in G}\partial_{s^{-1}}f(x) g(x) when f or g is finitely supported.

Proof. Fix the harmonic function f\colon G\to \mathbb{R} and let \phi\colon G \to [0,1] be defined by \phi(x) = \begin{cases} 1 & \text{if }\lVert x\rVert \le r,\\ 2 - \lVert x\rVert/r & \text{if }r < \lVert x\rVert < 2r, \\ 0 & \text{if }\lVert x\rVert \ge 2r.\end{cases}

For every s \in S, note that \partial_s (f\phi^2) = (\partial_s f)\phi^2 + f_s(\partial_s \phi^2) and \partial_s \phi^2 = (\partial_s \phi)\phi + \phi_s (\partial_s\phi) = (\partial_s \phi)(2\phi + \partial_s \phi). We obtain \begin{aligned}\partial_s f \partial_s (f\phi^2) & = (\partial_s f)^2 \phi^2 + (\partial_s f)f_s(\partial_s \phi)(2\phi + \partial_s\phi) \\ & \ge \tfrac{1}{2}(\partial_s f)^2\phi^2 - 2(f_s)^2(\partial_s \phi)^2 + (\partial_s f)f_s(\partial_s\phi)^2 \\ & = \tfrac{1}{2}(\partial_s f)^2\phi^2 - f_s(f_s + f)(\partial_s\phi)^2 \\ & \ge \tfrac{1}{2}(\partial_s f)^2\phi^2 - \tfrac{1}{2}(3f_s^2 + f^2)(\partial_s\phi)^2. \end{aligned}

When summing the above inequality over all s\in S and x \in G, by noticing that \sum_{s\in S}\sum_{x\in G}\partial_s f(x) \partial_s (f(x)\phi(x)^2) = \sum_{s\in S}\sum_{x\in G}(\partial_{s^{-1}}\partial_s f(x)) f(x)\phi(x)^2 \\ = \sum_{x\in G}\left(\sum_{s\in S}\partial_{s^{-1}}\partial_s f(x)\right) f(x)\phi^2(x) = 0, we get \sum_{x\in G}\sum_{s \in S}(\partial_s f(x))^2\phi(x)^2 \le \sum_{x\in G}\sum_{s \in S}(3f_s(xs)^2+f(x)^2)(\partial_s \phi(x))^2 .

The left hand side of the last inequality is at least \sum_{x,y\in B(r), x\sim y}(f(x)-f(y))^2, whereas the right hand side is at most 4\lvert S\rvert \tfrac{1}{r^2}\sum_{r \le \lVert x\rVert \le 2r}f(x)^2 because (\partial_s\phi)^2 \le 1/r^2 and \partial_s\phi(x)^2 > 0 only if both x and xs are in \{x \in G : r\le \lVert x\rVert \le 2r\}.

Colding–Minicozzi theorem

To simplify the presentation, we will assume the doubling constant \lvert B(2r) \rvert / \lvert B(r) \rvert is uniformly bounded at all scales r, which, for example, is indeed the case when \lvert B(r)\rvert = \Theta(r^d). In general, one needs the pigeonhole principle to select the correct radii for the argument below to work.

Proof assuming the doubling constant is uniformly bounded. Suppose for the sake of contradiction, the dimension of the linear space consisting of Lipschitz harmonic functions on G is at least n, where the parameter n will be determined later. Denote by V the n-dimensional linear subspace.

Let k be a natural number to be determined soon, and fix r for a moment. Let \mathcal{A}_r be a maximal collection of disjoint balls of radius r/2 with centers in B(kr), and let \mathcal{B}_r be the collection of balls with the same centers of the balls in \mathcal{A}_r, but of radius r. Let V_r be the linear subspace of V consisting of harmonic functions in V that average to 0 on each ball in \mathcal{B}_r. Note that the co-dimension of V_r as a subspace of V is at most \lvert \mathcal{B}_r \rvert = \lvert \mathcal{A}_r \rvert, which is at most \lvert B(kr+r/2) \rvert / \lvert B(r/2) \rvert = O(1) =: C.

For every harmonic function f in V_r, using the fact that B(kr) \subseteq \cup\mathcal{B}_r, the fact that each point in G is covered by 2B for at most \lvert B(2r+r/2) \rvert / \lvert B(r/2) \rvert = O(1) many B \in \mathcal{B}_r, the Poincaré inequality and the reverse Poincaré inequality, we get \begin{aligned}\sum_{x \in B(kr)}f(x)^2 & \le \sum_{B \in \mathcal{B}(r)}\sum_{x\in B}f(x)^2 \\ & \lesssim r^2\sum_{B \in \mathcal{B}(r)}\sum_{x,y \in 2B, x\sim y}(f(x)-f(y))^2 \\ & \lesssim r^2\sum_{x, y \in B(kr+2r), x\sim y}(f(x)-f(y))^2 \\ & \lesssim \frac{1}{(k+2)^2}\sum_{x \in B(2(k+2)r)}f(x)^2.\end{aligned} Now take k large enough (depending only on the group G) so that for all f\in V_r, 3^d\sum_{x\in B(kr)}f(x)^2 \le \sum_{x\in B(3kr)}f(x)^2.

Consider the quadratic form Q_r on V defined by Q_r(f) := \sum_{x\in B(r)}f(x)^2. Since the kernels of Q_r‘s form a descending chain of vector spaces, there exists r_0 such that Q_r is positive-definite for all r \ge r_0.

For every r \ge r_0, let q(r) be the volume of the ellipsoid E_r induced by Q_r. To be more precise, after fixing a basis \{f_1, \dots, f_n\} of V, the ellipsoid is defined by E_r := \{(c_1, \dots, c_n) \in \mathbb{R}^n : Q_r(c_1f_1 + \dots c_nf_n) \le 1\}. By scaling and translation, we may assume that f_i is 1-Lipschitz and f_i(e)=0 (whenever f_i is non-constant). Since \lvert B(r) \rvert is at most polynomial in r, there exists a natural number d (depending only on the group G) such that \sum_{x \in B(r)} f_i(x)^2 \le r^d for all i \in [n]. By Cauchy–Schwarz inequality, we have \begin{aligned}Q_r(c_1f_1 + \dots + c_nf_n) & = \sum_{x\in B(r)}(c_1f_1(x)+\dots+c_nf_n(x))^2 \\ & \le n\sum_{x\in B(r)}c_1^2f_1(x)^2 + \dots c_n^2f_n(x)^2 \\ & \le (c_1^2+\dots+c_n^2)n^2r^d.\end{aligned} Therefore E_r contains the ball of radius (nr^{d/2})^{-1}, hence q_r \ge v_n(nr^{d/2})^{-n}, where v_n is the volume of the n-dimensional Euclidean unit ball.

Although the volume q(r) of the ellipsoid is not intrinsic to Q_r, the ratio between q(r) and q(r') does not depend on the choice of the basis.

After a linear transformation, we may assume the symmetric matrices associated to Q_{kr} and Q_{3kr} are of the form \begin{pmatrix}A_1 & B_1 \\ B_1^T & C_1 \end{pmatrix}, \begin{pmatrix}A_3 & \\ & C_3\end{pmatrix}, where A_1, A_3 act on V_r. Using the Schur complement, we obtain that the volume ratio q_{3kr}/q_{kr} is \frac{\det Q_{kr}}{\det Q_{3kr}} = \frac{\det A_1 \det (C_1 - B_1^TA_1^{-1}B_1)}{\det A_3 \det C_3}. As B_1^TA_1^{-1}B_1 is positive semi-definite, we obtain \det(C_1 - B_1^TA_1^{-1}B_1) \le \det C_1 and so q_{3kr}/q_{kr} is at most \frac{\det A_1\det C_1}{\det A_3\det C_3}. Recall that 3^dQ_{kr}(f) \le Q_{3kr}(f) for all f \in V_r and clearly Q_{kr}(f) \le Q_{3kr}(f) for all f \in V. In other words, 3^dA_1 \preceq A_3 and C_1 \preceq C_3 and so q_{kr}/q_{3kr} \ge (3^d)^{\dim V_r} \ge (3^d)^{n-C}.

Choose n so that (3^d)^{n-C} \ge 2^{dn} hence 2^{dn}q(3kr) \le q(kr). Repeatedly apply the last inequality to obtain q(kr) \ge 2^{mdn}q(3^mkr) \ge 2^{mdn}v_n(n(3^mkr)^{d/2})^{-n} = \Omega_{d,n}((2/\sqrt{3})^{mdn}), which is absurd for m sufficiently large.


On the basis exchange property

One of the students in my class, Undergraduate Seminar on Discrete Mathematics, asks if the exchange property of a matroid can be strengthened to the following, of which I was completely unaware.

Strong basis exchange property. For every pair of bases B_1, B_2 and x_1 \in B_1, there exists x_2 \in B_2 such that both B_1 - x_1 + x_2 and B_2 + x_1 - x_2 are still bases.

As it turns out, Brualdi proved back in 1969 the strong exchange property (Theorem 2) in Comments on bases in dependence structures.

I record below the proof which showcases the benefit of viewing matroids both graphically and linear-algebraically.

As per Aaron Berger‘s suggestion, we use a graphic matroid to motivate the proof. Suppose T_1 and T_2 are two spanning trees of a connected graph G and e_1 is an edge in T_1. Clearly, when the edge e_1 is also in T_2, we can simply exchange e_1 with itself. Assume from now that e_1 is not in T_2. Note that T_2 + e_1 contains a unique cycle C, each edge of which can be “traded” to T_1. To be more precise, for every e_2 \in C, T_2 + e_1 - e_2 is still a spanning tree. It is then easy to see that adding back some edge from C - e_1 can reconnect the two connected components resulted from removing e_1 from T_1.

As the unique cycle (or circuit) C plays a central role in the above argument, one can show the same concept is valid in any matroid.

Lemma (Fundamental circuit). For every independent set I, if an element e satisfies that I + e is dependent, then I + e contains a unique circuit C, called the fundamental circuit. Moreover, for every f \in C, I + e - f is independent.

Proof of strong basis exchange property. Without loss of generality, we may assume that e_1 \not\in B_2. As B_2 is a maximal independent set and B_2 + e_1 is dependent, there is a unique circuit C \subseteq B_2 + e_1 such that B_2 + e_1 - e_2 is independent for every e_2 \in C.

Finally we resort to our intuition of representable matroids. Since e_1 \in \mathrm{span}(C - e_1) \subseteq \mathrm{span}(B_1 + C - e_1), we have \mathrm{span}(B_1 + C - e_1) = \mathrm{span}(B_1 + C), hence B_1 + C - e_1 contains a basis, say B_1'. By the ordinary exchange property for B_1 - e_1 and B_1', there is e_2 \in B_1' - (B_1 - e_1) \subseteq C - e_1 such that B_1 - e_1 + e_2 is independent.


Minimal Distance to Pi

Here is a problem from Week of Code 29 hosted by Hackerrank.

Problem. Given two integers q_1 and q_2 (1\le q_1 \le q_2 \le 10^{15}), find and print a common fraction p/q such that q_1 \le q \le q_2 and \left|p/q-\pi\right| is minimal. If there are several fractions having minimal distance to \pi, choose the one with the smallest denominator.

Note that checking all possible denominators does not work as iterating for 10^{15} times would exceed the time limit (2 seconds for C or 10 seconds for Ruby).

The problem setter suggested the following algorithm in the editorial of the problem:

  1. Given q, it is easy to compute p such that r(q) := p/q is the closest rational to \pi among all rationals with denominator q.
  2. Find the semiconvergents of the continued fraction of \pi with denominators \le 10^{15}.
  3. Start from q = q_1, and at each step increase q by the smallest denominator d of a semiconvergent such that r(q+d) is closer to \pi than r(q). Repeat until q exceeds q_2.

Given q, let d = d(q) be the smallest increment to the denominator q such that r(q+d) is closer to \pi than r(q). To justify the algorithm, one needs to prove the d is the denominator of one of the semiconvergents. The problem setter admits that he does not have a formal proof.

Inspired by the problem setter’s approach, here is a complete solution to the problem. Note that \pi should not be special in this problem, and can be replaced by any other irrational number \theta. Without loss of generality, we may assume that \theta\in(0,1).

We first introduce the Farey intervals of \theta.

  1. Start with the interval (0/1, 1/1).
  2. Suppose the last interval is (a/b, c/d). Cut it by the mediant of a/b and c/d and choose one of the intervals (a/b, (a+c)/(b+d)), ((a+c)/(b+d), c/d) that contains \theta as the next interval.

We call the intervals appeared in the above process Farey intervals of \theta. For example, take \theta = \pi - 3 = 0.1415926.... The Farey intervals are:

\begin{gathered}(0/1, 1/1), (0/1, 1/2), (0/1, 1/3), (0/1, 1/4), (0/1, 1/5), \\ (0/1, 1/6), (0/1, 1/7), (1/8, 1/7), (2/15, 1/7),\cdots\end{gathered}

The Farey sequence of order n, denoted by F_n, is the sequence of completely reduced fractions between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. Fractions which are neighboring terms in any Farey sequence are known as a Farey pair. For example, Farey sequence of order 5 is
F_5 = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1).

Using the Stern–Brocot tree, one can prove that

Lemma 1. For every Farey interval (a/b, c/d) of \theta, the pair (a/b, c/d) is a Farey pair. Conversely, for every Farey pair (a/b, c/d), if \theta \in (a/b, c/d), then (a/b, c/d) is a Farey interval.
We say p/q is a good rational approximation of \theta if every rational between p/q and \theta (exclusive) has a denominator greater than q.

By the definition of Farey sequence, incorporating with Lemma 1, we know that

Lemma 2. A rational is an endpoint of a Farey interval of \theta if and only if it is a good rational approximation of \theta.

In fact, one can show that the endpoints of Farey intervals and semiconvergents of continued fraction are the same thing! Thereof, the problem setter’s claim follows immediately from:

Proposition. Given q, let r(q) = p / q be the rational closest to \theta with denominator q. If d = d(q) is the minimal increment to q such that r(q + d) = (p + c) / (q + d) is closer to \theta than r(q), then c/d is a good rational approximation.

Remark. The proposition states that the increments to p/q always come from a good rational approximation. It is stronger than the problem setter’s statement, which only asserts that the increment to the q comes from a good rational approximation.

Proof. In (x, y)-plane, plot the trapezoid defined by

\left| y/x - \theta \right| < \left|p/q - \theta\right|, q < x < q + d.

Geometric interpretation

Also we interpret rational numbers p/q, (p+c)/(q+d) as points A = (q, p), B = (q+d, p+c). Let the line through (q, p) parallel to y=\theta x intersect the vertical line x = q+d at C = (q+d, p+\theta d). By the definition of d, we know that the trapezoid does not contain lattice points. In particular, there is no lattice point in the interior of the triangle ABC. In the coordinate system with origin at A, B has coordinate (d, c) and the line through A, C is y = \theta x. Since triangle ABC contains no lattice points, there is no (b, a) with b < d such that a/b is between \theta and c/d. In other words, c/d is a good rational approximation.

Here is a fine print of the algorithm. Because floats may not have enough precision for the purpose of computation, we will instead use a convergent of the continuous fraction of \pi instead. All the computations will then happen in \mathbb{Q}. Finally, we present the algorithm.

P = Rational(5706674932067741, 1816491048114374) - 3
# find endpoints of Farey intervals
a, b, c, d = 0, 1, 1, 1
farey = [[a,b],[c,d]]
while (f = b + d) <= max - min
  farey << [e = a + c, f]
  if P < Rational(e, f)
    c, d = e, f
    a, b = e, f
min, max =
p_min = (P * q).round
# increase p_min/min by frations in farey
while min <= max
  c, d = nil, nil
  farey.each do |a, b|
    break if min + b > max
    if (Rational(p_min + a, min + b) - P).abs < (Rational(p_min, min) - P).abs
      c, d = a, b
  break if d == nil
  p_min += c; min += d
puts "#{p_min + 3 * min}/#{min}"

A Short Proof of the Nash-Williams' Partition Theorem


  1. \mathbb{N} – the set of natural numbers;
  2. \binom{M}{k} – the family of all subsets of M of size k;
  3. \binom{M}{<\omega} – the family of all finite subsets of M;
  4. \binom{M}{\omega} – the family of all infinite subsets of M;

The infinite Ramsey theorem, in its simplest form, states that for every partition \binom{\mathbb{N}}{k} = \mathcal{F}_1 \sqcup \dots \sqcup \mathcal{F}_r, there exists an infinite set M\subset \mathbb{N} such that \binom{M}{k}\subset \mathcal{F}_i for some i\in [r]. The Nash-Williams‘ partition theorem can be seen as a strengthening of the infinite Ramsey theorem, which considers a partition of a subset of \binom{\mathbb{N}}{<\omega}.


  1. \mathcal{F}\restriction M\mathcal{F}\cap 2^M, that is, the set \{s\in\mathcal{F} : s\subset M\}.
  2. s \sqsubset t , where s,t are subsets of \mathbb{N}s is an initial segment of t, that is s = \{n\in t : n \le \max s\}.

Definition. Let set \mathcal{F} \subset \binom{\mathbb{N}}{<\omega}.

  1. \mathcal{F} is Ramsey if for every partition \mathcal{F}=\mathcal{F}_1\sqcup \dots\sqcup\mathcal{F}_r and every M\in\binom{\mathbb{N}}{\omega}, there is N\in\binom{M}{\omega} such that \mathcal{F}_i\restriction N = \emptyset for all but at most one i\in[r].
  2. \mathcal{F} is a Nash-Williams family if for all s, t\in\mathcal{F}, s\sqsubset t \implies s = t.

Theorem (Nash-Williams 1965). Every Nash-Williams family is Ramsey.

The proof presented here is based on the proof given by Prof. James Cummings in his Infinite Ramsey Theory class. The purpose of this rewrite is to have a proof that resembles the one of the infinite Ramsey theorem.

Notation. Let s\in\binom{\mathbb{N}}{<\omega} and M\in\binom{\mathbb{N}}{\omega}. Denote $$[s, M] = \left\{t \in \binom{\mathbb{N}}{<\omega} : t \sqsubset s \text{ or } (s \sqsubset t \text{ and } t\setminus s \subset M)\right\}.$$

Definition. Fix \mathcal{F}\subset \binom{\mathbb{N}}{<\omega} and s\in \binom{\mathbb{N}}{<\omega}.

  1. M accepts s if [s, M]\cap \mathcal{F}\neq \emptyset and M rejects s otherwise;
  2. M strongly accepts s if every infinite subset of M accepts s;
  3. M decides s if M either rejects s or strongly accepts it.

We list some properties that encapsulates the combinatorial characteristics of the definitions above.


  1. If M decides (or strongly accepts, or rejects) s and N\subset M, then N decides (respectively strongly accepts, rejects) s as well.
  2. For every M\in\binom{\mathbb{N}}{\omega} and s\in\binom{\mathbb{N}}{<\omega}, there is N_1\in\binom{M}{\omega} deciding s. Consequently, there is N_2\in\binom{M}{\omega} deciding every subset of s.

Proof of Theorem. Enough to show that if \mathcal{F} = \mathcal{F}_1\sqcup \mathcal{F}_2, then for every M\in\binom{\mathbb{N}}{\omega}, there is infinite N\in \binom{M}{\omega} such that F_i \restriction N = \emptyset for some i\in[2].

We are going to use \mathcal{F}_1 instead of \mathcal{F} in the definitions of “accept”, “reject”, “strongly accept” and “decide”. Find N\in \binom{M}{\omega} that decides \emptyset. If N rejects \emptyset, by definition \mathcal{F}_1\restriction N = [\emptyset, N]\cap \mathcal{F}_1 = \emptyset. Otherwise N strongly accepts \emptyset.

Inductively, we build a decreasing sequence of infinite sets N \supset N_1 \supset N_2\supset \dots , an increasing sequence of natural numbers n_1, n_2, \dots, and maintain that n_i\in N_i, n_i < \min N_{i+1} and that N_i strongly accepts every s\subset \{n_j : j < i\}. Initially, we take N_1 = N as N strongly accepts \emptyset.

A mental picture of the construction.

Suppose N_1 \supset \dots \supset N_i and n_1 < \dots < n_{i-1} have been constructed. In the following lemma, when taking M = N_i and s = \{n_j : j < i\}, it spits out m and N, which are exactly what we need for n_i and N_{i+1} to finish the inductive step.

Lemma. Suppose M\in\binom{\mathbb{N}}{\omega}, s\in\binom{\mathbb{N}}{<\omega} and \max s < \min M. If M strongly accepts every subset of s, then there are m \in M and N \in \binom{M}{\omega} such that n < \min N and N strongly accepts every subset of s\cup \{n\}

Proof of lemma. We can build M = M_0 \supset M_1\supset M_2 \supset \dots such that for every i, m_i := \min M_i < \min M_{i+1} and M_{i+1} decides every subset of s\cup \{m_i\}. It might happen that M_{i+1} rejects a subset of s\cup \{m_i\}. However, we claim that this cannot happen for infinitely many times.

Otherwise, by the pigeonhole principle, there is t\subset s such that I = \{i : M_{i+1} \text{ rejects }t\cup\{m_{i}\}\} is infinite. Let M' = \{m_i : i\in I\}. Note that [t, M'] \subset \cup_i [t\cup\{m_i\}, M_{i+1}], and so [t,M']\cap \mathcal{F}_1\subset \cup_i \left([t\cup\{m_i\}, M_{i+1}]\cap\mathcal{F}_1\right) = \emptyset. Hence M'\subset M rejects t\subset s, which is a contradiction.

Now we pick one i such that M_{i+1} strongly accepts every subset of s\cup\{m_i\}, and it is easy to check that m = m_i and N = M_{i+1} suffice.

Finally, we take N_\infty = \{n_1, n_2, \dots\}. For any s\in\binom{N_\infty}{<\omega}, there is i such that s\subset \{n_1, \dots, n_{i-1}\}. Note that N_i strongly accepts s and N_\infty\subset N_i. Therefore N_\infty (strongly) accepts s, that is [s, N_\infty]\cap \mathcal{F}_1 \neq \emptyset, and say t\in [s, N_\infty]\cap \mathcal{F}_1. Because t\in\mathcal{F}_1 and \mathcal{F} = \mathcal{F}_1 \sqcup \mathcal{F}_2 is a Nash-Williams family, s\notin \mathcal{F}_2.


Alternative to Beamer for Math Presentation

Although using blackboard and chalk is the best option for a math talk for various reasons, sometimes due to limit on the time, one has to make slides to save time on writing. The most common tools to create slides nowadays are LaTeX and Beamer.

When I was preparing for my talk at Vancouver for Connections in Discrete Mathematics in honor of the work of Ron Graham, as it is my first ever conference talk, I decided to ditch Beamer due to my lack of experience. Finally, I ended up using html+css+javascript to leverage my knowledge in web design.

The javascript framework I used is reveal.js. Though there are other options such as impress.js, reveal.js fits better for a math talk. One can easily create a text-based presentation with static images / charts. The framework also has incorporated with MathJax as an optional dependency, which can be added with a few lines of code. What I really like about reveal.js as well as impress.js is that they provide a smooth spatial transition between slides. However, one has to use other javascript library to draw and animate diagrams. For that, I chose raphael.js, a javascript library that uses SVG and VML for creating graphics so that users can easily, for example, create their own specific chart. The source code of the examples on the official website is really a good place to start.

To integrate reveal.js and raphael.js to achieve a step-by-step animation of a diagram, I hacked it by adding a dummy fragment element in my html document so that reveal.js can listen to the fragmentshown event and hence trigger raphael.js to animate the diagram. In cases where the diagrams are made of html elements, I used jQuery to control the animation. Here is my favorite animation in the slides generated by jQuery.

How does mathematics progress?

However, one has to make more effort to reverse the animation made by raphael.js or jQuery if one wants to go backwards in slides. I did not implement any reverse animation since I did not plan to go back in slides at all.

In case there is no internet access during the presentation, one has to have copies of all external javascript libraries (sometimes also fonts), which, in my case, are MathJax, raphael.js and jQuery. In order to use MathJax offline, one need to configure reveal.js.

Currently, my slides only work on Chrome correctly. There is another bug that I have not figured out yet. If I start afresh from the first slide, then my second diagram generated by Raphael is not rendered correctly. I got around it by refreshing the slide where the second diagram lives. This is still something annoying that I would like to resolve.

After all, I really like this alternative approach of making slides for math presentation because it enables me to implement whatever I imagine.


A Short Proof for Hausdorff Moment Problem

Hausdorff moment problem asks for necessary and sufficient conditions that a given sequence (m_n) with m_0=1 be the sequence of moments of a random variable X supported on [0,1], i.e., \operatorname{E}X^n=m_n for all n.

In 1921, Hausdorff showed that (m_n) is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the equation (D^r m)_s \ge 0 for all r, s \ge 0. Here D is the difference operator on the space of real sequences (a_n) given by D a = (a_{n} - a_{n+1}).

The proof under the fold follows the outline given in (E18.5 – E18.6) Probability with Martingales by David Williams.

Proof of Necessity. Suppose (m_n) is the moment sequence of a random variable X supported on [0,1]. By induction, one can show that (D^r m)_s = \operatorname{E}(1-X)^rX^s. Clearly, as X is supported on [0,1], the moment sequence is completely monotonic.

Proof of Sufficiency. Suppose (m_n) is a completely monotonic sequence with m_0 = 1.

Define F_n(x) := \sum_{i \le nx}{n\choose i}(D^{n-i}m)_i. Clearly, F_n is right-continuous and non-decreasing, and F_n(0^-) = 0. To prove F_n(1) = 1, one has to prove the identity \sum_{i}{n\choose i}(D^{n-i}m)_i = m_0.

A classical trick. Since the identity above is about vectors in the linear space (over the reals) spanned by (m_n) and the linear space spanned by (m_n) is isomorphic to the one spanned by (\theta^n), the identity is equivalent to \sum_{i}{n\choose i}(D^{n-i}\theta)_i = \theta^0, where \theta_n = \theta^n. Now, we take advantage of the ring structure of \mathbb{R}[\theta]. Notice that (D^{r}\theta)_s = (1-\theta)^r\theta^s. Using the binomial theorem, we obtain \sum_{i}{n\choose i}(D^{n-i}\theta)_i = \sum_{i}{n\choose i}(1-\theta)^{n-i}\theta^i = (1-\theta + \theta)^n = \theta^0.

Therefore F_n is a bona fide distribution function. Define m_{n, k} := \int_{[0,1]} x^kdF_n(x), i.e., m_{n,k} is the kth moment of F_n. We now find an explicit formula for m_{n,k}.

Noticing that F_n is constant, say c_{n,i}, on [\frac{i}{n}, \frac{i+1}{n}), for all i=0, \dots, n-1 and c_{n,i} is a linear combination of m_0, \dots, m_n, we know that m_{n,k} = \sum_{i=0}^n a_{n,k,i}m_i.

Just like what we did in proving the identity, we use the special case m_n = \theta^n to compute the coefficients a_i = a_{n,k,i}, where 0 \le \theta \le 1. In this case, F_n(x) = \sum_{i \le nx}{n\choose i}(D^{n-i}\theta)_i = \sum_{i\le nx}{n\choose i}(1-\theta)^{n-i}\theta^i, m_{n,k} = \sum_{i=0}^n a_{i}\theta^i.

Now consider the situation in which a coin with probability \theta is tossed at times 1,2,\dots. The random variable H_k is 1 if the kth toss produces heads, 0 otherwise. Define A_n := (H_1 + \dots + H_n)/n. It is immediate from the formula of F_n that F_n is the distribution function of A_n, and so m_{n,k} is the kth moment of A_n. However, one can calculate the kth moment of A_n explicitly. Let f\colon [k] \to [n] be chosen uniformly at random, Im_f be the cardinality of the image of f and denote by p_i = p_{n,k,i} := \operatorname{Pr}(Im_f = i). Using f, Im_f and p_i, we obtain \begin{aligned}\operatorname{E}A_n^k & = \operatorname{E}\left(\frac{H_1 + \dots + H_n}{n}\right)^k = \operatorname{E}H_{f(1)}\dots H_{f(k)} \\ & = \operatorname{E}\operatorname{E}[H_{f(1)}\dots H_{f(k)}\mid Im_f] = \operatorname{E}\theta^{Im_f} = \sum_{i=0}^n p_{i}\theta^i.\end{aligned} Therefore, for all \theta\in [0,1], we know that \sum_{i=0}^n a_i\theta^i = \sum_{i=0}^n p_i\theta^i, and so a_i = p_i for all i=0,\dots, n.

As both (a_i) and (p_i) do not depend on m_i, a_i = p_i holds in general. Since p_k = p_{n, k, k} = \prod_{i=0}^{k-1}(1-i/n)\to 1 as n\to\infty and p_i = 0 for all i > k, we know that \lim_n m_{n,k}= m_k.

Using the Helly–Bray Theorem, since (F_n) is tight, there exists a distribution function F and a subsequence (F_{k_n}) such that F_{k_n} converges weakly to F. The definition of weak convergence implies that \int_{[0,1]} x^k dF(x) = \lim_n \int_{[0,1]}x^k dF_{k_n}(x) = \lim_n m_{k_n,k} = m_k. Therefore, the random variable X with distribution function F is supported on [0,1] and its kth moment is m_k.

There are other two classical moment problems: the Hamburger moment problem and the Stieltjes moment problem.




欺诈猜数游戏在甲和乙之间进行,甲和乙都知道正整数 kn。游戏开始时,甲先选定两个整数 xN,其中 1\leq x\leq N。甲告诉乙 N 的值,但对 x 守口如瓶。乙试图通过提问来获得与 x 相关的信息:每次提问,乙任选一个由若干正整数组成的集合S(可以重复使用之前提问中使用过的集合),问甲“x是否属于S?”。乙可以提任意数量的问题。每次提问之后,甲立刻回答是或否,甲可以说谎话,但在任意连续 k+1 次回答中至少有一次回答是真话。

在乙问完所有想问的问题之后,乙必须指出一个至多包含 n 个正整数的集合 X,若 x 属于 X,则乙获胜;否则甲获胜。证明:对所有充分大的整数 k,存在整数 n\geq 1.99^k,使得乙无法保证获胜。

为了解决第二个问题,需要从甲的角度考虑问题,每一次回答问题后,使得乙无法缩小 x 范围。

为此我们考虑贪心策略:甲每次在乙提出的集合和其补集中选择元素较多的那个集合。很明显这个策略有一个弱点:如果乙每次提问都问“x 是不是 1 呢?”那甲如果每次只顾眼前利益,应当回答“不是。”但经过 k+1 轮之后,甲就能轻松排除 1,从而缩小 x 的范围。

于是甲的策略必须是一种折中的策略,既要顾及眼前的利益,又不能放弃长期的利益。我们设定 N=n+1

于是甲在觉得到底要选择乙提出的集合或是其补集前,对 1n+1 中每个数都进行打分。甲对 i 的打分是 a^{m_i},其中a = 1.999m_i 满足 i 在最近连续 m_i 次选择的集合中都不出现。根据每个元素的打分,甲在 SS^c 中选择会使得之后整个集合总分较低的那个集合。

为了让证明成立,设定 n = (2-a)a^{k+1} - 1。下面,我们通过归纳法证明每一次所有元素的总分不超过 a^{k+1}。根据打分规则,最开始总分是 n = (2-a)a^{k+1}-1 < a^{k+1},命题显然那成立。假设目前的总分是 y = \sum_{i=1}^{n+1} a^{m_i},乙给出集合 S,如果甲选 S,则总和变为 y_1 = \sum_{i\in S}1 + \sum_{i\notin S}a^{m_i+1},如果甲选 S^c,则总和变为 y_2 = \sum_{i\notin S}1 + \sum_{i\in S}a^{m_i+1}。由于甲会选择使得总和变得更小的那个策略,于是他回答后总和至多为 \frac{1}{2}(y_1 + y_2) = \frac{1}{2}(ay+n+1)。由于 y < a^{k+1},总和小于 \frac{1}{2}(a\cdot a^{k+1}+(2-a)a^{k+1}) = a^{k+1}。命题成立!

如果 i 连续 k+1 次没有被甲选中,则其得分已经为 a^{k+1}。这不可能,因为总分总是小于 a^{k+1}。也就是说在这样的策略下乙无法缩小 x 的可能范围。证毕!


An Upper Bound on Stirling Number of the Second Kind

We shall show an upper bound on the Stirling number of the second kind, a byproduct of a homework exercise of Probabilistic Combinatorics offered by Prof. Tom Bohman.

Definition. A Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S(n,k).

Proposition. For all n, k, we have S(n,k) \leq \frac{k^n}{k!}\left(1-(1-1/m)^k\right)^m.

Proof. Consider a random bipartite graph with partite sets U:=[n], V:=[k]. For each vertex u\in U, it (independently) connects to exactly one of the vertices in V uniformly at random. Suppose X is the set of non-isolated vertices in V. It is easy to see that \operatorname{Pr}\left(X=V\right) = \frac{\text{number of surjections from }U\text{ to }V}{k^n} = \frac{k!S(n,k)}{k^n}.

On the other hand, we claim that for any \emptyset \neq A \subset [k] and i \in [k]\setminus A, \operatorname{Pr}\left(i\in X \mid A\subset X\right) \leq \operatorname{Pr}\left(i\in X\right). Note that the claim is equivalent to \operatorname{Pr}\left(A\subset X \mid i\notin X\right) \geq \operatorname{Pr}\left(A\subset X\right). Consider the same random bipartite graph with V replaced by V':=[k]\setminus \{i\} and let X' be the set of non-isolated vertices in V'. The claim is justified since \operatorname{Pr}\left(A\subset X\mid i\notin X\right) = \operatorname{Pr}\left(A\subset X'\right) \geq \operatorname{Pr}\left(A\subset X\right).

Set A:=[i-1] in above for i = 2, \ldots, k. Using the multiplication rule with telescoping the conditional probability, we obtain \begin{aligned}\operatorname{Pr}\left(X=V\right) =& \operatorname{Pr}\left(1\in X\right)\operatorname{Pr}\left(2\in X \mid [1]\subset X\right) \\ & \ldots \operatorname{Pr}\left(k\in X\mid [k-1]\subset X\right)\\ \leq & \operatorname{Pr}\left(1\in X\right)\operatorname{Pr}\left(2\in X\right)\ldots\operatorname{Pr}\left(k\in X\right) \\ = & \left(1-(1-1/m)^k\right)^m.\end{aligned}


A Probabilistic Proof of Isoperimetric Inequality

This note is based on Nicolas Garcia Trillos’ talk, Some Problems and Techniques in Geometric Probability, at Carnegie Mellon University on January 29, 2015.

In particular, we demonstrate a probabilistic proof of the isoperimetric inequality. The proof can also be found in Integral Geometry and Geometric Probability by Luis A. Santaló.

Theorem. For a convex set with perimeter L and area A, the isoperimetric inequality states that 4\pi A\leq L^2, and that equality holds if and only if the convex set is a disk. (Assume that the boundary is a closed convex curve of class C^1.)

Proof. Let P(s) parametrize the boundary by its arc length s. Given s and \theta. Suppose the line through P(s) whose angle to the tangent line equals \theta intersects the boundary at another point Q(s). Let \sigma(s, \theta) be the length of the chord between the two intersections P(s), Q(s). Consider the integral \int (\sigma_1\sin\theta_2 - \sigma_2\sin\theta_1)^2 \mathrm{d}s_1\mathrm{d}\theta_1\mathrm{d}s_2\mathrm{d}\theta_2, where the integration extends over 0 \leq s_1, s_2 \leq L and 0 \leq \theta_1, \theta_2 \leq \pi.

Random variables.

Expanding the square in the integrand, we obtain that the integral is equal to \pi L \int \sigma^2\mathrm{d}s\mathrm{d}\theta - 2\left(\int \sigma\sin\theta\mathrm{d}s\mathrm{d}\theta\right)^2.
On one hand, we have \int \sigma^2\mathrm{d}s\mathrm{d}\theta = \int_0^L\int_0^\pi \sigma^2\mathrm{d}\theta\mathrm{d}s = \int_0^L 2A\mathrm{d}s = 2LA.

Change variables.

On the other hand, let p be the distance from the chord to the origin and \phi the angle from the x-axis to the chord. Suppose the angle from the x-axis to the tangent line is \tau. We have p = \langle x, y\rangle\cdot\langle \sin\phi, -\cos\phi \rangle = x\sin\phi - y\cos\phi. Differentiating the latter, we get \mathrm{d}p = \sin\phi\mathrm{d}x - \cos\phi\mathrm{d}y + (x\cos\phi + y\sin\phi)\mathrm{d}\phi. Moreover, we know that \mathrm{d}x = \cos\tau\mathrm{d}s, \mathrm{d}y = \sin\tau\mathrm{d}s. Therefore \mathrm{d}p = \sin\phi\cos\tau\mathrm{d}s - \cos\phi\sin\tau\mathrm{d}s + + (x\cos\phi + y\sin\phi)\mathrm{d}\phi, and so \mathrm{d}p\mathrm{d}\phi = \sin(\phi - \tau)\mathrm{d}s\mathrm{d}\phi. Since \theta + \phi = \tau and \mathrm{d}\theta + \mathrm{d}\phi = \tau'\mathrm{d}s, we have \mathrm{d}p\mathrm{d}\phi = -\sin\theta\mathrm{d}s\mathrm{d}\theta, and so \int\sigma\sin\theta\mathrm{d}s\mathrm{d}\theta = \int_0^{2\pi}\int_{-\infty}^\infty \sigma\mathrm{d}p\mathrm{d}\theta = 2\pi A.

Since the integral is non-negative, we have that 2\pi A(L^2 - 4\pi A)\geq 0, and so 4\pi A \leq L^2. The equality is achieved if and only if \sigma / \sin\theta is a constant, in which case the boundary is a circle.

Remark. The proof is considered a probabilistic proof because the differential form \mathrm{d}p\mathrm{d}\theta is the measure (invariant under rigid motion) of a random line.


On Extension of Rationals

This note is based on my talk An Expedition to the World of p-adic Numbers at Carnegie Mellon University on January 15, 2014.

Construction from Cauchy Sequences

A standard approach to construct the real numbers from the rational numbers is a procedure called completion, which forces all Cauchy sequences in a metric space by adding new points to the metric space.

Definition. A norm, denoted by |\cdot|, on the field F is a function from F to the set of nonnegative numbers in an ordered field R such that (1) |x|=0 if and only if x=0; (2) |xy|=|x||y|; (3) |x+y|\leq|x|+|y|.

Remark. The ordered field is usually chosen to be \mathbb{R}. However, to construct of \mathbb{R}, the ordered field is \mathbb{Q}.

A norm on the field F naturally gives rise to a metric d(x,y)=|x-y| on F. For example, the standard metric on the rationals is defined by the absolute value, namely, d(x, y) = |x-y|, where x,y\in\mathbb{Q}, and |\cdot| is the standard norm, i.e., the absolute value, on \mathbb{Q}.

Given a metric d on the field F, the completion procedure considers the set of all Cauchy sequences on F and an equivalence relation
(a_n)\sim (b_n)\text{ if and only if }d(a_n, b_n)\to 0\text{ as }n\to\infty.

Definition. Two norms on F, |\cdot|_1, |\cdot|_2, are equivalent if and only if for every sequence (a_n), it is a Cauchy sequence with respect to d_1 if and only if it is so with respect to d_2, where d_1, d_2 are the metrics determined by |\cdot|_1, |\cdot|_2 respectively.

Remark. It is reasonable to worry about the situation in which two norms |\cdot|_1 and |\cdot|_2 are equivalent, but they introduce different equivalent relationships on the set of Cauchy sequences. However, given two equivalent norms, |a_n|_1 converges to 0 if and only if it does so with respect to d_2. (Hint: prove by contradiction and consider the sequence (1/a_n).)

Definition. The trivial norm on F is a norm |\cdot| such that |0|=0 and |x|=1 for x\neq 0.

Since we are interested in norms that generate different completions of the field, it would be great if we can classify all nontrivial norms modulo the norm equivalence.

Alternative Norms on Rationals

Definition. Let p be any prime number. For any non-zero integer a, let \mathrm{ord}_pa be the highest power of p which divides a. For any rational x=a/b, define \mathrm{ord}_px=\mathrm{ord}_pa-\mathrm{ord}_pb. Further define a map |\cdot|_p on \mathbb{Q} as follows: |x|_p = \begin{cases}p^{-\mathrm{ord}_px} & \text{if }x\neq 0 \\ 0 & \text{if }x=0\end{cases}.

Proposition. |\cdot|_p is a norm on \mathbb{Q}. We call it the p-adic norm on \mathbb{Q}.

Proof (sketch). We only check the triangle inequality. Notice that \begin{aligned}\mathrm{ord}_p\left(\frac{a}{b}-\frac{c}{d}\right) & = \mathrm{ord}_p\left(\frac{ad-bc}{bd}\right) \\ & = \mathrm{ord}_p(ad-bc)-\mathrm{ord}_p(bd) \\ & \geq \min(\mathrm{ord}_p(ad), \mathrm{ord}_p(bc)) - \mathrm{ord}_p(bd) \\ &= \min(\mathrm{ord}_p(ad/bd), \mathrm{ord}_p(bc/bd)).\end{aligned} Therefore, we obtain |a/b-c/d|_p \leq \max(|a/b|_p, |c/d|_p) \leq |a/b|_p+|c/d|_p.

Remark. Some counterintuitive fact about the p-adic norm are
The following theorem due to Ostrowski classifies all possible norms on the rationals up to norm equivalence. We denote the standard absolute value by |\cdot|_\infty.

Theorem (Ostrowski 1916). Every nontrivial norm |\cdot| on \mathbb{Q} is equivalent to |\cdot|_p for some prime p or for p=\infty.

Proof. We consider two cases (i) There is n\in\{1,2,\ldots\} such that |n|>1; (ii) For all n\in\{1,2,\ldots\}, |n|\leq 1. As we shall see, in the 1st case, the norm is equivalent to |\cdot|_\infty, whereas, in the 2nd case, the norm is equivalent to |\cdot|_p for some prime p.

Case (i). Let n_0 be the least such n\in\{1,2,\ldots\} such that |n|>1. Let \alpha > 0 be such that |n_0|=n_0^\alpha.

For every positive integer n, if n_0^s \leq n < n_0^{s+1}, then we can write it in n_0-base: n = a_0 + a_1n_0 + \ldots + a_sn_0^s.

By the choice of n_0, we know that |a_i|\leq 1 for all i. Therefore, we obtain \begin{aligned}|n| & \leq |a_0| + |a_1||n_0| + \ldots + |a_s||n_0|^s \\ & \leq 1 + |n_0| + \ldots |n_0|^s \\ & \leq n_0^{s\alpha}\left(1+n_0^{-\alpha}+n_0^{-2\alpha}+\ldots\right) \\ & \leq Cn^\alpha,\end{aligned} where C does not depend on n. Replace n by n^N and get |n|^N = |n^N| \leq Cn^{N\alpha}, and so |n|\leq \sqrt[N]{C}n^\alpha. As we can choose N to be arbitrarily large, we obtain |n| \leq n^\alpha.

On the other hand, we have \begin{aligned}|n| & \geq |n_0^{s+1}| - |n_0^{s+1}-n| \\ & \geq n_0^{(s+1)\alpha} - (n_0^{s+1}-n_0^s)^\alpha\\ & = n_0^{(s+1)\alpha}\left[1-(1-1/n_0)^\alpha\right] \\ & > C'n^\alpha.\end{aligned} Using the same trick, we can actually take C'=1.

Therefore |n| = n^\alpha. It is easy to see it is equivalent to |\cdot|_\infty.

Case (ii). Since the norm is nontrivial, let n_0 be the least n such that |n|<1.

Claim 1. n_0=p is a prime.

Claim 2. |q|=1 if q is a prime other than p.

Suppose |q| < 1. Find M large enough so that both |p^M| and |q^M| are less than 1/2. By Bézout’s lemma, there exists a,b\in\mathbb{Z} such that ap^M + bq^M = 1. However, 1 = |1| \leq |a||p^M| + |b||q^M| < 1/2 + 1/2 = 1, a contradiction.

Therefore, we know |n|=|p|^{ord_p n}. It is easy to see it is equivalent to |\cdot|_p.

Non-Archimedean Norm

As one might have noticed, the p-adic norm satisfies an inequality stronger than the triangle inequality, namely |a\pm b|_p\leq \max(|x|_p, |y|_p).

Definition. A norm is non-Archimedean provided |x\pm y|\leq \max(|x|, |y|).

The world of non-Archimedean norm is good and weird. Here are two testimonies.

Proposition (no more scalene triangles). If |x|\neq |y|, then |x\pm y| = \max(|x|, |y|).

Proof. Suppose |x| < |y|. On one hand, we have |x\pm y| \leq |y|. On the other hand, |y| \leq \max(|x\pm y|, |x|). Since |x| < |y|, we must have |y| \leq |x\pm y|.

Proposition (all points are centers). D(a, r^-) = D(b, r^-) for all b\in D(a, r^-) and D(a, r) = D(b,r) for all b\in D(a,r), where D(c, r^-) = \{x : |x-c|<r\} and D(c,r)=\{x:|x-c|\leq r\}.

Construction of p-adic Numbers

The p-adic numbers are the completion of \mathbb{Q} via the p-adic norm.

Definition. The set of p-adic numbers is defined as \mathbb{Q}_p = \{\text{Cauchy sequences with respect to }|\cdot|_p\} / \sim_p, where (a_n)\sim_p(b_n) iff |a_n-b_n|_p\to 0 as n\to\infty.

We would like to extend |\cdot|_p from \mathbb{Q} to \mathbb{Q}_p. When extending |\cdot|_\infty from \mathbb{Q} to \mathbb{R}, we set |[(a_n)]|_\infty to be [(|a_n|)], an element in \mathbb{R}. However, the values that |\cdot|_p can take, after the extension, are still in \mathbb{Q}.

Definition. The extension of |\cdot|_p on \mathbb{Q}_p is defined by |[(a_n)]|_p = \lim_{n\to\infty}|a_n|_p.

Remark. Suppose (a_n)\sim_p (a_n'). Then \lim_{n\to\infty}|a_n-a_n'|_p=0, and so \lim_{n\to\infty}|a_n|_p=\lim_{n\to\infty}|a_n'|_p. Moreover, one can prove that \lim_{n\to\infty}|a_n|_p always exists provided that (a_n) is a Cauchy sequence. (Hint: Suppose \lim_{n\to\infty}|a_n|_p > 0. There exists \epsilon > 0 such that |a_n|_p > \epsilon infinitely often. Choose N enough so that |a_m - a_n|_p < \epsilon for all m,n>N. Use ‘no more scalene triangles!’ property to deduce a contradiction.)

Representation of p-adic Numbers

Even though each real number is an equivalence class of Cauchy sequences, each equivalence class has a canonical representative. For instance, the canonical representative of \pi is (3, 3.1, 3.14, 3.141, 3.1415, \ldots). The analog for \mathbb{Q}_p is the following.

Theorem. Every equivalence class a in \mathbb{Q}_p for which |a|_p\leq 1 has exactly one representative Cauchy sequence of the form (a_i) for which (1) 0\leq a_i < p^i for i=1,2,3,\ldots; (2) a_i = a_{i+1} (\pmod p^i) for i=1,2,3,\ldots.

Proof of uniqueness. Prove by definition chasing.

Proof of existence. We shall repeatedly apply the following lemma.

Lemma. For every b\in\mathbb{Q} for which |b|_p \leq 1 and i\in\mathbb{N}, there exists a\in\{0, \ldots, p^i-1\} such that |a-b|_p \leq p^{-i}.

Proof of Lemma. Suppose b=m/n is in the lowest form. As |b|_p\leq 1, we know that (n, p^i)=1. By Bézout’s lemma, an+a'p^i=m for some integers a,a'. We may assume a\in\{0,\ldots,p^i-1\}. Note that a-b=a'p^i/n, and so |a-b|_p \leq p^{-i}.

Suppose (c_i) is a representative of a. As (c_i) is a Cauchy sequence, we can extract a subsequence (b_i) such that |b_i - b_j| \leq p^{-i} for all i < j which is still a representative of a. Using the lemma above, for each b_i, we can find 0 \leq a_i < q^i such that |a_i-b_i|_p \leq q^{-i}. Therefore (a_i) is a representative of a as well. For all i<j, we have |a_i-a_j|_p \leq \max(|a_i - b_i|_p, |b_i-b_j|_p, |b_j-a_j|_p) \leq q^{-i}, Therefore q^i divides a_i - a_j.

For |a|_p\leq 1, we write a=b_0 + b_1p + b_2p^2 + \ldots, where (b_{n-1}b_{n-2}\ldots b_0)_p = a_n.

What if |a|_p > 1? As |ap^m|=|a|/p^m, |ap^m|\leq 1 for some natural number m. By the representation theorem, we can write ap^m = b_0 + b_1p + b_2p^2 + \ldots, and a = b_0p^{-m} + b_1p^{-m+1} + b_2p^{-m+2} + \ldots.

Using the representation of p-adic numbers, one can perform arithmetic operations such as addition, subtraction, multiplication and division.

Like \mathbb{R}, \mathbb{Q}_p is not algebraically closed. Though \mathbb{C}, the algebraic closure of \mathbb{R}, has degree 2 over \mathbb{R}, and it is complete with respect to the absolute value, it is not so for \overline{\mathbb{Q}_p}, the algebraic closure of \mathbb{Q}_p. In fact, \overline{\mathbb{Q}_p} has infinite degree over \mathbb{Q}_p and is, unfortunately, not complete with respect to proper extension of |\cdot|_p. The good news is that the completion of \overline{\mathbb{Q}_p}, denoted by \Omega is algebraically closed.