Vinogradov’s Three-Prime Theorem (with Sammy Luo and Ryan Alweiss)

This was my final paper for 18.099, seminar in discrete analysis, jointly with Sammy Luo and Ryan Alweiss.

We prove that every sufficiently large odd integer can be written as the sum of three primes, conditioned on a strong form of the prime number theorem.

1. Introduction

In this paper, we prove the following result:

Theorem 1 (Vinogradov)

Every sufficiently large odd integer {N} is the sum of three prime numbers.

In fact, the following result is also true, called the “weak Goldbach conjecture”.

Theorem 2 (Weak Goldbach conjecture)

Every odd integer {N \ge 7} is the sum of three prime numbers.

The proof of Vinogradov’s theorem becomes significantly simpler if one assumes the generalized Riemann hypothesis; this allows one to use a strong form of the prime number theorem (Theorem 9). This conditional proof was given by Hardy and Littlewood in the 1923’s. In 1997, Deshouillers, Effinger, te Riele and Zinoviev showed that the generalized Riemann hypothesis in fact also implies the weak Goldbach conjecture by improving the bound to {10^{20}} and then exhausting the remaining cases via a computer search.

As for unconditional proofs, Vinogradov was able to eliminate the dependency on the generalized Riemann hypothesis in 1937, which is why the Theorem 1 bears his name. However, Vinogradov’s bound used the ineffective Siegel-Walfisz theorem; his student K. Borozdin showed that {3^{3^{15}}} is large enough. Over the years the bound was improved, until recently in 2013 when Harald Helfgott claimed the first unconditional proof of Theorem 2, see here.

In this exposition we follow Hardy and Littlewood’s approach, i.e. we prove Theorem 1 assuming the generalized Riemann hypothesis, following the exposition of Rhee. An exposition of the unconditional proof by Vinogradov is given by Rouse.

2. Synopsis

We are going to prove that

\displaystyle  	\sum_{a+b+c = N} \Lambda(a) \Lambda(b) \Lambda(c) \asymp \frac12 N^2 \mathfrak G(N) 	 \ \ \ \ \ (1)

where

\displaystyle  \mathfrak G(N) 	\overset{\text{def}}{=} \prod_{p \mid N} \left( 1 - \frac{1}{(p-1)^2} \right) 	\prod_{p \nmid N} \left( 1 + \frac{1}{(p-1)^3} \right)

and {\Lambda} is the von Mangoldt function defined as usual. Then so long as {2 \nmid N}, the quantity {\mathfrak G(N)} will be bounded away from zero; thus (1) will imply that in fact there are many ways to write {N} as the sum of three distinct prime numbers.

The sum (1) is estimated using Fourier analysis. Let us define the following.

Definition 3

Let {\mathbb T = \mathbb R/\mathbb Z} denote the circle group, and let {e : \mathbb T \rightarrow \mathbb C} be the exponential function {\theta \mapsto \exp(2\pi i \theta)}. For {\alpha\in\mathbb T}, {\|\alpha\|} denotes the minimal distance from {\alpha} to an integer.

Note that {|e(\theta)-1|=\Theta(\|\theta\|)}.

Definition 4

For {\alpha \in \mathbb T} and {x > 0} we define

\displaystyle  S(x, \alpha) = \sum_{n \le x} \Lambda(n) e(n\alpha).

Then we can rewrite (1) using {S} as a “Fourier coefficient”:

Proposition 5

We have

\displaystyle  		\sum_{a+b+c = N} \Lambda(a) \Lambda(b) \Lambda(c) 		= \int_{\alpha \in \mathbb T} S(N, \alpha)^3 e(-N\alpha) \; d\alpha. 		 	\ \ \ \ \ (2)

Proof: We have

\displaystyle S(N,\alpha)^3=\sum_{a,b,c\leq N}\Lambda(a)\Lambda(b)\Lambda(c)e((a+b+c)\alpha),

so

\displaystyle  \begin{aligned} \int_{\alpha \in \mathbb T} S(N, \alpha)^3 e(-N\alpha) \; d\alpha &= \int_{\alpha \in \mathbb T} \sum_{a,b,c\leq N}\Lambda(a)\Lambda(b)\Lambda(c)e((a+b+c)\alpha) e(-N\alpha) \; d\alpha \\ &= \sum_{a,b,c\leq N}\Lambda(a)\Lambda(b)\Lambda(c)\int_{\alpha \in \mathbb T}e((a+b+c-N)\alpha) \; d\alpha \\ &= \sum_{a,b,c\leq N}\Lambda(a)\Lambda(b)\Lambda(c)I(a+b+c=N) \\ &= \sum_{a+b+c=N}\Lambda(a)\Lambda(b)\Lambda(c), \end{aligned}

as claimed. \Box

In order to estimate the integral in Proposition 5, we divide {\mathbb T} into the so-called “major” and “minor” arcs. Roughly,

  • The “major arcs” are subintervals of {\mathbb T} centered at a rational number with small denominator.
  • The “minor arcs” are the remaining intervals.

These will be made more precise later. This general method is called the Hardy-Littlewood circle method, because of the integral over the circle group {\mathbb T}.

The rest of the paper is structured as follows. In Section 3, we define the Dirichlet character and other number-theoretic objects, and state some estimates for the partial sums of these objects conditioned on the Riemann hypothesis. These bounds are then used in Section 4 to provide corresponding estimates on {S(x, \alpha)}. In Section 5 we then define the major and minor arcs rigorously and use the previous estimates to given an upper bound for the integral over both areas. Finally, we complete the proof in Section 6.

3. Prime number theorem type bounds

In this section, we collect the necessary number-theoretic results that we will need. It is in this section only that we will require the generalized Riemann hypothesis.

As a reminder, the notation {f(x)\ll g(x)}, where {f} is a complex function and {g} a nonnegative real one, means {f(x)=O(g(x))}, a statement about the magnitude of {f}. Likewise, {f(x)=g(x)+O(h(x))} simply means that for some {C}, {|f(x)-g(x)|\leq C|h(x)|} for all sufficiently large {x}.

3.1. Dirichlet characters

In what follows, {q} denotes a positive integer.

Definition 6

A Dirichlet character modulo {q} {\chi} is a homomorphism {\chi : (\mathbb Z/q)^\times \rightarrow \mathbb C^\times}. It is said to be trivial if {\chi = 1}; we denote this character by {\chi_0}.

By slight abuse of notation, we will also consider {\chi} as a function {\mathbb Z \rightarrow \mathbb C^\ast} by setting {\chi(n) = \chi(n \pmod q)} for {\gcd(n,q) = 1} and {\chi(n) = 0} for {\gcd(n,q) > 1}.

Remark 7

The Dirichlet characters form a multiplicative group of order {\phi(q)} under multiplication, with inverse given by complex conjugation. Note that {\chi(m)} is a primitive {\phi(q)}th root of unity for any {m \in (\mathbb Z/q)^\times}, thus {\chi} takes values in the unit circle.

Moreover, the Dirichlet characters satisfy an orthogonality relation

Experts may recognize that the Dirichlet characters are just the elements of the Pontryagin dual of {(\mathbb Z/q)^\times}. In particular, they satisfy an orthogonality relationship

\displaystyle  	\frac{1}{\phi(q)} 	\sum_{\chi \text{ mod } q} \chi(n) \overline{\chi(a)} 	= \begin{cases} 		1 & n = a \pmod q \\ 		0 & \text{otherwise} 	\end{cases} 	 \ \ \ \ \ (3)

and thus form an orthonormal basis for functions {(\mathbb Z/q)^\times \rightarrow \mathbb C}.

3.2. Prime number theorem for arithmetic progressions

Definition 8

The generalized Chebyshev function is defined by

\displaystyle  \psi(x, \chi) = \sum_{n \le x} \Lambda(n) \chi(n).

The Chebyshev function is studied extensively in analytic number theory, as it is the most convenient way to phrase the major results of analytic number theory. For example, the prime number theorem is equivalent to the assertion that

\displaystyle  \psi(x, \chi_0) = \sum_{n \le x} \Lambda(n) \asymp x

where {q = 1} (thus {\chi_0} is the constant function {1}). Similarly, Dirichlet’s theorem actually asserts that any {q \ge 1},

\displaystyle  	\psi(x, \chi) 	= \begin{cases} 		x + o_q(x) & \chi = \chi_0 \text{ trivial} \\ 		o_q(x) & \chi \neq \chi_0 \text{ nontrivial}. 	\end{cases}

However, the error term in these estimates is quite poor (more than {x^{1-\varepsilon}} for every {\varepsilon}). However, by assuming the Riemann Hypothesis for a certain “{L}-function” attached to {\chi}, we can improve the error terms substantially.

Theorem 9 (Prime number theorem for arithmetic progressions)

Let {\chi} be a Dirichlet character modulo {q}, and assume the Riemann hypothesis for the {L}-function attached to {\chi}.

  1. If {\chi} is nontrivial, then

    \displaystyle  \psi(x, \chi) \ll \sqrt{x} (\log qx)^2.

  2. If {\chi = \chi_0} is trivial, then

    \displaystyle  \psi(x, \chi_0) = x + O\left( \sqrt x (\log x)^2 + \log q \log x \right).

Theorem 9 is the strong estimate that we will require when putting good estimates on {S(x, \alpha)}, and is the only place in which the generalized Riemann Hypothesis is actually required.

3.3. Gauss sums

Definition 10

For {\chi} a Dirichlet character modulo {q}, the Gauss sum {\tau(\chi)} is defined by

\displaystyle \tau(\chi)=\sum_{a=0}^{q-1}\chi(a)e(a/q).

We will need the following fact about Gauss sums.

Lemma 11

Consider Dirichlet characters modulo {q}. Then:

  1. We have {\tau(\chi_0) = \mu(q)}.
  2. For any {\chi} modulo {q}, {\left\lvert \tau(\chi) \right\rvert \le \sqrt q}.

3.4. Dirichlet approximation

We finally require Dirichlet approximation theorem in the following form.

Theorem 12 (Dirichlet approximation)

Let {\alpha \in \mathbb R} be arbitrary, and {M} a fixed integer. Then there exists integers {a} and {q = q(\alpha)}, with {1 \le q \le M} and {\gcd(a,q) = 1}, satisfying

\displaystyle  \left\lvert \alpha - \frac aq \right\rvert \le \frac{1}{qM}.

4. Bounds on {S(x, \alpha)}

In this section, we use our number-theoretic results to bound {S(x,\alpha)}.

First, we provide a bound for {S(x,\alpha)} if {\alpha} is a rational number with “small” denominator {q}.

Lemma 13

Let {\gcd(a,q) = 1}. Assuming Theorem 9, we have

\displaystyle  S(x, a/q) 		= \frac{\mu(q)}{\phi(q)} x + O\left( \sqrt{qx} (\log qx)^2 \right)

where {\mu} denotes the Möbius function.

Proof: Write the sum as

\displaystyle  S(x, a/q) = \sum_{n \le x} \Lambda(n) e(na/q).

First we claim that the terms {\gcd(n,q) > 1} (and {\Lambda(n) \neq 0}) contribute a negligibly small {\ll \log q \log x}. To see this, note that

  • The number {q} has {\ll \log q} distinct prime factors, and
  • If {p^e \mid q}, then {\Lambda(p) + \dots + \Lambda(p^e) 			= e\log p = \log(p^e) < \log x}.

So consider only terms with {\gcd(n,q) = 1}. To bound the sum, notice that

\displaystyle  \begin{aligned} 		e(n \cdot a/q) &= \sum_{b \text{ mod } q} e(b/q) \cdot \mathbf 1(b \equiv an) \\ 		&= \sum_{b \text{ mod } q} e(b/q) \left( \frac{1}{\phi(q)} 			\sum_{\chi \text{ mod } q} \chi(b) \overline{\chi(an)} \right) 	\end{aligned}

by the orthogonality relations. Now we swap the order of summation to obtain a Gauss sum:

\displaystyle  \begin{aligned} 		e(n \cdot a/q) &= \frac{1}{\phi(q)} \sum_{\chi \text{ mod } q} \overline{\chi(an)} 			\left( \sum_{b \text{ mod } q} \chi(b) e(b/q) \right) \\ 		&= \frac{1}{\phi(q)} \sum_{\chi \text{ mod } q} \overline{\chi(an)} \tau(\chi). 	\end{aligned}

Thus, we swap the order of summation to obtain that

\displaystyle  \begin{aligned} 		S(x, \alpha) &= \sum_{\substack{n \le x \\ \gcd(n,q) = 1}} 			\Lambda(n) e(n \cdot a/q) \\ 		&= \frac{1}{\phi(q)} \sum_{\substack{n \le x \\ \gcd(n,q) = 1}} 			\sum_{\chi \text{ mod } q} \Lambda(n) \overline{\chi(an)} \tau(\chi) \\ 		&= \frac{1}{\phi(q)} \sum_{\chi \text{ mod } q} \tau(\chi) 			\sum_{\substack{n \le x \\ \gcd(n,q) = 1}} \Lambda(n) \overline{\chi(an)} \\ 		&= \frac{1}{\phi(q)} \sum_{\chi \text{ mod } q} \overline{\chi(a)} \tau(\chi) 			\sum_{\substack{n \le x \\ \gcd(n,q) = 1}} \Lambda(n)\overline{\chi(n)} \\ 		&= \frac{1}{\phi(q)} \sum_{\chi \text{ mod } q} \overline{\chi(a)} 			\tau(\chi) \psi(x, \overline\chi) \\ 		&= \frac{1}{\phi(q)} \left( \tau(\chi_0) \psi(x, \chi_0) 			+ \sum_{1 \neq \chi \text{ mod } q} \overline{\chi(a)} \tau(\chi) 			\psi(x, \overline\chi) \right). 	\end{aligned}

Now applying both parts of Lemma 11 in conjunction with Theorem 9 gives

\displaystyle  \begin{aligned} 		S(x,\alpha) 		&= \frac{\mu(q)}{\phi(q)} 			\left( x + O\left( \sqrt x (\log qx)^2 \right) \right) 			+ O\left( \sqrt x (\log x)^2 \right) \\ 		&= \frac{\mu(q)}{\phi(q)} x + O\left( \sqrt{qx} (\log qx)^2 \right) 	\end{aligned}

as desired. \Box

We then provide a bound when {\alpha} is “close to” such an {a/q}.

Lemma 14

Let {\gcd(a,q) = 1} and {\beta \in \mathbb T}. Assuming Theorem 9, we have

\displaystyle  		S(x, a/q + \beta) = 		\frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(\beta n) \right) 		+ O\left( (1+\|\beta\|x) \sqrt{qx} (\log qx)^2 \right).

Proof: For convenience let us assume {x \in \mathbb Z}. Let {\alpha = a/q + \beta}. Let us denote {\text{Err}(x, \alpha) 		= S(x,\alpha) - \frac{\mu(q)}{\phi(q)} x}, so by Lemma 13 we have {\text{Err}(x,\alpha) \ll \sqrt{qx}(\log x)^2}. We have

\displaystyle  \begin{aligned} 		S(x, \alpha) &= \sum_{n \le x} \Lambda(n) e(na/q) e(n\beta) \\ 		&= \sum_{n \le x} e(n\beta) \left( S(n, a/q) - S(n-1, a/q) \right) \\ 		&= \sum_{n \le x} e(n\beta) \left( 			\frac{\mu(q)}{\phi(q)} 			+ \text{Err}(n, \alpha) - \text{Err}(n-1, \alpha) \right) \\ 		&= \frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(n\beta) \right) 			+ \sum_{1 \le m \le x-1} \left( e( (m+1)\beta) - e( m\beta ) \right) 			\text{Err}(m, \alpha) \\ 		&\qquad + e(x\beta) \text{Err}(x, \alpha) - e(0) \text{Err}(0, \alpha) \\ 		&\le \frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(n\beta) \right) 			+ \left( \sum_{1 \le m \le x-1} \|\beta\| \text{Err}(m, \alpha) \right) 			+ \text{Err}(0, \alpha) + \text{Err}(x, \alpha) \\ 		&\ll \frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(n\beta) \right) 			+ \left( 1+x\left\| \beta \right\| \right) 			O\left( \sqrt{qx} (\log qx)^2 \right) 	\end{aligned}

as desired. \Box

Thus if {\alpha} is close to a fraction with small denominator, the value of {S(x, \alpha)} is bounded above. We can now combine this with the Dirichlet approximation theorem to obtain the following general result.

Corollary 15

Suppose {M = N^{2/3}} and suppose {\left\lvert \alpha - a/q \right\rvert \le \frac{1}{qM}} for some {\gcd(a,q) = 1} with {q \le M}. Assuming Theorem 9, we have

\displaystyle  S(x, \alpha) \ll \frac{x}{\varphi(q)} + x^{\frac56+\varepsilon}

for any {\varepsilon > 0}.

Proof: Apply Lemma 14 directly. \Box

5. Estimation of the arcs

We’ll write

\displaystyle  f(\alpha) \overset{\text{def}}{=} S(N,\alpha)=\sum_{n \le N} \Lambda(n)e(n\alpha)

for brevity in this section.

Recall that we wish to bound the right-hand side of (2) in Proposition 5. We split {[0,1]} into two sets, which we call the “major arcs” and the “minor arcs.” To do so, we use Dirichlet approximation, as hinted at earlier.

In what follows, fix

\displaystyle  \begin{aligned} 	M &= N^{2/3} \\ 	K &= (\log N)^{10}. \end{aligned}

5.1. Setting up the arcs

Definition 16

For {q \le K} and {\gcd(a,q) = 1}, {1 \le a \le q}, we define

\displaystyle  		\mathfrak M(a,q) = \left\{ \alpha \in \mathbb T 		\mid \left\lvert \alpha - \frac aq \right\rvert \le \frac 1M \right\}.

These will be the major arcs. The union of all major arcs is denoted by {\mathfrak M}. The complement is denoted by {\mathfrak m}.

Equivalently, for any {\alpha}, consider {q = q(\alpha) \le M} as in Theorem 12. Then {\alpha \in \mathfrak M} if {q \le K} and {\alpha \in \mathfrak m} otherwise.

Proposition 17

{\mathfrak M} is composed of finitely many disjoint intervals {\mathfrak M(a,q)} with {q \le K}. The complement {\mathfrak m} is nonempty.

Proof: Note that if {q_1, q_2 \le K} and {a/q_1 \neq b/q_2} then {\left\lvert \frac{a}{q_1} - \frac{b}{q_2} \right\rvert 	\ge \frac{1}{q_1q_2} \gg \frac{3}{qM}}. \Box

In particular both {\mathfrak M} and {\mathfrak m} are measurable. Thus we may split the integral in (2) over {\mathfrak M} and {\mathfrak m}. This integral will have large magnitude on the major arcs, and small magnitude on the minor arcs, so overall the whole interval {[0,1]} it will have large magnitude.

5.2. Estimate of the minor arcs

First, we note the well known fact {\phi(q) \gg q/\log q}. Note also that if {q=q(\alpha)} as in the last section and {\alpha} is on a minor arc, we have {q > (\log N)^{10}}, and thus {\phi(q) \gg (\log N)^{9}}.

As such Corollary 3.3 yields that {f(\alpha) \ll \frac{N}{\phi(q)}+N^{.834} \ll \frac{N}{(\log N)^9}}.

Now,

\displaystyle  \begin{aligned} 	\left\lvert \int_{\mathfrak m}f(\alpha)^3e(-N\alpha) \; d\alpha \right\rvert 	&\le \int_{\mathfrak m}\left\lvert f(\alpha)\right\rvert ^3 \; d\alpha \\ 	&\ll \frac{N}{(\log N)^9} \int_{0}^{1}\left\lvert f(\alpha)\right\rvert ^2 \;d\alpha \\ 	&=\frac{N}{(\log N)^9}\int_{0}^{1}f(\alpha)f(-\alpha) \; d\alpha \\ 	&=\frac{N}{(\log N)^9}\sum_{n \le N} \Lambda(n)^2 \\ 	&\ll \frac{N^2}{(\log N)^8}, \end{aligned}

using the well known bound {\sum_{n \le N} \Lambda(n)^2 \ll \frac{N}{\log N}}. This bound of {\frac{N^2}{(\log N)^8}} will be negligible compared to lower bounds for the major arcs in the next section.

5.3. Estimate on the major arcs

We show that

\displaystyle  \int_{\mathfrak M}f(\alpha)^3e(-N\alpha) d\alpha \asymp \frac{N^2}{2} \mathfrak G(N).

By Proposition 17 we can split the integral over each interval and write

\displaystyle  \int_{\mathfrak M} f(\alpha)^3e(-N\alpha) \; d\alpha 	= \sum_{q \le (\log N)^{10}}\sum_{\substack{1 \le a \le q \\ \gcd(a,q)=1}} 	\int_{-1/qM}^{1/qM}f(a/q+\beta)^3e(-N(a/q+\beta)) \; d\beta.

Then we apply Lemma 14, which gives

\displaystyle  \begin{aligned} 	f(a/q+\beta)^3 	&= \left(\frac{\mu(q)}{\phi(q)}\sum_{n \le N}e(\beta n) \right)^3 \\ 	&+\left(\frac{\mu(q)}{\phi(q)}\sum_{n \le N}e(\beta n)\right)^2 		O\left((1+\|\beta\|N)\sqrt{qN} \log^2 qN\right) \\ 	&+\left(\frac{\mu(q)}{\phi(q)}\sum_{n \le N}e(\beta n)\right) 		O\left((1+\|\beta\|N)\sqrt{qN} \log^2 qN\right)^2 \\ 	&+O\left((1+\|\beta\|N)\sqrt{qN} \log^2 qN\right)^3. \end{aligned}

Now, we can do casework on the side of {N^{-.9}} that {\|\beta\|} lies on.

  • If {\|\beta\| \gg N^{-.9}}, we have {\sum_{n \le N}e(\beta n) \ll \frac{2}{|e(\beta)-1|} 	\ll \frac{1}{\|\beta\|} \ll N^{.9}}, and {(1+\|\beta\|N)\sqrt{qN} \log^2 qN \ll N^{5/6+\varepsilon}}, because certainly we have {\|\beta\|<1/M=N^{-2/3}}.
  • If on the other hand {\|\beta\|\ll N^{-.9}}, we have {\sum_{n \le N}e(\beta n) \ll N} obviously, and {O(1+\|\beta\|N)\sqrt{qN} \log^2 qN) \ll N^{3/5+\varepsilon}}.

As such, we obtain

\displaystyle  f(a/q+\beta)^3 \ll \left( \frac{\mu(q)}{\phi(q)}\sum_{n \le N}e(\beta n) \right)^3 	+ O\left(N^{79/30+\varepsilon}\right)

in either case. Thus, we can write

\displaystyle  \begin{aligned} 	&\qquad \int_{\mathfrak M} f(\alpha)^3e(-N\alpha) \; d\alpha \\ 	&= \sum_{q \le (\log N)^{10}} \sum_{\substack{1 \le a \le q \\ \gcd(a,q)=1}} 	\int_{-1/qM}^{1/qM} f(a/q+\beta)^3e(-N(a/q+\beta)) \; d\beta \\ 	&= \sum_{q \le (\log N)^{10}} \sum_{\substack{1 \le a \le q \\ \gcd(a,q)=1}} 		\int_{-1/qM}^{1/qM}\left[\left(\frac{\mu(q)}{\phi(q)}\sum_{n \le N}e(\beta n)\right)^3 		+ O\left(N^{79/30+\varepsilon}\right)\right]e(-N(a/q+\beta)) \; d\beta \\ 	&=\sum_{q \le (\log N)^{10}} \frac{\mu(q)}{\phi(q)^3} S_q 		\left(\sum_{\substack{1 \le a \le q \\ \gcd(a,q)=1}} e(-N(a/q))\right) 		\left( \int_{-1/qM}^{1/qM}\left(\sum_{n \le N}e(\beta n)\right)^3e(-N\beta) 		\; d\beta \right ) \\ 	&\qquad +O\left(N^{59/30+\varepsilon}\right). \end{aligned}

just using {M \le N^{2/3}}. Now, we use

\displaystyle  \sum_{n \le N}e(\beta n) = \frac{1-e(\beta N)}{1-e(\beta)} 	\ll \frac{1}{\|\beta\|}.

This enables us to bound the expression

\displaystyle  \int_{1/qM}^{1-1/qM}\left (\sum_{n \le N}e(\beta n)\right) ^ 3 e(-N\beta)d\beta 	\ll \int_{1/qM}^{1-1/qM}\|\beta\|^{-3} d\beta = 2\int_{1/qM}^{1/2}\beta^{-3} d\beta 	\ll q^2M^2.

But the integral over the entire interval is

\displaystyle  \begin{aligned} 	\int_{0}^{1}\left(\sum_{n \le N}e(\beta n) \right)^3 e(-N\beta)d\beta 	&= \int_{0}^{1} \sum_{a,b,c \le N} e((a+b+c-N)\beta) \\ 	&\ll \sum_{a,b,c \le N} \mathbf 1(a+b+c=N) \\ 	&= \binom{N-1}{2}. \end{aligned}

Considering the difference of the two integrals gives

\displaystyle  \int_{-1/qM}^{1/qM}\left(\sum_{n \le N}e(\beta n) \right)^3 	e(-N\beta) \; d\beta - \frac{N^2}{2} \ll q^2 M^2 + N 	\ll (\log N)^c N^{4/3},

for some absolute constant {c}.

For brevity, let

\displaystyle  S_q = \sum_{\substack{1 \le a \le q \\ \gcd(a,q)=1}} e(-N(a/q)).

Then

\displaystyle  \begin{aligned} 	\int_{\mathfrak M} f(\alpha)^3e(-N\alpha) \; d\alpha &= \sum_{q \le (\log N)^{10}} \frac{\mu(q)}{\phi(q)^3}S_q 		\left( \int_{-1/qM}^{1/qM}\left(\sum_{n \le N}e(\beta n)\right)^3e(-N\beta) 		\; d\beta \right ) \\ 	&\qquad +O\left(N^{59/30+\varepsilon}\right) \\ &= \frac{N^2}{2}\sum_{q \le (\log N)^{10}} 	\frac{\mu(q)}{\phi(q)^3}S_q + O((\log N)^{10+c} N^{4/3}) 		+ O(N^{59/30+\varepsilon}) \\ &= \frac{N^2}{2}\sum_{q \le (\log N)^{10}} \frac{\mu(q)}{\phi(q)^3} 		+ O(N^{59/30+\varepsilon}). \end{aligned}

.

The inner sum is bounded by {\phi(q)}. So,

\displaystyle \left\lvert \sum_{q>(\log N)^{10}} 	\frac{\mu(q)}{\phi(q)^3} S_q \right\rvert 	 \le \sum_{q>(\log N)^{10}} \frac{1}{\phi(q)^2},

which converges since {\phi(q)^2 \gg q^c} for some {c > 1}. So

\displaystyle  \int_{\mathfrak M} f(\alpha)^3e(-N\alpha) \; d\alpha 	= \frac{N^2}{2}\sum_{q = 1}^\infty \frac{\mu(q)}{\phi(q)^3}S_q 	+ O(N^{59/30+\varepsilon}).

Now, since {\mu(q)}, {\phi(q)}, and {\sum_{\substack{1 \le a \le q \\ \gcd(a,q)=1}} e(-N(a/q))} are multiplicative functions of {q}, and {\mu(q)=0} unless {q} is squarefree,

\displaystyle  \begin{aligned} \sum_{q = 1}^\infty \frac{\mu(q)}{\phi(q)^3} S_q 	&= \prod_p \left(1+\frac{\mu(p)}{\phi(p)^3}S_p \right) \\ 	&= \prod_p \left(1-\frac{1}{(p-1)^3} 		\sum_{a=1}^{p-1} e(-N(a/p))\right) \\ 	&= \prod_p \left(1-\frac{1}{(p-1)^3}\sum_{a=1}^{p-1} 		(p\cdot \mathbf 1(p|N) - 1)\right) \\ 	&= \prod_{p|N}\left(1-\frac{1}{(p-1)^2}\right) 		\prod_{p \nmid N}\left(1+\frac{1}{(p-1)^3}\right) \\ 	&= \mathfrak G(N). \end{aligned}

So,

\displaystyle \int_{\mathfrak M} f(\alpha)^3e(-N\alpha) \; d\alpha = \frac{N^2}{2}\mathfrak{G}(N) + O(N^{59/30+\varepsilon}).

When {N} is odd,

\displaystyle  \mathfrak{G}(N) = \prod_{p|N}\left(1-\frac{1}{(p-1)^2}\right)\prod_{p \nmid N}\left(1+\frac{1}{(p-1)^3}\right)\geq \prod_{m\geq 3}\left(\frac{m-2}{m-1}\frac{m}{m-1}\right)=\frac{1}{2},

so that we have

\displaystyle \int_{\mathfrak M} f(\alpha)^3e(-N\alpha) \; d\alpha \asymp \frac{N^2}{2}\mathfrak{G}(N),

as desired.

6. Completing the proof

Because the integral over the minor arc is {o(N^2)}, it follows that

\displaystyle \sum_{a+b+c=N} \Lambda(a)\Lambda(b)\Lambda(c) = \int_{0}^{1} f(\alpha)^3 e(-N\alpha) d \alpha \asymp \frac{N^2}{2}\mathfrak{G}(N) \gg N^2.

Consider the set {S_N} of integers {p^k\leq N} with {k>1}. We must have {p \le N^{\frac{1}{2}}}, and for each such {p} there are at most {O(\log N)} possible values of {k}. As such, {|S_N| \ll\pi(N^{1/2}) \log N\ll N^{1/2}}.

Thus

\displaystyle \sum_{\substack{a+b+c=N \\ a\in S_N}} \Lambda(a)\Lambda(b)\Lambda(c) \ll (\log N)^3 |S|N \ll\log(N)^3 N^{3/2},

and similarly for {b\in S_N} and {c\in S_N}. Notice that summing over {a\in S_N} is equivalent to summing over composite {a}, so

\displaystyle  \sum_{p_1+p_2+p_3=N} \Lambda(p_1)\Lambda(p_2)\Lambda(p_3) 	=\sum_{a+b+c=N} \Lambda(a)\Lambda(b)\Lambda(c) + O(\log(N)^3 N^{3/2}) 	\gg N^2,

where the sum is over primes {p_i}. This finishes the proof.

Things Fourier

For some reason several classes at MIT this year involve Fourier analysis. I was always confused about this as a high schooler, because no one ever gave me the “orthonormal basis” explanation, so here goes. As a bonus, I also prove a form of Arrow’s Impossibility Theorem using binary Fourier analysis, and then talk about the fancier generalizations using Pontryagin duality and the Peter-Weyl theorem.

In what follows, we let {\mathbb T = \mathbb R/\mathbb Z} denote the “circle group”, thought of as the additive group of “real numbers modulo {1}”. There is a canonical map {e : \mathbb T \rightarrow \mathbb C} sending {\mathbb T} to the complex unit circle, given by {e(\theta) = \exp(2\pi i \theta)}.

Disclaimer: I will deliberately be sloppy with convergence issues, in part because I don’t fully understand them myself, and in part because I don’t care.

1. Synopsis

Suppose we have a domain {Z} and are interested in functions {f : Z \rightarrow \mathbb C}. Naturally, the set of such functions form a complex vector space. We like to equip the set of such functions with an positive definite inner product. The idea of Fourier analysis is to then select an orthonormal basis for this set of functions, say {(e_\xi)_{\xi}}, which we call the characters; the indexing {\xi} are called frequencies. In that case, since we have a basis, every function {f : Z \rightarrow \mathbb C} becomes a sum

\displaystyle  f(x) = \sum_{\xi} \widehat f(\xi) e_\xi

where {\widehat f(\xi)} are complex coefficients of the basis; appropriately we call {\widehat f} the Fourier coefficients. The variable {x \in Z} is referred to as the physical variable. This is generally good because the characters are deliberately chosen to be nice “symmetric” functions, like sine or cosine waves or other periodic functions. Thus {we} decompose an arbitrarily complicated function into a sum on nice ones.

For convenience, we record a few facts about orthonormal bases.

Proposition 1 (Facts about orthonormal bases)

Let {V} be a complex Hilbert space with inner form {\left< -,-\right>} and suppose {x = \sum_\xi a_\xi e_\xi} and {y = \sum_\xi b_\xi e_\xi} where {e_\xi} are an orthonormal basis. Then

\displaystyle  \begin{aligned} \left< x,x \right> &= \sum_\xi |a_\xi|^2 \\ a_\xi &= \left< x, e_\xi \right> \\ \left< x,y \right> &= \sum_\xi a_\xi \overline{b_\xi}. \end{aligned}

2. Common Examples

2.1. Binary Fourier analysis on {\{\pm1\}^n}

Let {Z = \{\pm 1\}^n} for some positive integer {n}, so we are considering functions {f(x_1, \dots, x_n)} accepting binary values. Then the functions {Z \rightarrow \mathbb C} form a {2^n}-dimensional vector space {\mathbb C^Z}, and we endow it with the inner form

\displaystyle  \left< f,g \right> = \frac{1}{2^n} \sum_{x \in Z} f(x) \overline{g(x)}.

In particular,

\displaystyle  \left< f,f \right> = \frac{1}{2^n} \sum_{x \in Z} \left\lvert f(x) \right\rvert^2

is the average of the squares; this establishes also that {\left< -,-\right>} is positive definite.

In that case, the multilinear polynomials form a basis of {\mathbb C^Z}, that is the polynomials

\displaystyle  \chi_S(x_1, \dots, x_n) = \prod_{s \in S} x_s.

Thus our frequency set is actually the subsets {S \subseteq \{1, \dots, n\}}. Thus, we have a decomposition

\displaystyle  f = \sum_{S \subseteq \{1, \dots, n\}} \widehat f(S) \chi_S.

Example 2 (An example of binary Fourier analysis)

Let {n = 2}. Then binary functions {\{ \pm 1\}^2 \rightarrow \mathbb C} have a basis given by the four polynomials

\displaystyle  1, \quad x_1, \quad x_2, \quad x_1x_2.

For example, consider the function {f} which is {1} at {(1,1)} and {0} elsewhere. Then we can put

\displaystyle  f(x_1, x_2) = \frac{x_1+1}{2} \cdot \frac{x_2+1}{2} = \frac14 \left( 1 + x_1 + x_2 + x_1x_2 \right).

So the Fourier coefficients are {\widehat f(S) = \frac 14} for each of the four {S}‘s.

This notion is useful in particular for binary functions {f : \{\pm1\}^n \rightarrow \{\pm1\}}; for these functions (and products thereof), we always have {\left< f,f \right> = 1}.

It is worth noting that the frequency {\varnothing} plays a special role:

Exercise 3

Show that

\displaystyle  \widehat f(\varnothing) = \frac{1}{|Z|} \sum_{x \in Z} f(x).

2.2. Fourier analysis on finite groups {Z}

This is the Fourier analysis used in this post and this post. Here, we have a finite abelian group {Z}, and consider functions {Z \rightarrow \mathbb C}; this is a {|Z|}-dimensional vector space. The inner product is the same as before:

\displaystyle  \left< f,g \right> = \frac{1}{|Z|} \sum_{x \in Z} f(x) \overline{g}(x).

Now here is how we generate the characters. We equip {Z} with a non-degenerate symmetric bilinear form

\displaystyle  Z \times Z \xrightarrow{\cdot} \mathbb T \qquad (\xi, x) \mapsto \xi \cdot x.

Experts may already recognize this as a choice of isomorphism between {Z} and its Pontryagin dual. This time the characters are given by

\displaystyle  \left( e_\xi \right)_{\xi \in Z} \qquad \text{where} \qquad e_\xi(x) = e(\xi \cdot x).

In this way, the set of frequencies is also {Z}, but the {\xi \in Z} play very different roles from the “physical” {x \in Z}. (It is not too hard to check these indeed form an orthonormal basis in the function space {\mathbb C^{\left\lvert Z \right\rvert}}, since we assumed that {\cdot} is non-degenerate.)

Example 4 (Cube roots of unity filter)

Suppose {Z = \mathbb Z/3\mathbb Z}, with the inner form given by {\xi \cdot x = (\xi x)/3}. Let {\omega = \exp(\frac 23 \pi i)} be a primitive cube root of unity. Note that

\displaystyle  e_\xi(x) = \begin{cases} 1 & \xi = 0 \\ \omega^x & \xi = 1 \\ \omega^{2x} & \xi = 2. \end{cases}

Then given {f : Z \rightarrow \mathbb C} with {f(0) = a}, {f(1) = b}, {f(2) = c}, we obtain

\displaystyle  f(x) = \frac{a+b+c}{3} \cdot 1 + \frac{a + \omega^2 b + \omega c}{3} \cdot \omega^x + \frac{a + \omega b + \omega^2 c}{3} \cdot \omega^{2x}.

In this way we derive that the transforms are

\displaystyle  \begin{aligned} \widehat f(0) &= \frac{a+b+c}{3} \\ \widehat f(1) &= \frac{a+\omega^2 b+ \omega c}{3} \\ \widehat f(2) &= \frac{a+\omega b+\omega^2c}{3}. \end{aligned}

Exercise 5

Show that

\displaystyle  \widehat f(0) = \frac{1}{|Z|} \sum_{x \in Z} f(x).

Olympiad contestants may recognize the previous example as a “roots of unity filter”, which is exactly the point. For concreteness, suppose one wants to compute

\displaystyle  \binom{1000}{0} + \binom{1000}{3} + \dots + \binom{1000}{999}.

In that case, we can consider the function

\displaystyle  w : \mathbb Z/3 \rightarrow \mathbb C.

such that {w(0) = 1} but {w(1) = w(2) = 0}. By abuse of notation we will also think of {w} as a function {w : \mathbb Z \twoheadrightarrow \mathbb Z/3 \rightarrow \mathbb C}. Then the sum in question is

\displaystyle  \begin{aligned} \sum_n \binom{1000}{n} w(n) &= \sum_n \binom{1000}{n} \sum_{k=0,1,2} \widehat w(k) \omega^{kn} \\ &= \sum_{k=0,1,2} \widehat w(k) \sum_n \binom{1000}{n} \omega^{kn} \\ &= \sum_{k=0,1,2} \widehat w(k) (1+\omega^k)^n. \end{aligned}

In our situation, we have {\widehat w(0) = \widehat w(1) = \widehat w(2) = \frac13}, and we have evaluated the desired sum. More generally, we can take any periodic weight {w} and use Fourier analysis in order to interchange the order of summation.

Example 6 (Binary Fourier analysis)

Suppose {Z = \{\pm 1\}^n}, viewed as an abelian group under pointwise multiplication hence isomorphic to {(\mathbb Z/2\mathbb Z)^{\oplus n}}. Assume we pick the dot product defined by

\displaystyle  \xi \cdot x = \frac{1}{2} \sum_i \xi_i x_i

where {\xi = (\xi_1, \dots, \xi_n)} and {x = (x_1, \dots, x_n)}.

We claim this coincides with the first example we gave. Indeed, let {S \subseteq \{1, \dots, n\}} and let {\xi \in \{\pm1\}^n} which is {-1} at positions in {S}, and {+1} at positions not in {S}. Then the character {\chi_S} form the previous example coincides with the character {e_\xi} in the new notation. In particular, {\widehat f(S) = \widehat f(\xi)}.

Thus Fourier analysis on a finite group {Z} subsumes binary Fourier analysis.

2.3. Fourier series for functions {L^2([-\pi, \pi])}

Now we consider the space {L^2([-\pi, \pi])} of square-integrable functions {[-\pi, \pi] \rightarrow \mathbb C}, with inner form

\displaystyle  \left< f,g \right> = \frac{1}{2\pi} \int_{[-\pi, \pi]} f(x) \overline{g(x)}.

Sadly, this is not a finite-dimensional vector space, but fortunately it is a Hilbert space so we are still fine. In this case, an orthonormal basis must allow infinite linear combinations, as long as the sum of squares is finite.

Now, it turns out in this case that

\displaystyle  (e_n)_{n \in \mathbb Z} \qquad\text{where}\qquad e_n(x) = \exp(inx)

is an orthonormal basis for {L^2([-\pi, \pi])}. Thus this time the frequency set {\mathbb Z} is infinite. So every function {f \in L^2([-\pi, \pi])} decomposes as

\displaystyle  f(x) = \sum_n \widehat f(n) \exp(inx)

for {\widehat f(n)}.

This is a little worse than our finite examples: instead of a finite sum on the right-hand side, we actually have an infinite sum. This is because our set of frequencies is now {\mathbb Z}, which isn’t finite. In this case the {\widehat f} need not be finitely supported, but do satisfy {\sum_n |\widehat f(n)|^2 < \infty}.

Since the frequency set is indexed by {\mathbb Z}, we call this a Fourier series to reflect the fact that the index is {n \in \mathbb Z}.

Exercise 7

Show once again

\displaystyle  \widehat f(0) = \frac{1}{2\pi} \int_{[-\pi, \pi]} f(x).

Often we require that the function {f} satisfies {f(-\pi) = f(\pi)}, so that {f} becomes a periodic function, and we can think of it as {f : \mathbb T \rightarrow \mathbb C}.

2.4. Summary

We summarize our various flavors of Fourier analysis in the following table.

\displaystyle  \begin{array}{llll} \hline \text{Type} & \text{Physical var} & \text{Frequency var} & \text{Basis functions} \\ \hline \textbf{Binary} & \{\pm1\}^n & \text{Subsets } S \subseteq \left\{ 1, \dots, n \right\} & \prod_{s \in S} x_s \\ \textbf{Finite group} & Z & \xi \in Z, \text{ choice of } \cdot, & e(\xi \cdot x) \\ \textbf{Fourier series} & \mathbb T \text{ or } [-\pi, \pi] & n \in \mathbb Z & \exp(inx) \\ \end{array}

In fact, we will soon see that all these examples are subsumed by Pontryagin duality for compact groups {G}.

3. Parseval and friends

The notion of an orthonormal basis makes several “big-name” results in Fourier analysis quite lucid. Basically, we can take every result from Proposition~1, translate it into the context of our Fourier analysis, and get a big-name result.

Corollary 8 (Parseval theorem)

Let {f : Z \rightarrow \mathbb C}, where {Z} is a finite abelian group. Then

\displaystyle  \sum_\xi |\widehat f(\xi)|^2 = \frac{1}{|Z|} \sum_{x \in Z} |f(x)|^2.

Similarly, if {f : [-\pi, \pi] \rightarrow \mathbb C} is square-integrable then its Fourier series satisfies

\displaystyle  \sum_n |\widehat f(n)|^2 = \frac{1}{2\pi} \int_{[-\pi, \pi]} |f(x)|^2.

Proof: Recall that {\left< f,f\right>} is equal to the square sum of the coefficients. \Box

Corollary 9 (Formulas for {\widehat f})

Let {f : Z \rightarrow \mathbb C}, where {Z} is a finite abelian group. Then

\displaystyle  \widehat f(\xi) = \frac{1}{|Z|} \sum_{x \in Z} f(x) \overline{e_\xi(x)}.

Similarly, if {f : [-\pi, \pi] \rightarrow \mathbb C} is square-integrable then its Fourier series is given by

\displaystyle  \widehat f(n) = \frac{1}{2\pi} \int_{[-\pi, \pi]} f(x) \exp(-inx).

Proof: Recall that in an orthonormal basis {(e_\xi)_\xi}, the coefficient of {e_\xi} in {f} is {\left< f, e_\xi\right>}. \Box
Note in particular what happens if we select {\xi = 0} in the above!

Corollary 10 (Plancherel theorem)

Let {f : Z \rightarrow \mathbb C}, where {Z} is a finite abelian group. Then

\displaystyle  \left< f,g \right> = \sum_{\xi \in Z} \widehat f(\xi) \overline{\widehat g(\xi)}.

Similarly, if {f : [-\pi, \pi] \rightarrow \mathbb C} is square-integrable then

\displaystyle  \left< f,g \right> = \sum_n \widehat f(\xi) \overline{\widehat g(\xi)}.

Proof: Guess! \Box

4. (Optional) Arrow’s Impossibility Theorem

As an application, we now prove a form of Arrow’s theorem. Consider {n} voters voting among {3} candidates {A}, {B}, {C}. Each voter specifies a tuple {v_i = (x_i, y_i, z_i) \in \{\pm1\}^3} as follows:

  • {x_i = 1} if {A} ranks {A} ahead of {B}, and {x_i = -1} otherwise.
  • {y_i = 1} if {A} ranks {B} ahead of {C}, and {y_i = -1} otherwise.
  • {z_i = 1} if {A} ranks {C} ahead of {A}, and {z_i = -1} otherwise.

Tacitly, we only consider {3! = 6} possibilities for {v_i}: we forbid “paradoxical” votes of the form {x_i = y_i = z_i} by assuming that people’s votes are consistent (meaning the preferences are transitive).

Then, we can consider a voting mechanism

\displaystyle  \begin{aligned} f : \{\pm1\}^n &\rightarrow \{\pm1\} \\ g : \{\pm1\}^n &\rightarrow \{\pm1\} \\ h : \{\pm1\}^n &\rightarrow \{\pm1\} \end{aligned}

such that {f(x_\bullet)} is the global preference of {A} vs. {B}, {g(y_\bullet)} is the global preference of {B} vs. {C}, and {h(z_\bullet)} is the global preference of {C} vs. {A}. We’d like to avoid situations where the global preference {(f(x_\bullet), g(y_\bullet), h(z_\bullet))} is itself paradoxical.

In fact, we will prove the following theorem:

Theorem 11 (Arrow Impossibility Theorem)

Assume that {(f,g,h)} always avoids paradoxical outcomes, and assume {\mathbf E f = \mathbf E g = \mathbf E h = 0}. Then {(f,g,h)} is either a dictatorship or anti-dictatorship: there exists a “dictator” {k} such that

\displaystyle  f(x_\bullet) = \pm x_k, \qquad g(y_\bullet) = \pm y_k, \qquad h(z_\bullet) = \pm z_k

where all three signs coincide.

The “irrelevance of independent alternatives” reflects that The assumption {\mathbf E f = \mathbf E g = \mathbf E h = 0} provides symmetry (and e.g. excludes the possibility that {f}, {g}, {h} are constant functions which ignore voter input). Unlike the usual Arrow theorem, we do not assume that {f(+1, \dots, +1) = +1} (hence possibility of anti-dictatorship).

To this end, we actually prove the following result:

Lemma 12

Assume the {n} voters vote independently at random among the {3! = 6} possibilities. The probability of a paradoxical outcome is exactly

\displaystyle  \frac14 + \frac14 \sum_{S \subseteq \{1, \dots, n\}} \left( -\frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S) \widehat g(S) + \widehat g(S) \widehat h(S) + \widehat h(S) \widehat f(S) \right) .

Proof: Define the Boolean function {D : \{\pm 1\}^3 \rightarrow \mathbb R} by

\displaystyle  D(a,b,c) = ab + bc + ca = \begin{cases} 3 & a,b,c \text{ all equal} \\ -1 & a,b,c \text{ not all equal}. \end{cases}.

Thus paradoxical outcomes arise when {D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) = 3}. Now, we compute that for randomly selected {x_\bullet}, {y_\bullet}, {z_\bullet} that

\displaystyle  \begin{aligned} \mathbf E D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) &= \mathbf E \sum_S \sum_T \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \left( \chi_S(x_\bullet)\chi_T(y_\bullet) \right) \\ &= \sum_S \sum_T \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \mathbf E\left( \chi_S(x_\bullet)\chi_T(y_\bullet) \right). \end{aligned}

Now we observe that:

  • If {S \neq T}, then {\mathbf E \chi_S(x_\bullet) \chi_T(y_\bullet) = 0}, since if say {s \in S}, {s \notin T} then {x_s} affects the parity of the product with 50% either way, and is independent of any other variables in the product.
  • On the other hand, suppose {S = T}. Then

    \displaystyle  \chi_S(x_\bullet) \chi_T(y_\bullet) = \prod_{s \in S} x_sy_s.

    Note that {x_sy_s} is equal to {1} with probability {\frac13} and {-1} with probability {\frac23} (since {(x_s, y_s, z_s)} is uniform from {3!=6} choices, which we can enumerate). From this an inductive calculation on {|S|} gives that

    \displaystyle  \prod_{s \in S} x_sy_s = \begin{cases} +1 & \text{ with probability } \frac{1}{2}(1+(-1/3)^{|S|}) \\ -1 & \text{ with probability } \frac{1}{2}(1-(-1/3)^{|S|}). \end{cases}

    Thus

    \displaystyle  \mathbf E \left( \prod_{s \in S} x_sy_s \right) = \left( -\frac13 \right)^{|S|}.

Piecing this altogether, we now have that

\displaystyle  \mathbf E D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) = \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \left( -\frac13 \right)^{|S|}.

Then, we obtain that

\displaystyle  \begin{aligned} &\mathbf E \frac14 \left( 1 + D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) \right) \\ =& \frac14 + \frac14\sum_S \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \widehat f(S)^2 \left( -\frac13 \right)^{|S|}. \end{aligned}

Comparing this with the definition of {D} gives the desired result. \Box

Now for the proof of the main theorem. We see that

\displaystyle  1 = \sum_{S \subseteq \{1, \dots, n\}} -\left( -\frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S) \widehat g(S) + \widehat g(S) \widehat h(S) + \widehat h(S) \widehat f(S) \right).

But now we can just use weak inequalities. We have {\widehat f(\varnothing) = \mathbf E f = 0} and similarly for {\widehat g} and {\widehat h}, so we restrict attention to {|S| \ge 1}. We then combine the famous inequality {|ab+bc+ca| \le a^2+b^2+c^2} (which is true across all real numbers) to deduce that

\displaystyle  \begin{aligned} 1 &= \sum_{S \subseteq \{1, \dots, n\}} -\left( -\frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S) \widehat g(S) + \widehat g(S) \widehat h(S) + \widehat h(S) \widehat f(S) \right) \\ &\le \sum_{S \subseteq \{1, \dots, n\}} \left( \frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S)^2 + \widehat g(S)^2 + \widehat h(S)^2 \right) \\ &\le \sum_{S \subseteq \{1, \dots, n\}} \left( \frac13 \right)^1 \left( \widehat f(S)^2 + \widehat g(S)^2 + \widehat h(S)^2 \right) \\ &= \frac13 (1+1+1) = 1. \end{aligned}

with the last step by Parseval. So all inequalities must be sharp, and in particular {\widehat f}, {\widehat g}, {\widehat h} are supported on one-element sets, i.e. they are linear in inputs. As {f}, {g}, {h} are {\pm 1} valued, each {f}, {g}, {h} is itself either a dictator or anti-dictator function. Since {(f,g,h)} is always consistent, this implies the final result.

5. Pontryagin duality

In fact all the examples we have covered can be subsumed as special cases of Pontryagin duality, where we replace the domain with a general group {G}. In what follows, we assume {G} is a locally compact abelian (LCA) group, which just means that:

  • {G} is a abelian topological group,
  • the topology on {G} is Hausdorff, and
  • the topology on {G} is locally compact: every point of {G} has a compact neighborhood.

Notice that our previous examples fall into this category:

Example 13 (Examples of locally compact abelian groups)

  • Any finite group {Z} with the discrete topology is LCA.
  • The circle group {\mathbb T} is LCA and also in fact compact.
  • The real numbers {\mathbb R} are an example of an LCA group which is not compact.

5.1. The Pontryagin dual

The key definition is:

Definition 14

Let {G} be an LCA group. Then its Pontryagin dual is the abelian group

\displaystyle  \widehat G \overset{\mathrm{def}}{=} \left\{ \text{continuous group homomorphisms } \xi : G \rightarrow \mathbb T \right\}.

The maps {\xi} are called characters. By equipping it with the compact-open topology, we make {\widehat G} into an LCA group as well.

Example 15 (Examples of Pontryagin duals)

  • {\widehat{\mathbb Z} \cong \mathbb T}.
  • {\widehat{\mathbb T} \cong \mathbb Z}. The characters are given by {\theta \mapsto n\theta} for {n \in \mathbb Z}.
  • {\widehat{\mathbb R} \cong \mathbb R}. This is because a nonzero continuous homomorphism {\mathbb R \rightarrow S^1} is determined by the fiber above {1 \in S^1}. (Covering projections, anyone?)
  • {\widehat{\mathbb Z/n\mathbb Z} \cong \mathbb Z/n\mathbb Z}, characters {\xi} being determined by the image {\xi(1) \in \mathbb T}.
  • {\widehat{G \times H} \cong \widehat G \times \widehat H}.
  • If {Z} is a finite abelian group, then previous two examples (and structure theorem for abelian groups) imply that {\widehat{Z} \cong Z}, though not canonically. You may now recognize that the bilinear form {\cdot : Z \times Z \rightarrow Z} is exactly a choice of isomorphism {Z \rightarrow \widehat Z}.
  • For any group {G}, the dual of {\widehat G} is canonically isomorphic to {G}, id est there is a natural isomorphism

    \displaystyle  G \cong \widehat{\widehat G} \qquad \text{by} \qquad x \mapsto \left( \xi \mapsto \xi(x) \right).

    This is the Pontryagin duality theorem. (It is an analogy to the isomorphism {(V^\vee)^\vee \cong V} for vector spaces {V}.)

5.2. The orthonormal basis in the compact case

Now assume {G} is LCA but also compact, and thus has a unique Haar measure {\mu} such that {\mu(G) = 1}; this lets us integrate over {G}. Let {L^2(G)} be the space of square-integrable functions to {\mathbb C}, i.e.

\displaystyle  L^2(G) = \left\{ f : G \rightarrow \mathbb C \quad\text{such that}\quad \int_G |f|^2 \; d\mu < \infty \right\}.

Thus we can equip it with the inner form

\displaystyle  \left< f,g \right> = \int_G f\overline{g} \; d\mu.

In that case, we get all the results we wanted before:

Theorem 16 (Characters of {\widehat G} forms an orthonormal basis)

Assume {G} is LCA and compact. Then {\widehat G} is discrete, and the characters

\displaystyle  (e_\xi)_{\xi \in \widehat G} \qquad\text{by}\qquad e_\xi(x) = e(\xi(x)) = \exp(2\pi i \xi(x))

form an orthonormal basis of {L^2(G)}. Thus for each {f \in L^2(G)} we have

\displaystyle  f = \sum_{\xi \in \widehat G} \widehat f(\xi) e_\xi

where

\displaystyle  \widehat f(\xi) = \left< f, e_\xi \right> = \int_G f(x) \exp(-2\pi i \xi(x)) \; d\mu.

The sum {\sum_{\xi \in \widehat G}} makes sense since {\widehat G} is discrete. In particular,

  • Letting {G = Z} gives “Fourier transform on finite groups”.
  • The special case {G = \mathbb Z/n\mathbb Z} has its own Wikipedia page.
  • Letting {G = \mathbb T} gives the “Fourier series” earlier.

5.3. The Fourier transform of the non-compact case

If {G} is LCA but not compact, then Theorem~16 becomes false. On the other hand, it is still possible to define a transform, but one needs to be a little more careful. The generic example to keep in mind in what follows is {G = \mathbb R}.

In what follows, we fix a Haar measure {\mu} for {G}. (This {\mu} is no longer unique up to scaling, since {\mu(G) = \infty}.)

One considers this time the space {L^1(G)} of absolutely integrable functions. Then one directly defines the Fourier transform of {f \in L^1(G)} to be

\displaystyle  \widehat f(\xi) = \int_G f \overline{e_\xi} \; d\mu

imitating the previous definitions in the absence of an inner product. This {\widehat f} may not be {L^1}, but it is at least bounded. Then we manage to at least salvage:

Theorem 17 (Fourier inversion on {L^1(G)})

Take an LCA group {G} and fix a Haar measure {\mu} on it. One can select a unique dual measure {\widehat \mu} on {\widehat G} such that if {f \in L^1(G)}, {\widehat f \in L^1(\widehat G)}, the “Fourier inversion formula”

\displaystyle  f(x) = \int_{\widehat G} \widehat f(\xi) e_\xi(x) \; d\widehat\mu.

holds almost everywhere. It holds everywhere if {f} is continuous.

Notice the extra nuance of having to select measures, because it is no longer the case that {G} has a single distinguished measure.

Despite the fact that the {e_\xi} no longer form an orthonormal basis, the transformed function {\widehat f : \widehat G \rightarrow \mathbb C} is still often useful. In particular, they have special names for a few special {G}:

5.4. Summary

In summary,

  • Given any LCA group {G}, we can transform sufficiently nice functions on {G} into functions on {\widehat G}.
  • If {G} is compact, then we have the nicest situation possible: {L^2(G)} is an inner product space with {\left< f,g \right> = \int_G f \overline{g} \; d\mu}, and {e_\xi} form an orthonormal basis across {\widehat \xi \in \widehat G}.
  • If {G} is not compact, then we no longer get an orthonormal basis or even an inner product space, but it is still possible to define the transform

    \displaystyle  \widehat f : \widehat G \rightarrow \mathbb C

    for {f \in L^1(G)}. If {\widehat f} is also in {L^1(G)} we still get a “Fourier inversion formula” expressing {f} in terms of {\widehat f}.

We summarize our various flavors of Fourier analysis for various {G} in the following. In the first half {G} is compact, in the second half {G} is not.

\displaystyle  \begin{array}{llll} \hline \text{Name} & \text{Domain }G & \text{Dual }\widehat G & \text{Characters} \\ \hline \textbf{Binary Fourier analysis} & \{\pm1\}^n & S \subseteq \left\{ 1, \dots, n \right\} & \prod_{s \in S} x_s \\ \textbf{Fourier transform on finite groups} & Z & \xi \in \widehat Z \cong Z & e( i \xi \cdot x) \\ \textbf{Discrete Fourier transform} & \mathbb Z/n\mathbb Z & \xi \in \mathbb Z/n\mathbb Z & e(\xi x / n) \\ \textbf{Fourier series} & \mathbb T \cong [-\pi, \pi] & n \in \mathbb Z & \exp(inx) \\ \hline \textbf{Continuous Fourier transform} & \mathbb R & \xi \in \mathbb R & e(\xi x) \\ \textbf{Discrete time Fourier transform} & \mathbb Z & \xi \in \mathbb T \cong [-\pi, \pi] & \exp(i \xi n) \\ \end{array}

You might notice that the various names are awful. This is part of the reason I got confused as a high school student: every type of Fourier series above has its own Wikipedia article. If it were up to me, we would just use the term “{G}-Fourier transform”, and that would make everyone’s lives a lot easier.

6. Peter-Weyl

In fact, if {G} is a Lie group, even if {G} is not abelian we can still give an orthonormal basis of {L^2(G)} (the square-integrable functions on {G}). It turns out in this case the characters are attached to complex irreducible representations of {G} (and in what follows all representations are complex).

The result is given by the Peter-Weyl theorem. First, we need the following result:

Lemma 18 (Compact Lie groups have unitary reps)

Any finite-dimensional (complex) representation {V} of a compact Lie group {G} is unitary, meaning it can be equipped with a {G}-invariant inner form. Consequently, {V} is completely reducible: it splits into the direct sum of irreducible representations of {G}.

Proof: Suppose {B : V \times V \rightarrow \mathbb C} is any inner product. Equip {G} with a right-invariant Haar measure {dg}. Then we can equip it with an “averaged” inner form

\displaystyle  \widetilde B(v,w) = \int_G B(gv, gw) \; dg.

Then {\widetilde B} is the desired {G}-invariant inner form. Now, the fact that {V} is completely reducible follows from the fact that given a subrepresentation of {V}, its orthogonal complement is also a subrepresentation. \Box

The Peter-Weyl theorem then asserts that the finite-dimensional irreducible unitary representations essentially give an orthonormal basis for {L^2(G)}, in the following sense. Let {V = (V, \rho)} be such a representation of {G}, and fix an orthonormal basis of {e_1}, \dots, {e_d} for {V} (where {d = \dim V}). The {(i,j)}th matrix coefficient for {V} is then given by

\displaystyle  G \xrightarrow{\rho} \mathop{\mathrm{GL}}(V) \xrightarrow{\pi_{ij}} \mathbb C

where {\pi_{ij}} is the projection onto the {(i,j)}th entry of the matrix. We abbreviate {\pi_{ij} \circ \rho} to {\rho_{ij}}. Then the theorem is:

Theorem 19 (Peter-Weyl)

Let {G} be a compact Lie group. Let {\Sigma} denote the (pairwise non-isomorphic) irreducible finite-dimensional unitary representations of {G}. Then

\displaystyle  \left\{ \sqrt{\dim V} \rho_{ij} \; \Big\vert \; (V, \rho) \in \Sigma, \text{ and } 1 \le i,j \le \dim V \right\}

is an orthonormal basis of {L^2(G)}.

Strictly, I should say {\Sigma} is a set of representatives of the isomorphism classes of irreducible unitary representations, one for each isomorphism class.

In the special case {G} is abelian, all irreducible representations are one-dimensional. A one-dimensional representation of {G} is a map {G \hookrightarrow \mathop{\mathrm{GL}}(\mathbb C) \cong \mathbb C^\times}, but the unitary condition implies it is actually a map {G \hookrightarrow S^1 \cong \mathbb T}, i.e. it is an element of {\widehat G}.