I’ve just uploaded to the arXiv my paper “Failure of the ${L^1}$ pointwise and maximal ergodic theorems for the free group“, submitted to Forum of Mathematics, Sigma. This paper concerns a variant of the pointwise ergodic theorem of Birkhoff, which asserts that if one has a measure-preserving shift map ${T: X \rightarrow X}$ on a probability space ${X = (X,\mu)}$, then for any ${f \in L^1(X)}$, the averages ${\frac{1}{N} \sum_{n=1}^N f \circ T^{-n}}$ converge pointwise almost everywhere. (In the important case when the shift map ${T}$ is ergodic, the pointwise limit is simply the mean ${\int_X f\ d\mu}$ of the original function ${f}$.)

The pointwise ergodic theorem can be extended to measure-preserving actions of other amenable groups, if one uses a suitably “tempered” Folner sequence of averages; see this paper of Lindenstrauss for more details. (I also wrote up some notes on that paper here, back in 2006 before I had started this blog.) But the arguments used to handle the amenable case break down completely for non-amenable groups, and in particular for the free non-abelian group ${F_2}$ on two generators.

Nevo and Stein studied this problem and obtained a number of pointwise ergodic theorems for ${F_2}$-actions ${(T_g)_{g \in F_2}}$ on probability spaces ${(X,\mu)}$. For instance, for the spherical averaging operators

$\displaystyle {\mathcal A}_n f := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f \circ T_g^{-1}$

(where ${|g|}$ denotes the length of the reduced word that forms ${g}$), they showed that ${{\mathcal A}_{2n} f}$ converged pointwise almost everywhere provided that ${f}$ was in ${L^p(X)}$ for some ${p>1}$. (The need to restrict to spheres of even radius can be seen by considering the action of ${F_2}$ on the two-element set ${\{0,1\}}$ in which both generators of ${F_2}$ act by interchanging the elements, in which case ${{\mathcal A}_n}$ is determined by the parity of ${n}$.) This result was reproven with a different and simpler proof by Bufetov, who also managed to relax the condition ${f \in L^p(X)}$ to the weaker condition ${f \in L \log L(X)}$.

The question remained open as to whether the pointwise ergodic theorem for ${F_2}$-actions held if one only assumed that ${f}$ was in ${L^1(X)}$. Nevo and Stein were able to establish this for the Cesáro averages ${\frac{1}{N} \sum_{n=1}^N {\mathcal A}_n}$, but not for ${{\mathcal A}_n}$ itself. About six years ago, Assaf Naor and I tried our hand at this problem, and was able to show an associated maximal inequality on ${\ell^1(F_2)}$, but due to the non-amenability of ${F_2}$, this inequality did not transfer to ${L^1(X)}$ and did not have any direct impact on this question, despite a fair amount of effort on our part to attack it.

Inspired by some recent conversations with Lewis Bowen, I returned to this problem. This time around, I tried to construct a counterexample to the ${L^1}$ pointwise ergodic theorem – something Assaf and I had not seriously attempted to do (perhaps due to being a bit too enamoured of our ${\ell^1(F_2)}$ maximal inequality). I knew of an existing counterexample of Ornstein regarding a failure of an ${L^1}$ ergodic theorem for iterates ${P^n}$ of a self-adjoint Markov operator – in fact, I had written some notes on this example back in 2007. Upon revisiting my notes, I soon discovered that the Ornstein construction was adaptable to the ${F_2}$ setting, thus settling the problem in the negative:

Theorem 1 (Failure of ${L^1}$ pointwise ergodic theorem) There exists a measure-preserving ${F_2}$-action on a probability space ${X}$ and a non-negative function ${f \in L^1(X)}$ such that ${\sup_n {\mathcal A}_{2n} f(x) = +\infty}$ for almost every ${x}$.

To describe the proof of this theorem, let me first briefly sketch the main ideas of Ornstein’s construction, which gave an example of a self-adjoint Markov operator ${P}$ on a probability space ${X}$ and a non-negative ${f \in L^1(X)}$ such that ${\sup_n P^n f(x) = +\infty}$ for almost every ${x}$. By some standard manipulations, it suffices to show that for any given ${\alpha > 0}$ and ${\varepsilon>0}$, there exists a self-adjoint Markov operator ${P}$ on a probability space ${X}$ and a non-negative ${f \in L^1(X)}$ with ${\|f\|_{L^1(X)} \leq \alpha}$, such that ${\sup_n P^n f \geq 1-\varepsilon}$ on a set of measure at least ${1-\varepsilon}$. Actually, it will be convenient to replace the Markov chain ${(P^n f)_{n \geq 0}}$ with an ancient Markov chain ${(f_n)_{n \in {\bf Z}}}$ – that is to say, a sequence of non-negative functions ${f_n}$ for both positive and negative ${f}$, such that ${f_{n+1} = P f_n}$ for all ${n \in {\bf Z}}$. The purpose of requiring the Markov chain to be ancient (that is, to extend infinitely far back in time) is to allow for the Markov chain to be shifted arbitrarily in time, which is key to Ornstein’s construction. (Technically, Ornstein’s original argument only uses functions that go back to a large negative time, rather than being infinitely ancient, but I will gloss over this point for sake of discussion, as it turns out that the ${F_2}$ version of the argument can be run using infinitely ancient chains.)

For any ${\alpha>0}$, let ${P(\alpha)}$ denote the claim that for any ${\varepsilon>0}$, there exists an ancient Markov chain ${(f_n)_{n \in {\bf Z}}}$ with ${\|f_n\|_{L^1(X)} = \alpha}$ such that ${\sup_{n \in {\bf Z}} f_n \geq 1-\varepsilon}$ on a set of measure at least ${1-\varepsilon}$. Clearly ${P(1)}$ holds since we can just take ${f_n=1}$ for all ${n}$. Our objective is to show that ${P(\alpha)}$ holds for arbitrarily small ${\alpha}$. The heart of Ornstein’s argument is then the implication

$\displaystyle P(\alpha) \implies P( \alpha (1 - \frac{\alpha}{4}) ) \ \ \ \ \ (1)$

for any ${0 < \alpha \leq 1}$, which upon iteration quickly gives the desired claim.

Let’s see informally how (1) works. By hypothesis, and ignoring epsilons, we can find an ancient Markov chain ${(f_n)_{n \in {\bf Z}}}$ on some probability space ${X}$ of total mass ${\|f_n\|_{L^1(X)} = \alpha}$, such that ${\sup_n f_n}$ attains the value of ${1}$ or greater almost everywhere. Assuming that the Markov process is irreducible, the ${f_n}$ will eventually converge as ${n \rightarrow \infty}$ to the constant value of ${\|f_n\|_{L^1(X)}}$, in particular its final state will essentially stay above ${\alpha}$ (up to small errors).

Now suppose we duplicate the Markov process by replacing ${X}$ with a double copy ${X \times \{1,2\}}$ (giving ${\{1,2\}}$ the uniform probability measure), and using the disjoint sum of the Markov operators on ${X \times \{1\}}$ and ${X \times \{2\}}$ as the propagator, so that there is no interaction between the two components of this new system. Then the functions ${f'_n(x,i) := f_n(x) 1_{i=1}}$ form an ancient Markov chain of mass at most ${\alpha/2}$ that lives solely in the first half ${X \times \{1\}}$ of this copy, and ${\sup_n f'_n}$ attains the value of ${1}$ or greater on almost all of the first half ${X \times \{1\}}$, but is zero on the second half. The final state of ${f'_n}$ will be to stay above ${\alpha}$ in the first half ${X \times \{1\}}$, but be zero on the second half.

Now we modify the above example by allowing an infinitesimal amount of interaction between the two halves ${X \times \{1\}}$, ${X \times \{2\}}$ of the system (I mentally think of ${X \times \{1\}}$ and ${X \times \{2\}}$ as two identical boxes that a particle can bounce around in, and now we wish to connect the boxes by a tiny tube). The precise way in which this interaction is inserted is not terribly important so long as the new Markov process is irreducible. Once one does so, then the ancient Markov chain ${(f'_n)_{n \in {\bf Z}}}$ in the previous example gets replaced by a slightly different ancient Markov chain ${(f''_n)_{n \in {\bf Z}}}$ which is more or less identical with ${f'_n}$ for negative times ${n}$, or for bounded positive times ${n}$, but for very large values of ${n}$ the final state is now constant across the entire state space ${X \times \{1,2\}}$, and will stay above ${\alpha/2}$ on this space.

Finally, we consider an ancient Markov chain ${F_n}$ which is basically of the form

$\displaystyle F_n(x,i) \approx f''_n(x,i) + (1 - \frac{\alpha}{2}) f_{n-M}(x) 1_{i=2}$

for some large parameter ${M}$ and for all ${n \leq M}$ (the approximation becomes increasingly inaccurate for ${n}$ much larger than ${M}$, but never mind this for now). This is basically two copies of the original Markov process in separate, barely interacting state spaces ${X \times \{1\}, X \times \{2\}}$, but with the second copy delayed by a large time delay ${M}$, and also attenuated in amplitude by a factor of ${1-\frac{\alpha}{2}}$. The total mass of this process is now ${\frac{\alpha}{2} + \frac{\alpha}{2} (1 -\frac{\alpha}{2}) = \alpha (1 - \alpha/4)}$. Because of the ${f''_n}$ component of ${F_n}$, we see that ${\sup_n F_n}$ basically attains the value of ${1}$ or greater on the first half ${X \times \{1\}}$. On the second half ${X \times \{2\}}$, we work with times ${n}$ close to ${M}$. If ${M}$ is large enough, ${f''_n}$ would have averaged out to about ${\alpha/2}$ at such times, but the ${(1 - \frac{\alpha}{2}) f_{n-M}(x)}$ component can get as large as ${1-\alpha/2}$ here. Summing (and continuing to ignore various epsilon losses), we see that ${\sup_n F_n}$ can get as large as ${1}$ on almost all of the second half of ${X \times \{2\}}$. This concludes the rough sketch of how one establishes the implication (1).

It was observed by Bufetov that the spherical averages ${{\mathcal A}_n}$ for a free group action can be lifted up to become powers ${P^n}$ of a Markov operator, basically by randomly assigning a “velocity vector” ${s \in \{a,b,a^{-1},b^{-1}\}}$ to one’s base point ${x}$ and then applying the Markov process that moves ${x}$ along that velocity vector (and then randomly changing the velocity vector at each time step to the “reduced word” condition that the velocity never flips from ${s}$ to ${s^{-1}}$). Thus the spherical average problem has a Markov operator interpretation, which opens the door to adapting the Ornstein construction to the setting of ${F_2}$ systems. This turns out to be doable after a certain amount of technical artifice; the main thing is to work with ${F_2}$-measure preserving systems that admit ancient Markov chains that are initially supported in a very small region in the “interior” of the state space, so that one can couple such systems to each other “at the boundary” in the fashion needed to establish the analogue of (1) without disrupting the ancient dynamics of such chains. The initial such system (used to establish the base case ${P(1)}$) comes from basically considering the action of ${F_2}$ on a (suitably renormalised) “infinitely large ball” in the Cayley graph, after suitably gluing together the boundary of this ball to complete the action. The ancient Markov chain associated to this system starts at the centre of this infinitely large ball at infinite negative time ${n=-\infty}$, and only reaches the boundary of this ball at the time ${n=0}$.

The lonely runner conjecture is the following open problem:

Conjecture 1 Suppose one has ${n \geq 1}$ runners on the unit circle ${{\bf R}/{\bf Z}}$, all starting at the origin and moving at different speeds. Then for each runner, there is at least one time ${t}$ for which that runner is “lonely” in the sense that it is separated by a distance at least ${1/n}$ from all other runners.

One can normalise the speed of the lonely runner to be zero, at which point the conjecture can be reformulated (after replacing ${n}$ by ${n+1}$) as follows:

Conjecture 2 Let ${v_1,\dots,v_n}$ be non-zero real numbers for some ${n \geq 1}$. Then there exists a real number ${t}$ such that the numbers ${tv_1,\dots,tv_n}$ are all a distance at least ${\frac{1}{n+1}}$ from the integers, thus ${\|tv_1\|_{{\bf R}/{\bf Z}},\dots,\|tv_n\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}}$ where ${\|x\|_{{\bf R}/{\bf Z}}}$ denotes the distance of ${x}$ to the nearest integer.

This conjecture has been proven for ${n \leq 7}$, but remains open for larger ${n}$. The bound ${\frac{1}{n+1}}$ is optimal, as can be seen by looking at the case ${v_i=i}$ and applying the Dirichlet approximation theorem. Note that for each non-zero ${v}$, the set ${\{ t \in {\bf R}: \|vt\|_{{\bf R}/{\bf Z}} \leq r \}}$ has (Banach) density ${2r}$ for any ${0 < r < 1/2}$, and from this and the union bound we can easily find ${t \in {\bf R}}$ for which

$\displaystyle \|tv_1\|_{{\bf R}/{\bf Z}},\dots,\|tv_n\|_{{\bf R}/{\bf Z}} \geq \frac{1}{2n}-\varepsilon$

for any ${\varepsilon>0}$, but it has proven to be quite challenging to remove the factor of ${2}$ to increase ${\frac{1}{2n}}$ to ${\frac{1}{n+1}}$. (As far as I know, even improving ${\frac{1}{2n}}$ to ${\frac{1+c}{2n}}$ for some absolute constant ${c>0}$ and sufficiently large ${n}$ remains open.)

The speeds ${v_1,\dots,v_n}$ in the above conjecture are arbitrary non-zero reals, but it has been known for some time that one can reduce without loss of generality to the case when the ${v_1,\dots,v_n}$ are rationals, or equivalently (by scaling) to the case where they are integers; see e.g. Section 4 of this paper of Bohman, Holzman, and Kleitman.

In this post I would like to remark on a slight refinement of this reduction, in which the speeds ${v_1,\dots,v_n}$ are integers of bounded size, where the bound depends on ${n}$. More precisely:

Proposition 3 In order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that the ${v_1,\dots,v_n}$ are integers of size at most ${n^{Cn^2}}$, where ${C}$ is an (explicitly computable) absolute constant. (More precisely: if this restricted version of the lonely runner conjecture is true for all ${n \leq n_0}$, then the original version of the conjecture is also true for all ${n \leq n_0}$.)

In principle, this proposition allows one to verify the lonely runner conjecture for a given ${n}$ in finite time; however the number of cases to check with this proposition grows faster than exponentially in ${n}$, and so this is unfortunately not a feasible approach to verifying the lonely runner conjecture for more values of ${n}$ than currently known.

One of the key tools needed to prove this proposition is the following additive combinatorics result. Recall that a generalised arithmetic progression (or ${GAP}$) in the reals ${{\bf R}}$ is a set of the form

$\displaystyle P = \{ n_1 v_1 + \dots + n_d v_d: n_1,\dots,n_d \in {\bf Z}; |n_1| \leq N_1, \dots, |n_d| \leq N_d \}$

for some ${v_1,\dots,v_d \in {\bf R}}$ and ${N_1,\dots,N_d > 0}$; the quantity ${d}$ is called the rank of the progression. If ${t>0}$, the progression ${P}$ is said to be ${t}$-proper if the sums ${n_1 v_1 + \dots + n_d v_d}$ with ${|n_i| \leq t N_i}$ for ${i=1,\dots,d}$ are all distinct. We have

Lemma 4 (Progressions lie inside proper progressions) Let ${P}$ be a GAP of rank ${d}$ in the reals, and let ${t \geq 1}$. Then ${P}$ is contained in a ${t}$-proper GAP ${Q}$ of rank at most ${d}$, with

$\displaystyle |Q| \leq (2t)^d d^{6d^2} \prod_{i=1}^d (2N_i+1).$

Proof: See Theorem 2.1 of this paper of Bilu. (Very similar results can also be found in Theorem 3.40 of my book with Van Vu, or Theorem 1.10 of this paper of mine with Van Vu.) $\Box$

Now let ${n \geq 1}$, and assume inductively that the lonely runner conjecture has been proven for all smaller values of ${n}$, as well as for the current value of ${n}$ in the case that ${v_1,\dots,v_n}$ are integers of size at most ${n^{Cn^2}}$ for some sufficiently large ${C}$. We will show that the lonely runner conjecture holds in general for this choice of ${n}$.

let ${v_1,\dots,v_n}$ be non-zero real numbers. Let ${C_0}$ be a large absolute constant to be chosen later. From the above lemma applied to the GAP ${\{ n_1 v_1 + \dots + n_d v_d: n_1,\dots,n_d \in \{-1,0,1\}\}}$, one can find a ${n^{C_0n}}$-proper GAP ${Q}$ of rank at most ${n}$ containing ${\{v_1,\dots,v_n\}}$ such that

$\displaystyle |Q| \leq (6n^{C_0 n})^n n^{6n^2};$

in particular ${|Q| \leq n^{Cn^2}}$ if ${C}$ is large enough depending on ${C_0}$.

We write

$\displaystyle Q = \{ n_1 w_1 + \dots + n_d w_d: n_1,\dots,n_d \in {\bf Z}; |n_1| \leq N_1,\dots,|n_d| \leq N_d \}$

for some ${d \leq n}$, ${w_1,\dots,w_d}$, and ${N_1,\dots,N_d \geq 0}$. We thus have ${v_i = \phi(a_i)}$ for ${i=1,\dots,n}$, where ${\phi: {\bf R}^d \rightarrow {\bf R}}$ is the linear map ${\phi(n_1,\dots,n_d) := n_1 w_1 + \dots + n_d w_d}$ and ${a_1,\dots,a_n \in {\bf Z}^d}$ are non-zero and lie in the box ${\{ (n_1,\dots,n_d) \in {\bf R}^d: |n_1| \leq N_1,\dots,|n_d| \leq N_d \}}$.

We now need an elementary lemma that allows us to create a “collision” between two of the ${a_1,\dots,a_n}$ via a linear projection, without making any of the ${a_i}$ collide with the origin:

Lemma 5 Let ${a_1,\dots,a_n \in {\bf R}^d}$ be non-zero vectors that are not all collinear with the origin. Then, after replacing one or more of the ${a_i}$ with their negatives ${-a_i}$ if necessary, there exists a pair ${a_i,a_j}$ such that ${a_i-a_j \neq 0}$, and such that none of the ${a_1,\dots,a_n}$ is a scalar multiple of ${a_i-a_j}$.

Proof: We may assume that ${d \geq 2}$, since the ${d \leq 1}$ case is vacuous. Applying a generic linear projection to ${{\bf R}^2}$ (which does not affect collinearity, or the property that a given ${a_k}$ is a scalar multiple of ${a_i-a_j}$), we may then reduce to the case ${d=2}$.

By a rotation and relabeling, we may assume that ${a_1}$ lies on the negative ${x}$-axis; by flipping signs as necessary we may then assume that all of the ${a_2,\dots,a_n}$ lie in the closed right half-plane. As the ${a_i}$ are not all collinear with the origin, one of the ${a_i}$ lies off of the ${x}$-axis, by relabeling, we may assume that ${a_2}$ lies off of the ${x}$ axis and makes a minimal angle with the ${x}$-axis. Then the angle of ${a_2-a_1}$ with the ${x}$-axis is non-zero but smaller than any non-zero angle that any of the ${a_i}$ make with this axis, and so none of the ${a_i}$ are a scalar multiple of ${a_2-a_1}$, and the claim follows. $\Box$

We now return to the proof of the proposition. If the ${a_1,\dots,a_n}$ are all collinear with the origin, then ${\phi(a_1),\dots,\phi(a_n)}$ lie in a one-dimensional arithmetic progression ${\{ mv: |m| \leq |Q| \}}$, and then by rescaling we may take the ${v_1,\dots,v_n}$ to be integers of magnitude at most ${|Q| \leq n^{Cn}}$, at which point we are done by hypothesis. Thus, we may assume that the ${a_1,\dots,a_n}$ are not all collinear with the origin, and so by the above lemma and relabeling we may assume that ${a_n-a_1}$ is non-zero, and that none of the ${a_i}$ are scalar multiples of ${a_n-a_1}$.

We write

$\displaystyle a_n-a_1 = (c_1,\dots,c_d) \ \ \ \ \ (1)$

with ${|c_i| \leq 2 N_i}$ for ${i=1,\dots,d}$; by relabeling we may assume without loss of generality that ${c_d}$ is non-zero, and furthermore that

$\displaystyle \frac{|c_i|}{N_i} \leq \frac{|c_d|}{N_d}$

for ${i=1,\dots,d}$. We can also factor

$\displaystyle (c_1,\dots,c_d) = q (c'_1,\dots,c'_d) \ \ \ \ \ (2)$

where ${q}$ is a natural number and ${c'_1,\dots,c'_d}$ have no common factor.

We now define a variant ${\tilde \phi: {\bf R}^d \rightarrow {\bf R}}$ of ${\phi: {\bf R}^d \rightarrow {\bf R}}$ by the map

$\displaystyle \tilde \phi(n_1,\dots,n_d) := n_1 \tilde w_1 + \dots + n_{d-1} \tilde w_{d-1} - \frac{n_d}{c_d} (c_1 \tilde w_1 + \dots + c_{d-1} \tilde w_{d-1}),$

where the ${\tilde w_1,\dots,\tilde w_{d-1}}$ are real numbers that are linearly independent over ${{\bf Q}}$, whose precise value will not be of importance in our argument. This is a linear map with the property that ${\tilde \phi(a_n-a_1)=0}$, so that ${\tilde \phi(a_1),\dots,\tilde \phi(a_n)}$ consists of at most ${n-1}$ distinct real numbers, which are non-zero since none of the ${a_i}$ are scalar multiples of ${a_n-a_1}$, and the ${\tilde w_i}$ are linearly independent over ${{\bf Q}}$. As we are assuming inductively that the lonely runner conjecture holds for ${n-1}$, we conclude (after deleting duplicates) that there exists at least one real number ${\tilde t}$ such that

$\displaystyle \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n}.$

We would like to “approximate” ${\phi}$ by ${\tilde \phi}$ to then conclude that there is at least one real number ${t}$ such that

$\displaystyle \| t \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| t \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}.$

It turns out that we can do this by a Fourier-analytic argument taking advantage of the ${n^{C_0 n}}$-proper nature of ${Q}$. Firstly, we see from the Dirichlet approximation theorem that one has

$\displaystyle \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \leq \frac{1}{10 n^2}$

for a set ${\tilde t}$ of reals of (Banach) density ${\gg n^{-O(n)}}$. Thus, by the triangle inequality, we have

$\displaystyle \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n} - \frac{1}{10n^2}$

for a set ${\tilde t}$ of reals of density ${\gg n^{-O(n)}}$.

Applying a smooth Fourier multiplier of Littlewood-Paley type, one can find a trigonometric polynomial

$\displaystyle \eta(x) = \sum_{m: |m| \leq n^{C_0 n/10}} b_m e^{2\pi i mx}$

which takes values in ${[0,1]}$, is ${\gg 1}$ for ${\|x\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n} - \frac{1}{10n^2}}$, and is no larger than ${O( n^{-100 C_0n} )}$ for ${\|x\|_{{\bf R}/{\bf Z}} \leq \frac{1}{n+1}}$. We then have

$\displaystyle \mathop{\bf E}_t \prod_{j=1}^n \eta( t \tilde \phi(a_j) ) \gg n^{-O(n)}$

where ${\mathop{\bf E}_t f(t)}$ denotes the mean value of a quasiperiodic function ${f}$ on the reals ${{\bf R}}$. We expand the left-hand side out as

$\displaystyle \sum_{m_1,\dots,m_n: m_1 \tilde \phi(a_1) + \dots + m_n \tilde \phi(a_n) = 0} b_{m_1} \dots b_{m_n}.$

From the genericity of ${\tilde w_1,\dots,\tilde w_n}$, we see that the constraint

$\displaystyle m_1 \tilde \phi(a_1) + \dots + m_n \tilde \phi(a_n) = 0$

occurs if and only if ${m_1 a_1 + \dots + m_n a_n}$ is a scalar multiple of ${a_n-a_1}$, or equivalently (by (1), (2)) an integer multiple of ${(c'_1,\dots,c'_d)}$. Thus

$\displaystyle \sum_{m_1,\dots,m_n: m_1 a_1 + \dots + m_n a_n \in {\bf Z} (c'_1,\dots,c'_d)} b_{m_1} \dots b_{m_n} \gg n^{-O(n)}. \ \ \ \ \ (3)$

Next, we consider the average

$\displaystyle \mathop{\bf E}_t \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \ \ \ \ \ (4)$

where

$\displaystyle \xi := c'_1 w_1 + \dots + c'_d w_d. \ \ \ \ \ (5)$

and ${\varphi}$ is the Dirichlet series

$\displaystyle \varphi(x) := \sum_{m: |m| \leq n^{C_0 n/2}} e^{2\pi i mx}.$

By Fourier expansion and writing ${v_j = \phi(a_j)}$, we may write (4) as

$\displaystyle \sum_{m,m_1,\dots,m_n: |m| \leq n^{C_0n/2}; m_1 \phi(a_1) + \dots + m_n \phi(a_n) = m \xi} b_{m_1} \dots b_{m_n}.$

The support of the ${b_{m_i}}$ implies that ${|m_i| \leq n^{C_0n/10}}$. Because of the ${n^{C_0 n}}$-properness of ${Q}$, we see (for ${n}$ large enough) that the equation

$\displaystyle m_1 \phi(a_1) + \dots + m_n \phi(a_n) = m \xi \ \ \ \ \ (6)$

implies that

$\displaystyle m_1 a_1 + \dots + m_n a_n \in {\bf Z} (c'_1,\dots,c'_d) \ \ \ \ \ (7)$

and conversely that (7) implies that (6) holds for some ${m}$ with ${|m| \leq n^{C_0 n/2}}$. From (3) we thus have

$\displaystyle \mathop{\bf E}_t \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \gg n^{-O(1)}.$

In particular, there exists a ${t}$ such that

$\displaystyle \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \gg n^{-O(1)}.$

Since ${\varphi}$ is bounded in magnitude by ${n^{C_0n/2}}$, and ${\eta}$ is bounded by ${1}$, we thus have

$\displaystyle \eta(t v_j) \gg n^{-C_0 n/2 - O(1)}$

for each ${1 \leq j \leq n}$, which by the size properties of ${\eta}$ implies that ${\|tv_j\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}}$ for all ${1 \leq j \leq n}$, giving the lonely runner conjecture for ${n}$.

Because of Euler’s identity ${e^{\pi i} + 1 = 0}$, the complex exponential is not injective: ${e^{z + 2\pi i k} = e^z}$ for any complex ${z}$ and integer ${k}$. As such, the complex logarithm ${z \mapsto \log z}$ is not well-defined as a single-valued function from ${{\bf C} \backslash \{0\}}$ to ${{\bf C}}$. However, after making a branch cut, one can create a branch of the logarithm which is single-valued. For instance, after removing the negative real axis ${(-\infty,0]}$, one has the standard branch ${\hbox{Log}: {\bf C} \backslash (-\infty,0] \rightarrow \{ z \in {\bf C}: |\hbox{Im} z| < \pi \}}$ of the logarithm, with ${\hbox{Log}(z)}$ defined as the unique choice of the complex logarithm of ${z}$ whose imaginary part has magnitude strictly less than ${\pi}$. This particular branch has a number of useful additional properties:

• The standard branch ${\hbox{Log}}$ is holomorphic on its domain ${{\bf C} \backslash (-\infty,0]}$.
• One has ${\hbox{Log}( \overline{z} ) = \overline{ \hbox{Log}(z) }}$ for all ${z}$ in the domain ${{\bf C} \backslash (-\infty,0]}$. In particular, if ${z \in {\bf C} \backslash (-\infty,0]}$ is real, then ${\hbox{Log} z}$ is real.
• One has ${\hbox{Log}( z^{-1} ) = - \hbox{Log}(z)}$ for all ${z}$ in the domain ${{\bf C} \backslash (-\infty,0]}$.

One can then also use the standard branch of the logarithm to create standard branches of other multi-valued functions, for instance creating a standard branch ${z \mapsto \exp( \frac{1}{2} \hbox{Log} z )}$ of the square root function. We caution however that the identity ${\hbox{Log}(zw) = \hbox{Log}(z) + \hbox{Log}(w)}$ can fail for the standard branch (or indeed for any branch of the logarithm).

One can extend this standard branch of the logarithm to ${n \times n}$ complex matrices, or (equivalently) to linear transformations ${T: V \rightarrow V}$ on an ${n}$-dimensional complex vector space ${V}$, provided that the spectrum of that matrix or transformation avoids the branch cut ${(-\infty,0]}$. Indeed, from the spectral theorem one can decompose any such ${T: V \rightarrow V}$ as the direct sum of operators ${T_\lambda: V_\lambda \rightarrow V_\lambda}$ on the non-trivial generalised eigenspaces ${V_\lambda}$ of ${T}$, where ${\lambda \in {\bf C} \backslash (-\infty,0]}$ ranges in the spectrum of ${T}$. For each component ${T_\lambda}$ of ${T}$, we define

$\displaystyle \hbox{Log}( T_\lambda ) = P_\lambda( T_\lambda )$

where ${P_\lambda}$ is the Taylor expansion of ${\hbox{Log}}$ at ${\lambda}$; as ${T_\lambda-\lambda}$ is nilpotent, only finitely many terms in this Taylor expansion are required. The logarithm ${\hbox{Log} T}$ is then defined as the direct sum of the ${\hbox{Log} T_\lambda}$.

The matrix standard branch of the logarithm has many pleasant and easily verified properties (often inherited from their scalar counterparts), whenever ${T: V \rightarrow V}$ has no spectrum in ${(-\infty,0]}$:

• (i) We have ${\exp( \hbox{Log} T ) = T}$.
• (ii) If ${T_1: V_1 \rightarrow V_1}$ and ${T_2: V_2 \rightarrow V_2}$ have no spectrum in ${(-\infty,0]}$, then ${\hbox{Log}( T_1 \oplus T_2 ) = \hbox{Log}(T_1) \oplus \hbox{Log}(T_2)}$.
• (iii) If ${T}$ has spectrum in a closed disk ${B(z,r)}$ in ${{\bf C} \backslash (-\infty,0]}$, then ${\hbox{Log}(T) = P_z(T)}$, where ${P_z}$ is the Taylor series of ${\hbox{Log}}$ around ${z}$ (which is absolutely convergent in ${B(z,r)}$).
• (iv) ${\hbox{Log}(T)}$ depends holomorphically on ${T}$. (Easily established from (ii), (iii), after covering the spectrum of ${T}$ by disjoint disks; alternatively, one can use the Cauchy integral representation ${\hbox{Log}(T) = \frac{1}{2\pi i} \int_\gamma \hbox{Log}(z)(z-T)^{-1}\ dz}$ for a contour ${\gamma}$ in the domain enclosing the spectrum of ${T}$.) In particular, the standard branch of the matrix logarithm is smooth.
• (v) If ${S: V \rightarrow W}$ is any invertible linear or antilinear map, then ${\hbox{Log}( STS^{-1} ) = S \hbox{Log}(T) S^{-1}}$. In particular, the standard branch of the logarithm commutes with matrix conjugations; and if ${T}$ is real with respect to a complex conjugation operation on ${V}$ (that is to say, an antilinear involution), then ${\hbox{Log}(T)}$ is real also.
• (vi) If ${T^*: V^* \rightarrow V^*}$ denotes the transpose of ${T}$ (with ${V^*}$ the complex dual of ${V}$), then ${\hbox{Log}(T^*) = \hbox{Log}(T)^*}$. Similarly, if ${T^\dagger: V^\dagger \rightarrow V^\dagger}$ denotes the adjoint of ${T}$ (with ${V^\dagger}$ the complex conjugate of ${V^*}$, i.e. ${V^*}$ with the conjugated multiplication map ${(c,z) \mapsto \overline{c} z}$), then ${\hbox{Log}(T^\dagger) = \hbox{Log}(T)^\dagger}$.
• (vii) One has ${\hbox{Log}(T^{-1}) = - \hbox{Log}( T )}$.
• (viii) If ${\sigma(T)}$ denotes the spectrum of ${T}$, then ${\sigma(\hbox{Log} T) = \hbox{Log} \sigma(T)}$.

As a quick application of the standard branch of the matrix logarithm, we have

Proposition 1 Let ${G}$ be one of the following matrix groups: ${GL_n({\bf C})}$, ${GL_n({\bf R})}$, ${U_n({\bf C})}$, ${O(Q)}$, ${Sp_{2n}({\bf C})}$, or ${Sp_{2n}({\bf R})}$, where ${Q: {\bf R}^n \rightarrow {\bf R}}$ is a non-degenerate real quadratic form (so ${O(Q)}$ is isomorphic to a (possibly indefinite) orthogonal group ${O(k,n-k)}$ for some ${0 \leq k \leq n}$. Then any element ${T}$ of ${G}$ whose spectrum avoids ${(-\infty,0]}$ is exponential, that is to say ${T = \exp(X)}$ for some ${X}$ in the Lie algebra ${{\mathfrak g}}$ of ${G}$.

Proof: We just prove this for ${G=O(Q)}$, as the other cases are similar (or a bit simpler). If ${T \in O(Q)}$, then (viewing ${T}$ as a complex-linear map on ${{\bf C}^n}$, and using the complex bilinear form associated to ${Q}$ to identify ${{\bf C}^n}$ with its complex dual ${({\bf C}^n)^*}$, then ${T}$ is real and ${T^{*-1} = T}$. By the properties (v), (vi), (vii) of the standard branch of the matrix logarithm, we conclude that ${\hbox{Log} T}$ is real and ${- \hbox{Log}(T)^* = \hbox{Log}(T)}$, and so ${\hbox{Log}(T)}$ lies in the Lie algebra ${{\mathfrak g} = {\mathfrak o}(Q)}$, and the claim now follows from (i). $\Box$

Exercise 2 Show that ${\hbox{diag}(-\lambda, -1/\lambda)}$ is not exponential in ${GL_2({\bf R})}$ if ${-\lambda \in (-\infty,0) \backslash \{-1\}}$. Thus we see that the branch cut in the above proposition is largely necessary. See this paper of Djokovic for a more complete description of the image of the exponential map in classical groups, as well as this previous blog post for some more discussion of the surjectivity (or lack thereof) of the exponential map in Lie groups.

For a slightly less quick application of the standard branch, we have the following result (recently worked out in the answers to this MathOverflow question):

Proposition 3 Let ${T}$ be an element of the split orthogonal group ${O(n,n)}$ which lies in the connected component of the identity. Then ${\hbox{det}(1+T) \geq 0}$.

The requirement that ${T}$ lie in the identity component is necessary, as the counterexample ${T = \hbox{diag}(-\lambda, -1/\lambda )}$ for ${\lambda \in (-\infty,-1) \cup (-1,0)}$ shows.

Proof: We think of ${T}$ as a (real) linear transformation on ${{\bf C}^{2n}}$, and write ${Q}$ for the quadratic form associated to ${O(n,n)}$, so that ${O(n,n) \equiv O(Q)}$. We can split ${{\bf C}^{2n} = V_1 \oplus V_2}$, where ${V_1}$ is the sum of all the generalised eigenspaces corresponding to eigenvalues in ${(-\infty,0]}$, and ${V_2}$ is the sum of all the remaining eigenspaces. Since ${T}$ and ${(-\infty,0]}$ are real, ${V_1,V_2}$ are real (i.e. complex-conjugation invariant) also. For ${i=1,2}$, the restriction ${T_i: V_i \rightarrow V_i}$ of ${T}$ to ${V_i}$ then lies in ${O(Q_i)}$, where ${Q_i}$ is the restriction of ${Q}$ to ${V_i}$, and

$\displaystyle \hbox{det}(1+T) = \hbox{det}(1+T_1) \hbox{det}(1+T_2).$

The spectrum of ${T_2}$ consists of positive reals, as well as complex pairs ${\lambda, \overline{\lambda}}$ (with equal multiplicity), so ${\hbox{det}(1+T_2) > 0}$. From the preceding proposition we have ${T_2 = \exp( X_2 )}$ for some ${X_2 \in {\mathfrak o}(Q_2)}$; this will be important later.

It remains to show that ${\hbox{det}(1+T_1) \geq 0}$. If ${T_1}$ has spectrum at ${-1}$ then we are done, so we may assume that ${T_1}$ has spectrum only at ${(-\infty,-1) \cup (-1,0)}$ (being invertible, ${T}$ has no spectrum at ${0}$). We split ${V_1 = V_3 \oplus V_4}$, where ${V_3,V_4}$ correspond to the portions of the spectrum in ${(-\infty,-1)}$, ${(-1,0)}$; these are real, ${T}$-invariant spaces. We observe that if ${V_\lambda, V_\mu}$ are generalised eigenspaces of ${T}$ with ${\lambda \mu \neq 1}$, then ${V_\lambda, V_\mu}$ are orthogonal with respect to the (complex-bilinear) inner product ${\cdot}$ associated with ${Q}$; this is easiest to see first for the actual eigenspaces (since ${ \lambda \mu u \cdot v = Tu \cdot Tv = u \cdot v}$ for all ${u \in V_\lambda, v \in V_\mu}$), and the extension to generalised eigenvectors then follows from a routine induction. From this we see that ${V_1}$ is orthogonal to ${V_2}$, and ${V_3}$ and ${V_4}$ are null spaces, which by the non-degeneracy of ${Q}$ (and hence of the restriction ${Q_1}$ of ${Q}$ to ${V_1}$) forces ${V_3}$ to have the same dimension as ${V_4}$, indeed ${Q}$ now gives an identification of ${V_3^*}$ with ${V_4}$. If we let ${T_3, T_4}$ be the restrictions of ${T}$ to ${V_3,V_4}$, we thus identify ${T_4}$ with ${T_3^{*-1}}$, since ${T}$ lies in ${O(Q)}$; in particular ${T_3}$ is invertible. Thus

$\displaystyle \hbox{det}(1+T_1) = \hbox{det}(1 + T_3) \hbox{det}( 1 + T_3^{*-1} ) = \hbox{det}(T_3)^{-1} \hbox{det}(1+T_3)^2$

and so it suffices to show that ${\hbox{det}(T_3) > 0}$.

At this point we need to use the hypothesis that ${T}$ lies in the identity component of ${O(n,n)}$. This implies (by a continuity argument) that the restriction of ${T}$ to any maximal-dimensional positive subspace has positive determinant (since such a restriction cannot be singular, as this would mean that ${T}$ positive norm vector would map to a non-positive norm vector). Now, as ${V_3,V_4}$ have equal dimension, ${Q_1}$ has a balanced signature, so ${Q_2}$ does also. Since ${T_2 = \exp(X_2)}$, ${T_2}$ already lies in the identity component of ${O(Q_2)}$, and so has positive determinant on any maximal-dimensional positive subspace of ${V_2}$. We conclude that ${T_1}$ has positive determinant on any maximal-dimensional positive subspace of ${V_1}$.

We choose a complex basis of ${V_3}$, to identify ${V_3}$ with ${V_3^*}$, which has already been identified with ${V_4}$. (In coordinates, ${V_3,V_4}$ are now both of the form ${{\bf C}^m}$, and ${Q( v \oplus w ) = v \cdot w}$ for ${v,w \in {\bf C}^m}$.) Then ${\{ v \oplus v: v \in V_3 \}}$ becomes a maximal positive subspace of ${V_1}$, and the restriction of ${T_1}$ to this subspace is conjugate to ${T_3 + T_3^{*-1}}$, so that

$\displaystyle \hbox{det}( T_3 + T_3^{*-1} ) > 0.$

But since ${\hbox{det}( T_3 + T_3^{*-1} ) = \hbox{det}(T_3) \hbox{det}( 1 + T_3^{-1} T_3^{*-1} )}$ and ${ 1 + T_3^{-1} T_3^{*-1}}$ is positive definite, so ${\hbox{det}(T_3)>0}$ as required. $\Box$

The Euler equations for three-dimensional incompressible inviscid fluid flow are

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

where ${u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3}$ is the velocity field, and ${p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}}$ is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as ${(-\Delta)^{-1}}$ or ${|\nabla|^{-1} := (-\Delta)^{-1/2}}$ formally, assuming that these inverses are well defined on the functions they are applied to.

Meanwhile, the surface quasi-geostrophic (SQG) equation is given by

$\displaystyle \partial_t \theta + (u \cdot \nabla) \theta = 0 \ \ \ \ \ (2)$

$\displaystyle u = ( -\partial_y |\nabla|^{-1}, \partial_x |\nabla|^{-1} ) \theta \ \ \ \ \ (3)$

where ${\theta: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ is the active scalar, and ${u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2}$ is the velocity field. The SQG equations are often used as a toy model for the 3D Euler equations, as they share many of the same features (e.g. vortex stretching); see this paper of Constantin, Majda, and Tabak for more discussion (or this previous blog post).

I recently found a more direct way to connect the two equations. We first recall that the Euler equations can be placed in vorticity-stream form by focusing on the vorticity ${\omega := \nabla \times u}$. Indeed, taking the curl of (1), we obtain the vorticity equation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (4)$

while the velocity ${u}$ can be recovered from the vorticity via the Biot-Savart law

$\displaystyle u = (-\Delta)^{-1} \nabla \times \omega. \ \ \ \ \ (5)$

The system (4), (5) has some features in common with the system (2), (3); in (2) it is a scalar field ${\theta}$ that is being transported by a divergence-free vector field ${u}$, which is a linear function of the scalar field as per (3), whereas in (4) it is a vector field ${\omega}$ that is being transported (in the Lie derivative sense) by a divergence-free vector field ${u}$, which is a linear function of the vector field as per (5). However, the system (4), (5) is in three dimensions whilst (2), (3) is in two spatial dimensions, the dynamical field is a scalar field ${\theta}$ for SQG and a vector field ${\omega}$ for Euler, and the relationship between the velocity field and the dynamical field is given by a zeroth order Fourier multiplier in (3) and a ${-1^{th}}$ order operator in (5).

However, we can make the two equations more closely resemble each other as follows. We first consider the generalisation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (6)$

$\displaystyle u = T (-\Delta)^{-1} \nabla \times \omega \ \ \ \ \ (7)$

where ${T}$ is an invertible, self-adjoint, positive-definite zeroth order Fourier multiplier that maps divergence-free vector fields to divergence-free vector fields. The Euler equations then correspond to the case when ${T}$ is the identity operator. As discussed in this previous blog post (which used ${A}$ to denote the inverse of the operator denoted here as ${T}$), this generalised Euler system has many of the same features as the original Euler equation, such as a conserved Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\bf R}^3} u \cdot T^{-1} u,$

the Kelvin circulation theorem, and conservation of helicity

$\displaystyle \int_{{\bf R}^3} \omega \cdot T^{-1} u.$

Also, if we require ${\omega}$ to be divergence-free at time zero, it remains divergence-free at all later times.

Let us consider “two-and-a-half-dimensional” solutions to the system (6), (7), in which ${u,\omega}$ do not depend on the vertical coordinate ${z}$, thus

$\displaystyle \omega(t,x,y,z) = \omega(t,x,y)$

and

$\displaystyle u(t,x,y,z) = u(t,x,y)$

but we allow the vertical components ${u_z, \omega_z}$ to be non-zero. For this to be consistent, we also require ${T}$ to commute with translations in the ${z}$ direction. As all derivatives in the ${z}$ direction now vanish, we can simplify (6) to

$\displaystyle D_t \omega = (\omega_x \partial_x + \omega_y \partial_y) u \ \ \ \ \ (8)$

where ${D_t}$ is the two-dimensional material derivative

$\displaystyle D_t := \partial_t + u_x \partial_x + u_y \partial_y.$

Also, divergence-free nature of ${\omega,u}$ then becomes

$\displaystyle \partial_x \omega_x + \partial_y \omega_y = 0$

and

$\displaystyle \partial_x u_x + \partial_y u_y = 0. \ \ \ \ \ (9)$

In particular, we may (formally, at least) write

$\displaystyle (\omega_x, \omega_y) = (\partial_y \theta, -\partial_x \theta)$

for some scalar field ${\theta(t,x,y,z) = \theta(t,x,y)}$, so that (7) becomes

$\displaystyle u = T ( (- \Delta)^{-1} \partial_y \omega_z, - (-\Delta^{-1}) \partial_x \omega_z, \theta ). \ \ \ \ \ (10)$

The first two components of (8) become

$\displaystyle D_t \partial_y \theta = \partial_y \theta \partial_x u_x - \partial_x \theta \partial_y u_x$

$\displaystyle - D_t \partial_x \theta = \partial_y \theta \partial_x u_y - \partial_x \theta \partial_y u_y$

which rearranges using (9) to

$\displaystyle \partial_y D_t \theta = \partial_x D_t \theta = 0.$

Formally, we may integrate this system to obtain the transport equation

$\displaystyle D_t \theta = 0, \ \ \ \ \ (11)$

Finally, the last component of (8) is

$\displaystyle D_t \omega_z = \partial_y \theta \partial_x u_z - \partial_x \theta \partial_y u_z. \ \ \ \ \ (12)$

At this point, we make the following choice for ${T}$:

$\displaystyle T ( U_x, U_y, \theta ) = \alpha (U_x, U_y, \theta) + (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, 0) \ \ \ \ \ (13)$

$\displaystyle + P( 0, 0, |\nabla|^{-1} (\partial_y U_x - \partial_x U_y) )$

where ${\alpha > 0}$ is a real constant and ${Pu := (-\Delta)^{-1} (\nabla \times (\nabla \times u))}$ is the Leray projection onto divergence-free vector fields. One can verify that for large enough ${\alpha}$, ${T}$ is a self-adjoint positive definite zeroth order Fourier multiplier from divergence free vector fields to divergence-free vector fields. With this choice, we see from (10) that

$\displaystyle u_z = \alpha \theta - |\nabla|^{-1} \omega_z$

so that (12) simplifies to

$\displaystyle D_t \omega_z = - \partial_y \theta \partial_x |\nabla|^{-1} \omega_z + \partial_x \theta \partial_y |\nabla|^{-1} \omega_z.$

This implies (formally at least) that if ${\omega_z}$ vanishes at time zero, then it vanishes for all time. Setting ${\omega_z=0}$, we then have from (10) that

$\displaystyle (u_x,u_y,u_z) = (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, \alpha \theta )$

and from (11) we then recover the SQG system (2), (3). To put it another way, if ${\theta(t,x,y)}$ and ${u(t,x,y)}$ solve the SQG system, then by setting

$\displaystyle \omega(t,x,y,z) := ( \partial_y \theta(t,x,y), -\partial_x \theta(t,x,y), 0 )$

$\displaystyle \tilde u(t,x,y,z) := ( u_x(t,x,y), u_y(t,x,y), \alpha \theta(t,x,y) )$

then ${\omega,\tilde u}$ solve the modified Euler system (6), (7) with ${T}$ given by (13).

We have ${T^{-1} \tilde u = (0, 0, \theta)}$, so the Hamiltonian ${\frac{1}{2} \int_{{\bf R}^3} \tilde u \cdot T^{-1} \tilde u}$ for the modified Euler system in this case is formally a scalar multiple of the conserved quantity ${\int_{{\bf R}^2} \theta^2}$. The momentum ${\int_{{\bf R}^3} x \cdot \tilde u}$ for the modified Euler system is formally a scalar multiple of the conserved quantity ${\int_{{\bf R}^2} \theta}$, while the vortex stream lines that are preserved by the modified Euler flow become the level sets of the active scalar that are preserved by the SQG flow. On the other hand, the helicity ${\int_{{\bf R}^3} \omega \cdot T^{-1} \tilde u}$ vanishes, and other conserved quantities for SQG (such as the Hamiltonian ${\int_{{\bf R}^2} \theta |\nabla|^{-1} \theta}$) do not seem to correspond to conserved quantities of the modified Euler system. This is not terribly surprising; a low-dimensional flow may well have a richer family of conservation laws than the higher-dimensional system that it is embedded in.

An extremely large portion of mathematics is concerned with locating solutions to equations such as

$\displaystyle f(x) = 0$

or

$\displaystyle \Phi(x) = x \ \ \ \ \ (1)$

for ${x}$ in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps ${f}$ or ${\Phi}$. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial approximate solution ${x_0}$ to (1), so that ${\Phi(x_0) \approx x_0}$, and then recursively constructs the sequence ${x_1, x_2, x_3, \dots}$ by ${x_n := \Phi(x_{n-1})}$. If ${\Phi}$ behaves enough like a “contraction”, and the domain is complete, then one can expect the ${x_n}$ to converge to a limit ${x}$, which should then be a solution to (1). For instance, if ${\Phi: X \rightarrow X}$ is a map from a metric space ${X = (X,d)}$ to itself, which is a contraction in the sense that

$\displaystyle d( \Phi(x), \Phi(y) ) \leq (1-\eta) d(x,y)$

for all ${x,y \in X}$ and some ${\eta>0}$, then with ${x_n}$ as above we have

$\displaystyle d( x_{n+1}, x_n ) \leq (1-\eta) d(x_n, x_{n-1} )$

for any ${n}$, and so the distances ${d(x_n, x_{n-1} )}$ between successive elements of the sequence decay at at least a geometric rate. This leads to the contraction mapping theorem, which has many important consequences, such as the inverse function theorem and the Picard existence theorem.

A slightly more complicated instance of this strategy arises when trying to linearise a complex map ${f: U \rightarrow {\bf C}}$ defined in a neighbourhood ${U}$ of a fixed point. For simplicity we normalise the fixed point to be the origin, thus ${0 \in U}$ and ${f(0)=0}$. When studying the complex dynamics ${f^2 = f \circ f}$, ${f^3 = f \circ f \circ f}$, ${\dots}$ of such a map, it can be useful to try to conjugate ${f}$ to another function ${g = \psi^{-1} \circ f \circ \psi}$, where ${\psi}$ is a holomorphic function defined and invertible near ${0}$ with ${\psi(0)=0}$, since the dynamics of ${g}$ will be conjguate to that of ${f}$. Note that if ${f(0)=0}$ and ${f'(0)=\lambda}$, then from the chain rule any conjugate ${g}$ of ${f}$ will also have ${g(0)=0}$ and ${g'(0)=\lambda}$. Thus, the “simplest” function one can hope to conjugate ${f}$ to is the linear function ${z \mapsto \lambda z}$. Let us say that ${f}$ is linearisable (around ${0}$) if it is conjugate to ${z \mapsto \lambda z}$ in some neighbourhood of ${0}$. Equivalently, ${f}$ is linearisable if there is a solution to the Schröder equation

$\displaystyle f( \psi(z) ) = \psi(\lambda z) \ \ \ \ \ (2)$

for some ${\psi: U' \rightarrow {\bf C}}$ defined and invertible in a neighbourhood ${U'}$ of ${0}$ with ${\psi(0)=0}$, and all ${z}$ sufficiently close to ${0}$. (The Schröder equation is normalised somewhat differently in the literature, but this form is equivalent to the usual form, at least when ${\lambda}$ is non-zero.) Note that if ${\psi}$ solves the above equation, then so does ${z \mapsto \psi(cz)}$ for any non-zero ${c}$, so we may normalise ${\psi'(0)=1}$ in addition to ${\psi(0)=0}$, which also ensures local invertibility from the inverse function theorem. (Note from winding number considerations that ${\psi}$ cannot be invertible near zero if ${\psi'(0)}$ vanishes.)

We have the following basic result of Koenigs:

Theorem 1 (Koenig’s linearisation theorem) Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function defined near ${0}$ with ${f(0)=0}$ and ${f'(0)=\lambda}$. If ${0 < |\lambda| < 1}$ (attracting case) or ${1 < |\lambda| < \infty}$ (repelling case), then ${f}$ is linearisable near zero.

Proof: Observe that if ${f, \psi, \lambda}$ solve (2), then ${f^{-1}, \psi^{-1}, \lambda^{-1}}$ solve (2) also (in a sufficiently small neighbourhood of zero). Thus we may reduce to the attractive case ${0 < |\lambda| < 1}$.

Let ${r>0}$ be a sufficiently small radius, and let ${X}$ denote the space of holomorphic functions ${\psi: B(0,r) \rightarrow {\bf C}}$ on the complex disk ${B(0,r) := \{z \in {\bf C}: |z| < r \}}$ with ${\psi(0)=0}$ and ${\psi'(0)=1}$. We can view the Schröder equation (2) as a fixed point equation

$\displaystyle \psi = \Phi(\psi)$

where ${\Phi: X' \rightarrow X}$ is the partially defined function on ${X}$ that maps a function ${\psi: B(0,r) \rightarrow {\bf C}}$ to the function ${\Phi(\psi): B(0,r) \rightarrow {\bf C}}$ defined by

$\displaystyle \Phi(\psi)(z) := f^{-1}( \psi( \lambda z ) ),$

assuming that ${f^{-1}}$ is well-defined on the range of ${\psi(B(0,\lambda r))}$ (this is why ${\Phi}$ is only partially defined).

We can solve this equation by the fixed point iteration method, if ${r}$ is small enough. Namely, we start with ${\psi_0: B(0,r) \rightarrow {\bf C}}$ being the identity map, and set ${\psi_1 := \Phi(\psi_0), \psi_2 := \Phi(\psi_1)}$, etc. We equip ${X}$ with the uniform metric ${d( \psi, \tilde \psi ) := \sup_{z \in B(0,r)} |\psi(z) - \tilde \psi(z)|}$. Observe that if ${d( \psi, \psi_0 ), d(\tilde \psi, \psi_0) \leq r}$, and ${r}$ is small enough, then ${\psi, \tilde \psi}$ takes values in ${B(0,2r)}$, and ${\Phi(\psi), \Phi(\tilde \psi)}$ are well-defined and lie in ${X}$. Also, since ${f^{-1}}$ is smooth and has derivative ${\lambda^{-1}}$ at ${0}$, we have

$\displaystyle |f^{-1}(z) - f^{-1}(w)| \leq (1+\varepsilon) |\lambda|^{-1} |z-w|$

if ${z, w \in B(0,r)}$, ${\varepsilon>0}$ and ${r}$ is sufficiently small depending on ${\varepsilon}$. This is not yet enough to establish the required contraction (thanks to Mario Bonk for pointing this out); but observe that the function ${\frac{\psi(z)-\tilde \psi(z)}{z^2}}$ is holomorphic on ${B(0,r)}$ and bounded by ${d(\psi,\tilde \psi)/r^2}$ on the boundary of this ball (or slightly within this boundary), so by the maximum principle we see that

$\displaystyle |\frac{\psi(z)-\tilde \psi(z)}{z^2}| \leq \frac{1}{r^2} d(\psi,\tilde \psi)$

on all of ${B(0,r)}$, and in particular

$\displaystyle |\psi(z)-\tilde \psi(z)| \leq |\lambda|^2 d(\psi,\tilde \psi)$

on ${B(0,\lambda r)}$. Putting all this together, we see that

$\displaystyle d( \Phi(\psi), \Phi(\tilde \psi)) \leq (1+\varepsilon) |\lambda| d(\psi, \tilde \psi);$

since ${|\lambda|<1}$, we thus obtain a contraction on the ball ${\{ \psi \in X: d(\psi,\psi_0) \leq r \}}$ if ${\varepsilon}$ is small enough (and ${r}$ sufficiently small depending on ${\varepsilon}$). From this (and the completeness of ${X}$, which follows from Morera’s theorem) we see that the iteration ${\psi_n}$ converges (exponentially fast) to a limit ${\psi \in X}$ which is a fixed point of ${\Phi}$, and thus solves Schröder’s equation, as required. $\Box$

Koenig’s linearisation theorem leaves open the indifferent case when ${|\lambda|=1}$. In the rationally indifferent case when ${\lambda^n=1}$ for some natural number ${n}$, there is an obvious obstruction to linearisability, namely that ${f^n = 1}$ (in particular, linearisation is not possible in this case when ${f}$ is a non-trivial rational function). An obstruction is also present in some irrationally indifferent cases (where ${|\lambda|=1}$ but ${\lambda^n \neq 1}$ for any natural number ${n}$), if ${\lambda}$ is sufficiently close to various roots of unity; the first result of this form is due to Cremer, and the optimal result of this type for quadratic maps was established by Yoccoz. In the other direction, we have the following result of Siegel:

Theorem 2 (Siegel’s linearisation theorem) Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function defined near ${0}$ with ${f(0)=0}$ and ${f'(0)=\lambda}$. If ${|\lambda|=1}$ and one has the Diophantine condition ${\frac{1}{|\lambda^n-1|} \leq C n^C}$ for all natural numbers ${n}$ and some constant ${C>0}$, then ${f}$ is linearisable at ${0}$.

The Diophantine condition can be relaxed to a more general condition involving the rational exponents of the phase ${\theta}$ of ${\lambda = e^{2\pi i \theta}}$; this was worked out by Brjuno, with the condition matching the one later obtained by Yoccoz. Amusingly, while the set of Diophantine numbers (and hence the set of linearisable ${\lambda}$) has full measure on the unit circle, the set of non-linearisable ${\lambda}$ is generic (the complement of countably many nowhere dense sets) due to the above-mentioned work of Cremer, leading to a striking disparity between the measure-theoretic and category notions of “largeness”.

Siegel’s theorem does not seem to be provable using a fixed point iteration method. However, it can be established by modifying another basic method to solve equations, namely Newton’s method. Let us first review how this method works to solve the equation ${f(x)=0}$ for some smooth function ${f: I \rightarrow {\bf R}}$ defined on an interval ${I}$. We suppose we have some initial approximant ${x_0 \in I}$ to this equation, with ${f(x_0)}$ small but not necessarily zero. To make the analysis more quantitative, let us suppose that the interval ${[x_0-r_0,x_0+r_0]}$ lies in ${I}$ for some ${r_0>0}$, and we have the estimates

$\displaystyle |f(x_0)| \leq \delta_0 r_0$

$\displaystyle |f'(x)| \geq \eta_0$

$\displaystyle |f''(x)| \leq \frac{1}{\eta_0 r_0}$

for some ${\delta_0 > 0}$ and ${0 < \eta_0 < 1/2}$ and all ${x \in [x_0-r_0,x_0+r_0]}$ (the factors of ${r_0}$ are present to make ${\delta_0,\eta_0}$ “dimensionless”).

Lemma 3 Under the above hypotheses, we can find ${x_1}$ with ${|x_1 - x_0| \leq \eta_0 r_0}$ such that

$\displaystyle |f(x_1)| \ll \delta_0^2 \eta_0^{-O(1)} r_0.$

In particular, setting ${r_1 := (1-\eta_0) r_0}$, ${\eta_1 := \eta_0/2}$, and ${\delta_1 = O(\delta_0^2 \eta_0^{-O(1)})}$, we have ${[x_1-r_1,x_1+r_1] \subset [x_0-r_0,x_0+r_0] \subset I}$, and

$\displaystyle |f(x_1)| \leq \delta_1 r_1$

$\displaystyle |f'(x)| \geq \eta_1$

$\displaystyle |f''(x)| \leq \frac{1}{\eta_1 r_1}$

for all ${x \in [x_1-r_1,x_1+r_1]}$.

The crucial point here is that the new error ${\delta_1}$ is roughly the square of the previous error ${\delta_0}$. This leads to extremely fast (double-exponential) improvement in the error upon iteration, which is more than enough to absorb the exponential losses coming from the ${\eta_0^{-O(1)}}$ factor.

Proof: If ${\delta_0 > c \eta_0^{C}}$ for some absolute constants ${C,c>0}$ then we may simply take ${x_0=x_1}$, so we may assume that ${\delta_0 \leq c \eta_0^{C}}$ for some small ${c>0}$ and large ${C>0}$. Using the Newton approximation ${f(x_0+h) \approx f(x_0) + h f'(x_0)}$ we are led to the choice

$\displaystyle x_1 := x_0 - \frac{f(x_0)}{f'(x_0)}$

for ${x_1}$. From the hypotheses on ${f}$ and the smallness hypothesis on ${\delta}$ we certainly have ${|x_1-x_0| \leq \eta_0 r_0}$. From Taylor’s theorem with remainder we have

$\displaystyle f(x_1) = f(x_0) - \frac{f(x_0)}{f'(x_0)} f'(x_0) + O( \frac{1}{\eta_0 r_0} |\frac{f(x_0)}{f'(x_0)}|^2 )$

$\displaystyle = O( \frac{1}{\eta_0 r_0} (\frac{\delta_0 r_0}{\eta_0})^2 )$

and the claim follows. $\Box$

We can iterate this procedure; starting with ${x_0,\eta_0,r_0,\delta_0}$ as above, we obtain a sequence of nested intervals ${[x_n-r_n,x_n+r_n]}$ with ${f(x_n)| \leq \delta_n}$, and with ${\eta_n,r_n,\delta_n,x_n}$ evolving by the recursive equations and estimates

$\displaystyle \eta_n = \eta_{n-1} / 2$

$\displaystyle r_n = (1 - \eta_{n-1}) r_{n-1}$

$\displaystyle \delta_n = O( \delta_{n-1}^2 \eta_{n-1}^{-O(1)} )$

$\displaystyle |x_n - x_{n-1}| \leq \eta_{n-1} r_{n-1}.$

If ${\delta_0}$ is sufficiently small depending on ${\eta_0}$, we see that ${\delta_n}$ converges rapidly to zero (indeed, we can inductively obtain a bound of the form ${\delta_n \leq \eta_0^{C (2^n + n)}}$ for some large absolute constant ${C}$ if ${\delta_0}$ is small enough), and ${x_n}$ converges to a limit ${x \in I}$ which then solves the equation ${f(x)=0}$ by the continuity of ${f}$.

As I recently learned from Zhiqiang Li, a similar scheme works to prove Siegel’s theorem, as can be found for instance in this text of Carleson and Gamelin. The key is the following analogue of Lemma 3.

Lemma 4 Let ${\lambda}$ be a complex number with ${|\lambda|=1}$ and ${\frac{1}{|\lambda^n-1|} \ll n^{O(1)}}$ for all natural numbers ${n}$. Let ${r_0>0}$, and let ${f_0: B(0,r_0) \rightarrow {\bf C}}$ be a holomorphic function with ${f_0(0)=0}$, ${f'_0(0)=\lambda}$, and

$\displaystyle |f_0(z) - \lambda z| \leq \delta_0 r_0 \ \ \ \ \ (3)$

for all ${z \in B(0,r_0)}$ and some ${\delta_0>0}$. Let ${0 < \eta_0 \leq 1/2}$, and set ${r_1 := (1-\eta_0) r_0}$. Then there exists an injective holomorphic function ${\psi_0: B(0, r_1) \rightarrow B(0, r_0)}$ and a holomorphic function ${f_1: B(0,r_1) \rightarrow {\bf C}}$ such that

$\displaystyle f_0( \psi_1(z) ) = \psi_1(f_1(z)) \ \ \ \ \ (4)$

for all ${z \in B(0,r_1)}$, and such that

$\displaystyle |\psi_1(z) - z| \ll \delta_0 \eta_0^{-O(1)} r_1$

and

$\displaystyle |f_1(z) - \lambda z| \leq \delta_1 r_1$

for all ${z \in B(0,r_1)}$ and some ${\delta_1 = O(\delta_0^2 \eta_0^{-O(1)})}$.

Proof: By scaling we may normalise ${r_0=1}$. If ${\delta_0 > c \eta_0^C}$ for some constants ${c,C>0}$, then we can simply take ${\psi_1}$ to be the identity and ${f_1=f_0}$, so we may assume that ${\delta_0 \leq c \eta_0^C}$ for some small ${c>0}$ and large ${C>0}$.

To motivate the choice of ${\psi_1}$, we write ${f_0(z) = \lambda z + \hat f_0(z)}$ and ${\psi_1(z) = z + \hat \psi(z)}$, with ${\hat f_0}$ and ${\hat \psi_1}$ viewed as small. We would like to have ${f_0(\psi_1(z)) \approx \psi_1(\lambda z)}$, which expands as

$\displaystyle \lambda z + \lambda \hat \psi_1(z) + \hat f_0( z + \hat \psi_1(z) ) \approx \lambda z + \hat \psi_1(\lambda z).$

As ${\hat f_0}$ and ${\hat \psi}$ are both small, we can heuristically approximate ${\hat f_0(z + \hat \psi_1(z) ) \approx \hat f_0(z)}$ up to quadratic errors (compare with the Newton approximation ${f(x_0+h) \approx f(x_0) + h f'(x_0)}$), and arrive at the equation

$\displaystyle \hat \psi_1(\lambda z) - \lambda \hat \psi_1(z) = \hat f_0(z). \ \ \ \ \ (5)$

This equation can be solved by Taylor series; the function ${\hat f_0}$ vanishes to second order at the origin and thus has a Taylor expansion

$\displaystyle \hat f_0(z) = \sum_{n=2}^\infty a_n z^n$

and then ${\hat \psi_1}$ has a Taylor expansion

$\displaystyle \hat \psi_1(z) = \sum_{n=2}^\infty \frac{a_n}{\lambda^n - \lambda} z^n.$

We take this as our definition of ${\hat \psi_1}$, define ${\psi_1(z) := z + \hat \psi_1(z)}$, and then define ${f_1}$ implicitly via (4).

Let us now justify that this choice works. By (3) and the generalised Cauchy integral formula, we have ${|a_n| \leq \delta_0}$ for all ${n}$; by the Diophantine assumption on ${\lambda}$, we thus have ${|\frac{a_n}{\lambda^n - \lambda}| \ll \delta_0 n^{O(1)}}$. In particular, ${\hat \psi_1}$ converges on ${B(0,1)}$, and on the disk ${B(0, (1-\eta_0/4))}$ (say) we have the bounds

$\displaystyle |\hat \psi_1(z)|, |\hat \psi'_1(z)| \ll \delta_0 \sum_{n=2}^\infty n^{O(1)} (1-\eta_0/4)^n \ll \eta_0^{-O(1)} \delta_0. \ \ \ \ \ (6)$

In particular, as ${\delta_0}$ is so small, we see that ${\psi_1}$ maps ${B(0, (1-\eta_0/4))}$ injectively to ${B(0,1)}$ and ${B(0,1-\eta_0)}$ to ${B(0,1-3\eta_0/4)}$, and the inverse ${\psi_1^{-1}}$ maps ${B(0, (1-\eta_0/2))}$ to ${B(0, (1-\eta_0/4))}$. From (3) we see that ${f_0}$ maps ${B(0,1-3\eta_0/4)}$ to ${B(0,1-\eta_0/2)}$, and so if we set ${f_1: B(0,1-\eta_0) \rightarrow B(0,1-\eta_0/4)}$ to be the function ${f_1 := \psi_1^{-1} \circ f_0 \circ \psi_1}$, then ${f_1}$ is a holomorphic function obeying (4). Expanding (4) in terms of ${\hat f_0}$ and ${\hat \psi_1}$ as before, and also writing ${f_1(z) = \lambda z + \hat f_1(z)}$, we have

$\displaystyle \lambda z + \lambda \hat \psi_1(z) + \hat f_0( z + \hat \psi_1(z) ) = \lambda z + \hat f_1(z) + \hat \psi_1(\lambda z + \hat f_1(z))$

for ${z \in B(0, 1-\eta_0)}$, which by (5) simplifies to

$\displaystyle \hat f_1(z) = \hat f_0( z + \hat \psi_1(z) ) - \hat f_0(z) + \hat \psi_1(\lambda z) - \hat \psi_1(\lambda z + \hat f_1(z)).$

From (6), the fundamental theorem of calculus, and the smallness of ${\delta_0}$ we have

$\displaystyle |\hat \psi_1(\lambda z) - \hat \psi_1(\lambda z + \hat f_1(z))| \leq \frac{1}{2} |\hat f_1(z)|$

and thus

$\displaystyle |\hat f_1(z)| \leq 2 |\hat f_0( z + \hat \psi_1(z) ) - \hat f_0(z)|.$

From (3) and the Cauchy integral formula we have ${\hat f'_0(z) = O( \delta_0 \eta_0^{-O(1)})}$ on (say) ${B(0,1-\eta_0/4)}$, and so from (6) and the fundamental theorem of calculus we conclude that

$\displaystyle |\hat f_1(z)| \ll \delta_0^2 \eta_0^{-O(1)}$

on ${B(0,1-\eta_0)}$, and the claim follows. $\Box$

If we set ${\eta_0 := 1/2}$, ${f_0 := f}$, and ${\delta_0>0}$ to be sufficiently small, then (since ${f(z)-\lambda z}$ vanishes to second order at the origin), the hypotheses of this lemma will be obeyed for some sufficiently small ${r_0}$. Iterating the lemma (and halving ${\eta_0}$ repeatedly), we can then find sequences ${\eta_n, \delta_n, r_n > 0}$, injective holomorphic functions ${\psi_n: B(0,r_n) \rightarrow B(0,r_{n-1})}$ and holomorphic functions ${f_n: B(0,r_n) \rightarrow {\bf C}}$ such that one has the recursive identities and estimates

$\displaystyle \eta_n = \eta_{n-1} / 2$

$\displaystyle r_n = (1 - \eta_{n-1}) r_{n-1}$

$\displaystyle \delta_n = O( \delta_{n-1}^2 \eta_{n-1}^{-O(1)} )$

$\displaystyle |\psi_n(z) - z| \ll \delta_{n-1} \eta_{n-1}^{-O(1)} r_n$

$\displaystyle |f_n(z) - \lambda z| \leq \delta_n r_n$

$\displaystyle f_{n-1}( \psi_n(z) ) = \psi_n(f_n(z))$

for all ${n \geq 1}$ and ${z \in B(0,r_n)}$. By construction, ${r_n}$ decreases to a positive radius ${r_\infty}$ that is a constant multiple of ${r_0}$, while (for ${\delta_0}$ small enough) ${\delta_n}$ converges double-exponentially to zero, so in particular ${f_n(z)}$ converges uniformly to ${\lambda z}$ on ${B(0,r_\infty)}$. Also, ${\psi_n}$ is close enough to the identity, the compositions ${\Psi_n := \psi_1 \circ \dots \circ \psi_n}$ are uniformly convergent on ${B(0,r_\infty/2)}$ with ${\Psi_n(0)=0}$ and ${\Psi'_n(0)=1}$. From this we have

$\displaystyle f( \Psi_n(z) ) = \Psi_n(f_n(z))$

on ${B(0,r_\infty/4)}$, and on taking limits using Morera’s theorem we obtain a holomorphic function ${\Psi}$ defined near ${0}$ with ${\Psi(0)=0}$, ${\Psi'(0)=1}$, and

$\displaystyle f( \Psi(z) ) = \Psi(\lambda z),$

obtaining the required linearisation.

Remark 5 The idea of using a Newton-type method to obtain error terms that decay double-exponentially, and can therefore absorb exponential losses in the iteration, also occurs in KAM theory and in Nash-Moser iteration, presumably due to Siegel’s influence on Moser. (I discuss Nash-Moser iteration in this note that I wrote back in 2006.)

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if ${U: H \rightarrow H}$ is a unitary operator on a Hilbert space ${H}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \pi_{H^U} v$

in the strong topology, where ${H^U := \{ w \in H: Uw = w \}}$ is the ${U}$-invariant subspace of ${H}$, and ${\pi_{H^U}}$ is the orthogonal projection to ${H^U}$. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if ${G}$ is a countable amenable group acting on a Hilbert space ${H}$ by unitary transformations ${T^g: H \rightarrow H}$ for ${g \in G}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \mathop{\bf E}_{g \in \Phi_N} T^g v = \pi_{H^G} v \ \ \ \ \ (1)$

for any Folner sequence ${\Phi_N}$ of ${G}$, where ${H^G := \{ w \in H: T^g w = w \hbox{ for all }g \in G \}}$ is the ${G}$-invariant subspace, and ${\mathop{\bf E}_{a \in A} f(a) := \frac{1}{|A|} \sum_{a \in A} f(a)}$ is the average of ${f}$ on ${A}$. Thus one can interpret ${\pi_{H^G} v}$ as a certain average of elements of the orbit ${Gv := \{ T^g v: g \in G \}}$ of ${v}$.

In a previous blog post, I noted a variant of this ergodic theorem (due to Alaoglu and Birkhoff) that holds even when the group ${G}$ is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem) Let ${G}$ be an arbitrary group acting unitarily on a Hilbert space ${H}$, and let ${v}$ be a vector in ${H}$. Then ${\pi_{H^G} v}$ is the element in the closed convex hull of ${Gv := \{ T^g v: g \in G \}}$ of minimal norm, and is also the unique element of ${H^G}$ in this closed convex hull.

I recently stumbled upon a different way to think about this theorem, in the additive case ${G = (G,+)}$ when ${G}$ is abelian, which has a closer resemblance to the classical mean ergodic theorem. Given an arbitrary additive group ${G = (G,+)}$ (not necessarily discrete, or countable), let ${{\mathcal F}}$ denote the collection of finite non-empty multisets in ${G}$ – that is to say, unordered collections ${\{a_1,\dots,a_n\}}$ of elements ${a_1,\dots,a_n}$ of ${G}$, not necessarily distinct, for some positive integer ${n}$. Given two multisets ${A = \{a_1,\dots,a_n\}}$, ${B = \{b_1,\dots,b_m\}}$ in ${{\mathcal F}}$, we can form the sum set ${A + B := \{ a_i + b_j: 1 \leq i \leq n, 1 \leq j \leq m \}}$. Note that the sum set ${A+B}$ can contain multiplicity even when ${A, B}$ do not; for instance, ${\{ 1,2\} + \{1,2\} = \{2,3,3,4\}}$. Given a multiset ${A = \{a_1,\dots,a_n\}}$ in ${{\mathcal F}}$, and a function ${f: G \rightarrow H}$ from ${G}$ to a vector space ${H}$, we define the average ${\mathop{\bf E}_{a \in A} f(a)}$ as

$\displaystyle \mathop{\bf E}_{a \in A} f(a) = \frac{1}{n} \sum_{j=1}^n f(a_j).$

Note that the multiplicity function of the set ${A}$ affects the average; for instance, we have ${\mathop{\bf E}_{a \in \{1,2\}} a = \frac{3}{2}}$, but ${\mathop{\bf E}_{a \in \{1,2,2\}} a = \frac{5}{3}}$.

We can define a directed set on ${{\mathcal F}}$ as follows: given two multisets ${A,B \in {\mathcal F}}$, we write ${A \geq B}$ if we have ${A = B+C}$ for some ${C \in {\mathcal F}}$. Thus for instance we have ${\{ 1, 2, 2, 3\} \geq \{1,2\}}$. It is easy to verify that this operation is transitive and reflexive, and is directed because any two elements ${A,B}$ of ${{\mathcal F}}$ have a common upper bound, namely ${A+B}$. (This is where we need ${G}$ to be abelian.) The notion of convergence along a net, now allows us to define the notion of convergence along ${{\mathcal F}}$; given a family ${x_A}$ of points in a topological space ${X}$ indexed by elements ${A}$ of ${{\mathcal F}}$, and a point ${x}$ in ${X}$, we say that ${x_A}$ converges to ${x}$ along ${{\mathcal F}}$ if, for every open neighbourhood ${U}$ of ${x}$ in ${X}$, one has ${x_A \in U}$ for sufficiently large ${A}$, that is to say there exists ${B \in {\mathcal F}}$ such that ${x_A \in U}$ for all ${A \geq B}$. If the topological space ${V}$ is Hausdorff, then the limit ${x}$ is unique (if it exists), and we then write

$\displaystyle x = \lim_{A \rightarrow G} x_A.$

When ${x_A}$ takes values in the reals, one can also define the limit superior or limit inferior along such nets in the obvious fashion.

We can then give an alternate formulation of the abstract ergodic theorem in the abelian case:

Theorem 2 (Abelian abstract ergodic theorem) Let ${G = (G,+)}$ be an arbitrary additive group acting unitarily on a Hilbert space ${H}$, and let ${v}$ be a vector in ${H}$. Then we have

$\displaystyle \pi_{H^G} v = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v$

in the strong topology of ${H}$.

Proof: Suppose that ${A \geq B}$, so that ${A=B+C}$ for some ${C \in {\mathcal F}}$, then

$\displaystyle \mathop{\bf E}_{a \in A} T^a v = \mathop{\bf E}_{c \in C} T^c ( \mathop{\bf E}_{b \in B} T^b v )$

so by unitarity and the triangle inequality we have

$\displaystyle \| \mathop{\bf E}_{a \in A} T^a v \|_H \leq \| \mathop{\bf E}_{b \in B} T^b v \|_H,$

thus ${\| \mathop{\bf E}_{a \in A} T^a v \|_H^2}$ is monotone non-increasing in ${A}$. Since this quantity is bounded between ${0}$ and ${\|v\|_H}$, we conclude that the limit ${\lim_{A \rightarrow G} \| \mathop{\bf E}_{a \in A} T^a v \|_H^2}$ exists. Thus, for any ${\varepsilon > 0}$, we have for sufficiently large ${A}$ that

$\displaystyle \| \mathop{\bf E}_{b \in B} T^b v \|_H^2 \geq \| \mathop{\bf E}_{a \in A} T^a v \|_H^2 - \varepsilon$

for all ${B \geq A}$. In particular, for any ${g \in G}$, we have

$\displaystyle \| \mathop{\bf E}_{b \in A + \{0,g\}} T^b v \|_H^2 \geq \| \mathop{\bf E}_{a \in A} T^a v \|_H^2 - \varepsilon.$

We can write

$\displaystyle \mathop{\bf E}_{b \in A + \{0,g\}} T^b v = \frac{1}{2} \mathop{\bf E}_{a \in A} T^a v + \frac{1}{2} T^g \mathop{\bf E}_{a \in A} T^a v$

and so from the parallelogram law and unitarity we have

$\displaystyle \| \mathop{\bf E}_{a \in A} T^a v - T^g \mathop{\bf E}_{a \in A} T^a v \|_H^2 \leq 4 \varepsilon$

for all ${g \in G}$, and hence by the triangle inequality (averaging ${g}$ over a finite multiset ${C}$)

$\displaystyle \| \mathop{\bf E}_{a \in A} T^a v - \mathop{\bf E}_{b \in A+C} T^b v \|_H^2 \leq 4 \varepsilon$

for any ${C \in {\mathcal F}}$. This shows that ${\mathop{\bf E}_{a \in A} T^a v}$ is a Cauchy sequence in ${H}$ (in the strong topology), and hence (by the completeness of ${H}$) tends to a limit. Shifting ${A}$ by a group element ${g}$, we have

$\displaystyle \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A + \{g\}} T^a v = T^g \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v$

and hence ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v}$ is invariant under shifts, and thus lies in ${H^G}$. On the other hand, for any ${w \in H^G}$ and ${A \in {\mathcal F}}$, we have

$\displaystyle \langle \mathop{\bf E}_{a \in A} T^a v, w \rangle_H = \mathop{\bf E}_{a \in A} \langle v, T^{-a} w \rangle_H = \langle v, w \rangle_H$

and thus on taking strong limits

$\displaystyle \langle \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v, w \rangle_H = \langle v, w \rangle_H$

and so ${v - \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v}$ is orthogonal to ${H^G}$. Combining these two facts we see that ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v}$ is equal to ${\pi_{H^G} v}$ as claimed. $\Box$

To relate this result to the classical ergodic theorem, we observe

Lemma 3 Let ${G}$ be a countable additive group, with a F{\o}lner sequence ${\Phi_n}$, and let ${f_g}$ be a bounded sequence in a normed vector space indexed by ${G}$. If ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} f_a}$ exists, then ${\lim_{n \rightarrow \infty} \mathop{\bf E}_{a \in \Phi_n} f_a}$ exists, and the two limits are equal.

Proof: From the F{\o}lner property, we see that for any ${A}$ and any ${\varepsilon>0}$, the averages ${\mathop{\bf E}_{a \in \Phi_n} f_a}$ and ${\mathop{\bf E}_{a \in A+\Phi_n} f_a}$ differ by at most ${\varepsilon}$ in norm if ${n}$ is sufficiently large depending on ${A}$, ${\varepsilon}$ (and the ${f_a}$). On the other hand, by the existence of the limit ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} f_a}$, the averages ${\mathop{\bf E}_{a \in A} f_a}$ and ${\mathop{\bf E}_{a \in A + \Phi_n} f_a}$ differ by at most ${\varepsilon}$ in norm if ${A}$ is sufficiently large depending on ${\varepsilon}$ (regardless of how large ${n}$ is). The claim follows. $\Box$

It turns out that this approach can also be used as an alternate way to construct the GowersHost-Kra seminorms in ergodic theory, which has the feature that it does not explicitly require any amenability on the group ${G}$ (or separability on the underlying measure space), though, as pointed out to me in comments, even uncountable abelian groups are amenable in the sense of possessing an invariant mean, even if they do not have a F{\o}lner sequence.

Given an arbitrary additive group ${G}$, define a ${G}$-system ${({\mathrm X}, T)}$ to be a probability space ${{\mathrm X} = (X, {\mathcal X}, \mu)}$ (not necessarily separable or standard Borel), together with a collection ${T^g: X \rightarrow X}$ of invertible, measure-preserving maps, such that ${T^0}$ is the identity and ${T^g T^h = T^{g+h}}$ (modulo null sets) for all ${g,h \in G}$. This then gives isomorphisms ${T^g: L^p({\mathrm X}) \rightarrow L^p({\mathrm X})}$ for ${1 \leq p \leq \infty}$ by setting ${T^g f(x) := f(T^{-g} x)}$. From the above abstract ergodic theorem, we see that

$\displaystyle {\mathbf E}( f | {\mathcal X}^G ) = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^g f$

in the strong topology of ${L^2({\mathrm X})}$ for any ${f \in L^2({\mathrm X})}$, where ${{\mathcal X}^G}$ is the collection of measurable sets ${E}$ that are essentially ${G}$-invariant in the sense that ${T^g E = E}$ modulo null sets for all ${g \in G}$, and ${{\mathbf E}(f|{\mathcal X}^G)}$ is the conditional expectation of ${f}$ with respect to ${{\mathcal X}^G}$.

In a similar spirit, we have

Theorem 4 (Convergence of Gowers-Host-Kra seminorms) Let ${({\mathrm X},T)}$ be a ${G}$-system for some additive group ${G}$. Let ${d}$ be a natural number, and for every ${\omega \in\{0,1\}^d}$, let ${f_\omega \in L^{2^d}({\mathrm X})}$, which for simplicity we take to be real-valued. Then the expression

$\displaystyle \langle (f_\omega)_{\omega \in \{0,1\}^d} \rangle_{U^d({\mathrm X})} := \lim_{A_1,\dots,A_d \rightarrow G}$

$\displaystyle \mathop{\bf E}_{h_1 \in A_1-A_1,\dots,h_d \in A_d-A_d} \int_X \prod_{\omega \in \{0,1\}^d} T^{\omega_1 h_1 + \dots + \omega_d h_d} f_\omega\ d\mu$

converges, where we write ${\omega = (\omega_1,\dots,\omega_d)}$, and we are using the product direct set on ${{\mathcal F}^d}$ to define the convergence ${A_1,\dots,A_d \rightarrow G}$. In particular, for ${f \in L^{2^d}({\mathrm X})}$, the limit

$\displaystyle \| f \|_{U^d({\mathrm X})}^{2^d} = \lim_{A_1,\dots,A_d \rightarrow G}$

$\displaystyle \mathop{\bf E}_{h_1 \in A_1-A_1,\dots,h_d \in A_d-A_d} \int_X \prod_{\omega \in \{0,1\}^d} T^{\omega_1 h_1 + \dots + \omega_d h_d} f\ d\mu$

converges.

We prove this theorem below the fold. It implies a number of other known descriptions of the Gowers-Host-Kra seminorms ${\|f\|_{U^d({\mathrm X})}}$, for instance that

$\displaystyle \| f \|_{U^d({\mathrm X})}^{2^d} = \lim_{A \rightarrow G} \mathop{\bf E}_{h \in A-A} \| f T^h f \|_{U^{d-1}({\mathrm X})}^{2^{d-1}}$

for ${d > 1}$, while from the ergodic theorem we have

$\displaystyle \| f \|_{U^1({\mathrm X})} = \| {\mathbf E}( f | {\mathcal X}^G ) \|_{L^2({\mathrm X})}.$

This definition also manifestly demonstrates the cube symmetries of the Host-Kra measures ${\mu^{[d]}}$ on ${X^{\{0,1\}^d}}$, defined via duality by requiring that

$\displaystyle \langle (f_\omega)_{\omega \in \{0,1\}^d} \rangle_{U^d({\mathrm X})} = \int_{X^{\{0,1\}^d}} \bigotimes_{\omega \in \{0,1\}^d} f_\omega\ d\mu^{[d]}.$

In a subsequent blog post I hope to present a more detailed study of the ${U^2}$ norm and its relationship with eigenfunctions and the Kronecker factor, without assuming any amenability on ${G}$ or any separability or topological structure on ${{\mathrm X}}$.

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices ${M_n}$ of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power ${n^{-A}}$ of ${n}$. This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices ${M_n}$ (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on ${M_n}$) a result of the form

$\displaystyle {\bf P} (\lambda_{i+1}(M_n) - \lambda_i(M_n) \leq n^{-A} ) \leq n^{-B}$

for any ${B}$ and ${i}$, if ${A}$ is sufficiently large depending on ${B}$ (in a linear fashion), and ${n}$ is sufficiently large depending on ${B}$. The point here is that ${B}$ can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)

In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of ${M_n}$ have unit variance, so that the eigenvalues of ${M_n}$ are ${O(\sqrt{n})}$ with high probability), giving the bound

$\displaystyle {\bf P} (\lambda_{i+1}(M_n) - \lambda_i(M_n) \leq \delta / \sqrt{n} ) \ll \delta \ \ \ \ \ (1)$

for ${\delta \geq n^{-O(1)}}$ (one also has good results of this type for smaller values of ${\delta}$). This is only optimal in the regime ${\delta \sim 1}$; we expect to establish some eigenvalue repulsion, improving the RHS to ${\delta^2}$ for real matrices and ${\delta^3}$ for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form

$\displaystyle {\bf P} (\lambda_{i+k}(M_n) - \lambda_i(M_n) \leq \delta / \sqrt{n} ) \ll \delta^{ck^2}$

for any fixed ${k \geq 1}$ and some absolute constant ${c>0}$ (which we can asymptotically make to be ${1/3}$ for large ${k}$, though it ought to be as large as ${1}$), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.

In the case of Erdös-Renyi graphs, we don’t have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn’t quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of ${n^{o(1)}}$ on the RHS.

As a sample applications of the eigenvalue separation results, we can now obtain some information about eigenvectors; for instance, we can show that the components of the eigenvectors all have magnitude at least ${n^{-A}}$ for some ${A}$ with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)

We have seen in previous notes that the operation of forming a Dirichlet series

$\displaystyle {\mathcal D} f(n) := \sum_n \frac{f(n)}{n^s}$

or twisted Dirichlet series

$\displaystyle {\mathcal D} \chi f(n) := \sum_n \frac{f(n) \chi(n)}{n^s}$

is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions ${n \mapsto \frac{1}{n^s}}$ and ${n \mapsto \frac{\chi(n)}{n^s}}$ are multiplicative characters.

Similarly, it turns out that the operation of forming an additive Fourier series

$\displaystyle \hat f(\theta) := \sum_n f(n) e(-n \theta),$

where ${\theta}$ lies on the (additive) unit circle ${{\bf R}/{\bf Z}}$ and ${e(\theta) := e^{2\pi i \theta}}$ is the standard additive character, is an incredibly useful tool for additive number theory, particularly when studying additive problems involving three or more variables taking values in sets such as the primes; the deployment of this tool is generally known as the Hardy-Littlewood circle method. (In the analytic number theory literature, the minus sign in the phase ${e(-n\theta)}$ is traditionally omitted, and what is denoted by ${\hat f(\theta)}$ here would be referred to instead by ${S_f(-\theta)}$, ${S(f;-\theta)}$ or just ${S(-\theta)}$.) We list some of the most classical problems in this area:

• (Even Goldbach conjecture) Is it true that every even natural number ${N}$ greater than two can be expressed as the sum ${p_1+p_2}$ of two primes?
• (Odd Goldbach conjecture) Is it true that every odd natural number ${N}$ greater than five can be expressed as the sum ${p_1+p_2+p_3}$ of three primes?
• (Waring problem) For each natural number ${k}$, what is the least natural number ${g(k)}$ such that every natural number ${N}$ can be expressed as the sum of ${g(k)}$ or fewer ${k^{th}}$ powers?
• (Asymptotic Waring problem) For each natural number ${k}$, what is the least natural number ${G(k)}$ such that every sufficiently large natural number ${N}$ can be expressed as the sum of ${G(k)}$ or fewer ${k^{th}}$ powers?
• (Partition function problem) For any natural number ${N}$, let ${p(N)}$ denote the number of representations of ${N}$ of the form ${N = n_1 + \dots + n_k}$ where ${k}$ and ${n_1 \geq \dots \geq n_k}$ are natural numbers. What is the asymptotic behaviour of ${p(N)}$ as ${N \rightarrow \infty}$?

The Waring problem and its asymptotic version will not be discussed further here, save to note that the Vinogradov mean value theorem (Theorem 13 from Notes 5) and its variants are particularly useful for getting good bounds on ${G(k)}$; see for instance the ICM article of Wooley for recent progress on these problems. Similarly, the partition function problem was the original motivation of Hardy and Littlewood in introducing the circle method, but we will not discuss it further here; see e.g. Chapter 20 of Iwaniec-Kowalski for a treatment.

Instead, we will focus our attention on the odd Goldbach conjecture as our model problem. (The even Goldbach conjecture, which involves only two variables instead of three, is unfortunately not amenable to a circle method approach for a variety of reasons, unless the statement is replaced with something weaker, such as an averaged statement; see this previous blog post for further discussion. On the other hand, the methods here can obtain weaker versions of the even Goldbach conjecture, such as showing that “almost all” even numbers are the sum of two primes; see Exercise 34 below.) In particular, we will establish the following celebrated theorem of Vinogradov:

Theorem 1 (Vinogradov’s theorem) Every sufficiently large odd number ${N}$ is expressible as the sum of three primes.

Recently, the restriction that ${n}$ be sufficiently large was replaced by Helfgott with ${N > 5}$, thus establishing the odd Goldbach conjecture in full. This argument followed the same basic approach as Vinogradov (based on the circle method), but with various estimates replaced by “log-free” versions (analogous to the log-free zero-density theorems in Notes 7), combined with careful numerical optimisation of constants and also some numerical work on the even Goldbach problem and on the generalised Riemann hypothesis. We refer the reader to Helfgott’s text for details.

We will in fact show the more precise statement:

Theorem 2 (Quantitative Vinogradov theorem) Let ${N \geq 2}$ be an natural number. Then

$\displaystyle \sum_{a,b,c: a+b+c=N} \Lambda(a) \Lambda(b) \Lambda(c) = G_3(N) \frac{N^2}{2} + O_A( N^2 \log^{-A} N )$

for any ${A>0}$, where

$\displaystyle G_3(N) = \prod_{p|N} (1-\frac{1}{(p-1)^2}) \times \prod_{p \not | N} (1 + \frac{1}{(p-1)^3}). \ \ \ \ \ (1)$

The implied constants are ineffective.

We dropped the hypothesis that ${N}$ is odd in Theorem 2, but note that ${G_3(N)}$ vanishes when ${N}$ is even. For odd ${N}$, we have

$\displaystyle 1 \ll G_3(N) \ll 1.$

Exercise 3 Show that Theorem 2 implies Theorem 1.

Unfortunately, due to the ineffectivity of the constants in Theorem 2 (a consequence of the reliance on the Siegel-Walfisz theorem in the proof of that theorem), one cannot quantify explicitly what “sufficiently large” means in Theorem 1 directly from Theorem 2. However, there is a modification of this theorem which gives effective bounds; see Exercise 32 below.

Exercise 4 Obtain a heuristic derivation of the main term ${G_3(N) \frac{N^2}{2}}$ using the modified Cramér model (Section 1 of Supplement 4).

To prove Theorem 2, we consider the more general problem of estimating sums of the form

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c)$

for various integers ${N}$ and functions ${f,g,h: {\bf Z} \rightarrow {\bf C}}$, which we will take to be finitely supported to avoid issues of convergence.

Suppose that ${f,g,h}$ are supported on ${\{1,\dots,N\}}$; for simplicity, let us first assume the pointwise bound ${|f(n)|, |g(n)|, |h(n)| \ll 1}$ for all ${n}$. (This simple case will not cover the case in Theorem 2, when ${f,g,h}$ are truncated versions of the von Mangoldt function ${\Lambda}$, but will serve as a warmup to that case.) Then we have the trivial upper bound

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) \ll N^2. \ \ \ \ \ (2)$

A basic observation is that this upper bound is attainable if ${f,g,h}$ all “pretend” to behave like the same additive character ${n \mapsto e(\theta n)}$ for some ${\theta \in {\bf R}/{\bf Z}}$. For instance, if ${f(n)=g(n)=h(n) = e(\theta n) 1_{n \leq N}}$, then we have ${f(a)g(b)h(c) = e(\theta N)}$ when ${a+b+c=N}$, and then it is not difficult to show that

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) = (\frac{1}{2}+o(1)) e(\theta N) N^2$

as ${N \rightarrow \infty}$.

The key to the success of the circle method lies in the converse of the above statement: the only way that the trivial upper bound (2) comes close to being sharp is when ${f,g,h}$ all correlate with the same character ${n \mapsto e(\theta n)}$, or in other words ${\hat f(\theta), \hat g(\theta), \hat h(\theta)}$ are simultaneously large. This converse is largely captured by the following two identities:

Exercise 5 Let ${f,g,h: {\bf Z} \rightarrow {\bf C}}$ be finitely supported functions. Then for any natural number ${N}$, show that

$\displaystyle \sum_{a,b,c: a+b+c=N} f(a) g(b) h(c) = \int_{{\bf R}/{\bf Z}} \hat f(\theta) \hat g(\theta) \hat h(\theta) e(\theta N)\ d\theta \ \ \ \ \ (3)$

and

$\displaystyle \sum_n |f(n)|^2 = \int_{{\bf R}/{\bf Z}} |\hat f(\theta)|^2\ d\theta.$

The traditional approach to using the circle method to compute sums such as ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$ proceeds by invoking (3) to express this sum as an integral over the unit circle, then dividing the unit circle into “major arcs” where ${\hat f(\theta), \hat g(\theta),\hat h(\theta)}$ are large but computable with high precision, and “minor arcs” where one has estimates to ensure that ${\hat f(\theta), \hat g(\theta),\hat h(\theta)}$ are small in both ${L^\infty}$ and ${L^2}$ senses. For functions ${f,g,h}$ of number-theoretic significance, such as truncated von Mangoldt functions, the “major arcs” typically consist of those ${\theta}$ that are close to a rational number ${\frac{a}{q}}$ with ${q}$ not too large, and the “minor arcs” consist of the remaining portions of the circle. One then obtains lower bounds on the contributions of the major arcs, and upper bounds on the contribution of the minor arcs, in order to get good lower bounds on ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$.

This traditional approach is covered in many places, such as this text of Vaughan. We will emphasise in this set of notes a slightly different perspective on the circle method, coming from recent developments in additive combinatorics; this approach does not quite give the sharpest quantitative estimates, but it allows for easier generalisation to more combinatorial contexts, for instance when replacing the primes by dense subsets of the primes, or replacing the equation ${a+b+c=N}$ with some other equation or system of equations.

From Exercise 5 and Hölder’s inequality, we immediately obtain

Corollary 6 Let ${f,g,h: {\bf Z} \rightarrow {\bf C}}$ be finitely supported functions. Then for any natural number ${N}$, we have

$\displaystyle |\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)| \leq (\sum_n |f(n)|^2)^{1/2} (\sum_n |g(n)|^2)^{1/2}$

$\displaystyle \times \sup_\theta |\sum_n h(n) e(n\theta)|.$

Similarly for permutations of the ${f,g,h}$.

In the case when ${f,g,h}$ are supported on ${[1,N]}$ and bounded by ${O(1)}$, this corollary tells us that we have ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$ is ${o(N^2)}$ whenever one has ${\sum_n h(n) e(n\theta) = o(N)}$ uniformly in ${\theta}$, and similarly for permutations of ${f,g,h}$. From this and the triangle inequality, we obtain the following conclusion: if ${f}$ is supported on ${[1,N]}$ and bounded by ${O(1)}$, and ${f}$ is Fourier-approximated by another function ${g}$ supported on ${[1,N]}$ and bounded by ${O(1)}$ in the sense that

$\displaystyle \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)$

uniformly in ${\theta}$, then we have

$\displaystyle \sum_{a,b,c: a+b+c=N} f(a) f(b) f(c) = \sum_{a,b,c: a+b+c=N} g(a) g(b) g(c) + o(N^2). \ \ \ \ \ (4)$

Thus, one possible strategy for estimating the sum ${\sum_{a,b,c: a+b+c=N} f(a) f(b) f(c)}$ is, one can effectively replace (or “model”) ${f}$ by a simpler function ${g}$ which Fourier-approximates ${g}$ in the sense that the exponential sums ${\sum_n f(n) e(n\theta), \sum_n g(n) e(n\theta)}$ agree up to error ${o(N)}$. For instance:

Exercise 7 Let ${N}$ be a natural number, and let ${A}$ be a random subset of ${\{1,\dots,N\}}$, chosen so that each ${n \in \{1,\dots,N\}}$ has an independent probability of ${1/2}$ of lying in ${A}$.

• (i) If ${f := 1_A}$ and ${g := \frac{1}{2} 1_{[1,N]}}$, show that with probability ${1-o(1)}$ as ${N \rightarrow \infty}$, one has ${\sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)}$ uniformly in ${\theta}$. (Hint: for any fixed ${\theta}$, this can be accomplished with quite a good probability (e.g. ${1-o(N^{-2})}$) using a concentration of measure inequality, such as Hoeffding’s inequality. To obtain the uniformity in ${\theta}$, round ${\theta}$ to the nearest multiple of (say) ${1/N^2}$ and apply the union bound).
• (ii) Show that with probability ${1-o(1)}$, one has ${(\frac{1}{16}+o(1))N^2}$ representations of the form ${N=a+b+c}$ with ${a,b,c \in A}$ (with ${(a,b,c)}$ treated as an ordered triple, rather than an unordered one).

In the case when ${f}$ is something like the truncated von Mangoldt function ${\Lambda(n) 1_{n \leq N}}$, the quantity ${\sum_n |f(n)|^2}$ is of size ${O( N \log N)}$ rather than ${O( N )}$. This costs us a logarithmic factor in the above analysis, however we can still conclude that we have the approximation (4) whenever ${g}$ is another sequence with ${\sum_n |g(n)|^2 \ll N \log N}$ such that one has the improved Fourier approximation

$\displaystyle \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(\frac{N}{\log N}) \ \ \ \ \ (5)$

uniformly in ${\theta}$. (Later on we will obtain a “log-free” version of this implication in which one does not need to gain a factor of ${\frac{1}{\log N}}$ in the error term.)

This suggests a strategy for proving Vinogradov’s theorem: find an approximant ${g}$ to some suitable truncation ${f}$ of the von Mangoldt function (e.g. ${f(n) = \Lambda(n) 1_{n \leq N}}$ or ${f(n) = \Lambda(n) \eta(n/N)}$) which obeys the Fourier approximation property (5), and such that the expression ${\sum_{a+b+c=N} g(a) g(b) g(c)}$ is easily computable. It turns out that there are a number of good options for such an approximant ${g}$. One of the quickest ways to obtain such an approximation (which is used in Chapter 19 of Iwaniec and Kowalski) is to start with the standard identity ${\Lambda = -\mu L * 1}$, that is to say

$\displaystyle \Lambda(n) = - \sum_{d|n} \mu(d) \log d,$

and obtain an approximation by truncating ${d}$ to be less than some threshold ${R}$ (which, in practice, would be a small power of ${N}$):

$\displaystyle \Lambda(n) \approx - \sum_{d \leq R: d|n} \mu(d) \log d. \ \ \ \ \ (6)$

Thus, for instance, if ${f(n) = \Lambda(n) 1_{n \leq N}}$, the approximant ${g}$ would be taken to be

$\displaystyle g(n) := - \sum_{d \leq R: d|n} \mu(d) \log d 1_{n \leq N}.$

One could also use the slightly smoother approximation

$\displaystyle \Lambda(n) \approx \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} \ \ \ \ \ (7)$

in which case we would take

$\displaystyle g(n) := \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} 1_{n \leq N}.$

The function ${g}$ is somewhat similar to the continuous Selberg sieve weights studied in Notes 4, with the main difference being that we did not square the divisor sum as we will not need to take ${g}$ to be non-negative. As long as ${z}$ is not too large, one can use some sieve-like computations to compute expressions like ${\sum_{a+b+c=N} g(a)g(b)g(c)}$ quite accurately. The approximation (5) can be justified by using a nice estimate of Davenport that exemplifies the Mobius pseudorandomness heuristic from Supplement 4:

Theorem 8 (Davenport’s estimate) For any ${A>0}$ and ${x \geq 2}$, we have

$\displaystyle \sum_{n \leq x} \mu(n) e(\theta n) \ll_A x \log^{-A} x$

uniformly for all ${\theta \in {\bf R}/{\bf Z}}$. The implied constants are ineffective.

This estimate will be proven by splitting into two cases. In the “major arc” case when ${\theta}$ is close to a rational ${a/q}$ with ${q}$ small (of size ${O(\log^{O(1)} x)}$ or so), this estimate will be a consequence of the Siegel-Walfisz theorem ( from Notes 2); it is the application of this theorem that is responsible for the ineffective constants. In the remaining “minor arc” case, one proceeds by using a combinatorial identity (such as Vaughan’s identity) to express the sum ${\sum_{n \leq x} \mu(n) e(\theta n)}$ in terms of bilinear sums of the form ${\sum_n \sum_m a_n b_m e(\theta nm)}$, and use the Cauchy-Schwarz inequality and the minor arc nature of ${\theta}$ to obtain a gain in this case. This will all be done below the fold. We will also use (a rigorous version of) the approximation (6) (or (7)) to establish Vinogradov’s theorem.

A somewhat different looking approximation for the von Mangoldt function that also turns out to be quite useful is

$\displaystyle \Lambda(n) \approx \sum_{q \leq Q} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} ) \ \ \ \ \ (8)$

for some ${Q}$ that is not too large compared to ${N}$. The methods used to establish Theorem 8 can also establish a Fourier approximation that makes (8) precise, and which can yield an alternate proof of Vinogradov’s theorem; this will be done below the fold.

The approximation (8) can be written in a way that makes it more similar to (7):

Exercise 9 Show that the right-hand side of (8) can be rewritten as

$\displaystyle \sum_{d \leq Q: d|n} \mu(d) \rho_d$

where

$\displaystyle \rho_d := \frac{d}{\phi(d)} \sum_{m \leq Q/d: (m,d)=1} \frac{\mu^2(m)}{\phi(m)}.$

Then, show the inequalities

$\displaystyle \sum_{m \leq Q/d} \frac{\mu^2(m)}{\phi(m)} \leq \rho_d \leq \sum_{m \leq Q} \frac{\mu^2(m)}{\phi(m)}$

and conclude that

$\displaystyle \log \frac{Q}{d} - O(1) \leq \rho_d \leq \log Q + O(1).$

(Hint: for the latter estimate, use Theorem 27 of Notes 1.)

The coefficients ${\rho_d}$ in the above exercise are quite similar to optimised Selberg sieve coefficients (see Section 2 of Notes 4).

Another approximation to ${\Lambda}$, related to the modified Cramér random model (see Model 10 of Supplement 4) is

$\displaystyle \Lambda(n) \approx \frac{W}{\phi(W)} 1_{(n,W)=1} \ \ \ \ \ (9)$

where ${W := \prod_{p \leq w} p}$ and ${w}$ is a slowly growing function of ${N}$ (e.g. ${w = \log\log N}$); a closely related approximation is

$\displaystyle \frac{\phi(W)}{W} \Lambda(Wn+b) \approx 1 \ \ \ \ \ (10)$

for ${W,w}$ as above and ${1 \leq b \leq W}$ coprime to ${W}$. These approximations (closely related to a device known as the “${W}$-trick”) are not as quantitatively accurate as the previous approximations, but can still suffice to establish Vinogradov’s theorem, and also to count many other linear patterns in the primes or subsets of the primes (particularly if one injects some additional tools from additive combinatorics, and specifically the inverse conjecture for the Gowers uniformity norms); see this paper of Ben Green and myself for more discussion (and this more recent paper of Shao for an analysis of this approach in the context of Vinogradov-type theorems). The following exercise expresses the approximation (9) in a form similar to the previous approximation (8):

Exercise 10 With ${W}$ as above, show that

$\displaystyle \frac{W}{\phi(W)} 1_{(n,W)=1} = \sum_{q|W} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} )$

for all natural numbers ${n}$.

Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper “An averaged form of Chowla’s conjecture“. This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that

$\displaystyle \sum_{n \leq X} \lambda(n+h_1) \dots \lambda(n+h_k) = o(X)$

as ${X \rightarrow \infty}$ for any distinct natural numbers ${h_1,\dots,h_k}$, where ${\lambda}$ denotes the Liouville function. (One could also replace the Liouville function here by the Möbius function ${\mu}$ and obtain a morally equivalent conjecture.) This conjecture remains open for any ${k \geq 2}$; for instance the assertion

$\displaystyle \sum_{n \leq X} \lambda(n) \lambda(n+2) = o(X)$

is a variant of the twin prime conjecture (though possibly a tiny bit easier to prove), and is subject to the notorious parity barrier (as discussed in this previous post).

Our main result asserts, roughly speaking, that Chowla’s conjecture can be established unconditionally provided one has non-trivial averaging in the ${h_1,\dots,h_k}$ parameters. More precisely, one has

Theorem 1 (Chowla on the average) Suppose ${H = H(X) \leq X}$ is a quantity that goes to infinity as ${X \rightarrow \infty}$ (but it can go to infinity arbitrarily slowly). Then for any fixed ${k \geq 1}$, we have

$\displaystyle \sum_{h_1,\dots,h_k \leq H} |\sum_{n \leq X} \lambda(n+h_1) \dots \lambda(n+h_k)| = o( H^k X ).$

In fact, we can remove one of the averaging parameters and obtain

$\displaystyle \sum_{h_2,\dots,h_k \leq H} |\sum_{n \leq X} \lambda(n) \lambda(n+h_2) \dots \lambda(n+h_k)| = o( H^{k-1} X ).$

Actually we can make the decay rate a bit more quantitative, gaining about ${\frac{\log\log H}{\log H}}$ over the trivial bound. The key case is ${k=2}$; while the unaveraged Chowla conjecture becomes more difficult as ${k}$ increases, the averaged Chowla conjecture does not increase in difficulty due to the increasing amount of averaging for larger ${k}$, and we end up deducing the higher ${k}$ case of the conjecture from the ${k=2}$ case by an elementary argument.

The proof of the theorem proceeds as follows. By exploiting the Fourier-analytic identity

$\displaystyle \int_{{\mathbf T}} (\int_{\mathbf R} |\sum_{x \leq n \leq x+H} f(n) e(\alpha n)|^2 dx)^2\ d\alpha$

$\displaystyle = \sum_{|h| \leq H} (H-|h|)^2 |\sum_n f(n) \overline{f}(n+h)|^2$

(related to a standard Fourier-analytic identity for the Gowers ${U^2}$ norm) it turns out that the ${k=2}$ case of the above theorem can basically be derived from an estimate of the form

$\displaystyle \int_0^X |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o( H X )$

uniformly for all ${\alpha \in {\mathbf T}}$. For “major arc” ${\alpha}$, close to a rational ${a/q}$ for small ${q}$, we can establish this bound from a generalisation of a recent result of Matomaki and Radziwill (discussed in this previous post) on averages of multiplicative functions in short intervals. For “minor arc” ${\alpha}$, we can proceed instead from an argument of Katai and Bourgain-Sarnak-Ziegler (discussed in this previous post).

The argument also extends to other bounded multiplicative functions than the Liouville function. Chowla’s conjecture was generalised by Elliott, who roughly speaking conjectured that the ${k}$ copies of ${\lambda}$ in Chowla’s conjecture could be replaced by arbitrary bounded multiplicative functions ${g_1,\dots,g_k}$ as long as these functions were far from a twisted Dirichlet character ${n \mapsto \chi(n) n^{it}}$ in the sense that

$\displaystyle \sum_p \frac{1 - \hbox{Re} g(p) \overline{\chi(p) p^{it}}}{p} = +\infty. \ \ \ \ \ (1)$

(This type of distance is incidentally now a fundamental notion in the Granville-Soundararajan “pretentious” approach to multiplicative number theory.) During our work on this project, we found that Elliott’s conjecture is not quite true as stated due to a technicality: one can cook up a bounded multiplicative function ${g}$ which behaves like ${n^{it_j}}$ on scales ${n \sim N_j}$ for some ${N_j}$ going to infinity and some slowly varying ${t_j}$, and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations ${\sum_{n \leq N_j} g(n+1) \overline{g(n)}}$ will be large). In our paper we propose a technical “fix” to Elliott’s conjecture (replacing (1) by a truncated variant), and show that this repaired version of Elliott’s conjecture is true on the average in much the same way that Chowla’s conjecture is. (If one restricts attention to real-valued multiplicative functions, then this technical issue does not show up, basically because one can assume without loss of generality that ${t=0}$ in this case; we discuss this fact in an appendix to the paper.)

A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function ${\zeta}$ in the critical strip ${\{ \sigma+it: 0 < \sigma < 1; t \in {\bf R} \}}$ in the limit ${t \rightarrow +\infty}$. For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude ${\log |\zeta|: {\bf C} \rightarrow [-\infty,+\infty]}$ of the zeta function. (In principle, one can reconstruct a branch of ${\log \zeta}$, and hence ${\zeta}$ itself, from ${\log |\zeta|}$ using the Cauchy-Riemann equations, or tools such as the Borel-Carathéodory theorem, see Exercise 40 of Supplement 2.)

One has the classical estimate

$\displaystyle \zeta(\sigma+it) = O( t^{O(1)} )$

when ${\sigma = O(1)}$ and ${t \geq 10}$ (say), so that

$\displaystyle \log |\zeta(\sigma+it)| \leq O( \log t ). \ \ \ \ \ (1)$

(See e.g. Exercise 37 from Supplement 3.) In view of this, let us define the normalised log-magnitudes ${F_T: {\bf C} \rightarrow [-\infty,+\infty]}$ for any ${T \geq 10}$ by the formula

$\displaystyle F_T( \sigma + it ) := \frac{1}{\log T} \log |\zeta( \sigma + i(T + t) )|;$

informally, this is a normalised window into ${\log |\zeta|}$ near ${iT}$. One can rephrase several assertions about the zeta function in terms of the asymptotic behaviour of ${F_T}$. For instance:

• (i) The bound (1) implies that ${F_T}$ is asymptotically locally bounded from above in the limit ${T \rightarrow \infty}$, thus for any compact set ${K \subset {\bf C}}$ we have ${F_T(\sigma+it) \leq O_K(1)}$ for ${\sigma+it \in K}$ and ${T}$ sufficiently large. In fact the implied constant in ${K}$ only depends on the projection of ${K}$ to the real axis.
• (ii) For ${\sigma > 1}$, we have the bounds

$\displaystyle |\zeta(\sigma+it)|, \frac{1}{|\zeta(\sigma+it)|} \leq \zeta(\sigma)$

which implies that ${F_T}$ converges locally uniformly as ${T \rightarrow +\infty}$ to zero in the region ${\{ \sigma+it: \sigma > 1, t \in {\bf R} \}}$.

• (iii) The functional equation, together with the symmetry ${\zeta(\sigma-it) = \overline{\zeta(\sigma+it)}}$, implies that

$\displaystyle |\zeta(\sigma+it)| = 2^\sigma \pi^{\sigma-1} |\sin \frac{\pi(\sigma+it)}{2}| |\Gamma(1-\sigma-it)| |\zeta(1-\sigma+it)|$

which by Exercise 17 of Supplement 3 shows that

$\displaystyle F_T( 1-\sigma+it ) = \frac{1}{2}-\sigma + F_T(\sigma+it) + o(1)$

as ${T \rightarrow \infty}$, locally uniformly in ${\sigma+it}$. In particular, when combined with the previous item, we see that ${F_T(\sigma+it)}$ converges locally uniformly as ${T \rightarrow +\infty}$ to ${\frac{1}{2}-\sigma}$ in the region ${\{ \sigma+it: \sigma < 0, t \in {\bf R}\}}$.

• (iv) From Jensen’s formula (Theorem 16 of Supplement 2) we see that ${\log|\zeta|}$ is a subharmonic function, and thus ${F_T}$ is subharmonic as well. In particular we have the mean value inequality

$\displaystyle F_T( z_0 ) \leq \frac{1}{\pi r^2} \int_{z: |z-z_0| \leq r} F_T(z)$

for any disk ${\{ z: |z-z_0| \leq r \}}$, where the integral is with respect to area measure. From this and (ii) we conclude that

$\displaystyle \int_{z: |z-z_0| \leq r} F_T(z) \geq O_{z_0,r}(1)$

for any disk with ${\hbox{Re}(z_0)>1}$ and sufficiently large ${T}$; combining this with (i) we conclude that ${F_T}$ is asymptotically locally bounded in ${L^1}$ in the limit ${T \rightarrow \infty}$, thus for any compact set ${K \subset {\bf C}}$ we have ${\int_K |F_T| \ll_K 1}$ for sufficiently large ${T}$.

From (v) and the usual Arzela-Ascoli diagonalisation argument, we see that the ${F_T}$ are asymptotically compact in the topology of distributions: given any sequence ${T_n}$ tending to ${+\infty}$, one can extract a subsequence such that the ${F_T}$ converge in the sense of distributions. Let us then define a normalised limit profile of ${\log|\zeta|}$ to be a distributional limit ${F}$ of a sequence of ${F_T}$; they are analogous to limiting profiles in PDE, and also to the more recent introduction of “graphons” in the theory of graph limits. Then by taking limits in (i)-(iv) we can say a lot about such normalised limit profiles ${F}$ (up to almost everywhere equivalence, which is an issue we will address shortly):

• (i) ${F}$ is bounded from above in the critical strip ${\{ \sigma+it: 0 \leq \sigma \leq 1 \}}$.
• (ii) ${F}$ vanishes on ${\{ \sigma+it: \sigma \geq 1\}}$.
• (iii) We have the functional equation ${F(1-\sigma+it) = \frac{1}{2}-\sigma + F(\sigma+it)}$ for all ${\sigma+it}$. In particular ${F(\sigma+it) = \frac{1}{2}-\sigma}$ for ${\sigma<0}$.
• (iv) ${F}$ is subharmonic.

Unfortunately, (i)-(iv) fail to characterise ${F}$ completely. For instance, one could have ${F(\sigma+it) = f(\sigma)}$ for any convex function ${f(\sigma)}$ of ${\sigma}$ that equals ${0}$ for ${\sigma \geq 1}$, ${\frac{1}{2}-\sigma}$ for ${\sigma \leq 1}$, and obeys the functional equation ${f(1-\sigma) = \frac{1}{2}-\sigma+f(\sigma)}$, and this would be consistent with (i)-(iv). One can also perturb such examples in a region where ${f}$ is strictly convex to create further examples of functions obeying (i)-(iv). Note from subharmonicity that the function ${\sigma \mapsto \sup_t F(\sigma+it)}$ is always going to be convex in ${\sigma}$; this can be seen as a limiting case of the Hadamard three-lines theorem (Exercise 41 of Supplement 2).

We pause to address one minor technicality. We have defined ${F}$ as a distributional limit, and as such it is a priori only defined up to almost everywhere equivalence. However, due to subharmonicity, there is a unique upper semi-continuous representative of ${F}$ (taking values in ${[-\infty,+\infty)}$), defined by the formula

$\displaystyle F(z_0) = \lim_{r \rightarrow 0^+} \frac{1}{\pi r^2} \int_{B(z_0,r)} F(z)\ dz$

for any ${z_0 \in {\bf C}}$ (note from subharmonicity that the expression in the limit is monotone nonincreasing as ${r \rightarrow 0}$, and is also continuous in ${z_0}$). We will now view this upper semi-continuous representative of ${F}$ as the canonical representative of ${F}$, so that ${F}$ is now defined everywhere, rather than up to almost everywhere equivalence.

By a classical theorem of Riesz, a function ${F}$ is subharmonic if and only if the distribution ${-\Delta F}$ is a non-negative measure, where ${\Delta := \frac{\partial^2}{\partial \sigma^2} + \frac{\partial^2}{\partial t^2}}$ is the Laplacian in the ${\sigma,t}$ coordinates. Jensen’s formula (or Greens’ theorem), when interpreted distributionally, tells us that

$\displaystyle -\Delta \log |\zeta| = \frac{1}{2\pi} \sum_\rho \delta_\rho$

away from the real axis, where ${\rho}$ ranges over the non-trivial zeroes of ${\zeta}$. Thus, if ${F}$ is a normalised limit profile for ${\log |\zeta|}$ that is the distributional limit of ${F_{T_n}}$, then we have

$\displaystyle -\Delta F = \nu$

where ${\nu}$ is a non-negative measure which is the limit in the vague topology of the measures

$\displaystyle \nu_{T_n} := \frac{1}{2\pi \log T_n} \sum_\rho \delta_{\rho - T_n}.$

Thus ${\nu}$ is a normalised limit profile of the zeroes of the Riemann zeta function.

Using this machinery, we can recover many classical theorems about the Riemann zeta function by “soft” arguments that do not require extensive calculation. Here are some examples:

Theorem 1 The Riemann hypothesis implies the Lindelöf hypothesis.

Proof: It suffices to show that any limiting profile ${F}$ (arising as the limit of some ${F_{T_n}}$) vanishes on the critical line ${\{1/2+it: t \in {\bf R}\}}$. But if the Riemann hypothesis holds, then the measures ${\nu_{T_n}}$ are supported on the critical line ${\{1/2+it: t \in {\bf R}\}}$, so the normalised limit profile ${\nu}$ is also supported on this line. This implies that ${F}$ is harmonic outside of the critical line. By (ii) and unique continuation for harmonic functions, this implies that ${F}$ vanishes on the half-space ${\{ \sigma+it: \sigma \geq \frac{1}{2} \}}$ (and equals ${\frac{1}{2}-\sigma}$ on the complementary half-space, by (iii)), giving the claim. $\Box$

In fact, we have the following sharper statement:

Theorem 2 (Backlund) The Lindelöf hypothesis is equivalent to the assertion that for any fixed ${\sigma_0 > \frac{1}{2}}$, the number of zeroes in the region ${\{ \sigma+it: \sigma > \sigma_0, T \leq t \leq T+1 \}}$ is ${o(\log T)}$ as ${T \rightarrow \infty}$.

Proof: If the latter claim holds, then for any ${T_n \rightarrow \infty}$, the measures ${\nu_{T_n}}$ assign a mass of ${o(1)}$ to any region of the form ${\{ \sigma+it: \sigma > \sigma_0; t_0 \leq t \leq t_0+1 \}}$ as ${n \rightarrow \infty}$ for any fixed ${\sigma_0>\frac{1}{2}}$ and ${t_0 \in {\bf R}}$. Thus the normalised limiting profile measure ${\nu}$ is supported on the critical line, and we can repeat the previous argument.

Conversely, suppose the claim fails, then we can find a sequence ${T_n}$ and ${\sigma_0>0}$ such that ${\nu_{T_n}}$ assigns a mass of ${\gg 1}$ to the region ${\{ \sigma+it: \sigma > \sigma_0; 0\leq t \leq 1 \}}$. Extracting a normalised limiting profile, we conclude that the normalised limiting profile measure ${\nu}$ is non-trivial somewhere to the right of the critical line, so the associated subharmonic function ${F}$ is not harmonic everywhere to the right of the critical line. From the maximum principle and (ii) this implies that ${F}$ has to be positive somewhere on the critical line, but this contradicts the Lindelöf hypothesis. (One has to take a bit of care in the last step since ${F_{T_n}}$ only converges to ${F}$ in the sense of distributions, but it turns out that the subharmonicity of all the functions involved gives enough regularity to justify the argument; we omit the details here.) $\Box$

Theorem 3 (Littlewood) Assume the Lindelöf hypothesis. Then for any fixed ${\alpha>0}$, the number of zeroes in the region ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ is ${(2\pi \alpha+o(1)) \log T}$ as ${T \rightarrow +\infty}$.

Proof: By the previous arguments, the only possible normalised limiting profile for ${\log |\zeta|}$ is ${\max( 0, \frac{1}{2}-\sigma )}$. Taking distributional Laplacians, we see that the only possible normalised limiting profile for the zeroes is Lebesgue measure on the critical line. Thus, ${\nu_T( \{\sigma+it: T \leq t \leq T+\alpha \} )}$ can only converge to ${\alpha}$ as ${T \rightarrow +\infty}$, and the claim follows. $\Box$

Even without the Lindelöf hypothesis, we have the following result:

Theorem 4 (Titchmarsh) For any fixed ${\alpha>0}$, there are ${\gg_\alpha \log T}$ zeroes in the region ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ for sufficiently large ${T}$.

Among other things, this theorem recovers a classical result of Littlewood that the gaps between the imaginary parts of the zeroes goes to zero, even without assuming unproven conjectures such as the Riemann or Lindelöf hypotheses.

Proof: Suppose for contradiction that this were not the case, then we can find ${\alpha > 0}$ and a sequence ${T_n \rightarrow \infty}$ such that ${\{ \sigma+it: T_n \leq t \leq T_n+\alpha \}}$ contains ${o(\log T)}$ zeroes. Passing to a subsequence to extract a limit profile, we conclude that the normalised limit profile measure ${\nu}$ assigns no mass to the horizontal strip ${\{ \sigma+it: 0 \leq t \leq\alpha \}}$. Thus the associated subharmonic function ${F}$ is actually harmonic on this strip. But by (ii) and unique continuation this forces ${F}$ to vanish on this strip, contradicting the functional equation (iii). $\Box$

Exercise 5 Use limiting profiles to obtain the matching upper bound of ${O_\alpha(\log T)}$ for the number of zeroes in ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ for sufficiently large ${T}$.

Remark 6 One can remove the need to take limiting profiles in the above arguments if one can come up with quantitative (or “hard”) substitutes for qualitative (or “soft”) results such as the unique continuation property for harmonic functions. This would also allow one to replace the qualitative decay rates ${o(1)}$ with more quantitative decay rates such as ${1/\log \log T}$ or ${1/\log\log\log T}$. Indeed, the classical proofs of the above theorems come with quantitative bounds that are typically of this form (see e.g. the text of Titchmarsh for details).

Exercise 7 Let ${S(T)}$ denote the quantity ${S(T) := \frac{1}{\pi} \hbox{arg} \zeta(\frac{1}{2}+iT)}$, where the branch of the argument is taken by using a line segment connecting ${\frac{1}{2}+iT}$ to (say) ${2+iT}$, and then to ${2}$. If we have a sequence ${T_n \rightarrow \infty}$ producing normalised limit profiles ${F, \nu}$ for ${\log|\zeta|}$ and the zeroes respectively, show that ${t \mapsto \frac{1}{\log T_n} S(T_n + t)}$ converges in the sense of distributions to the function ${t \mapsto \frac{1}{\pi} \int_{1/2}^1 \frac{\partial F}{\partial t}(\sigma+it)\ d\sigma}$, or equivalently

$\displaystyle t \mapsto \frac{1}{2\pi} \frac{\partial}{\partial t} \int_0^1 F(\sigma+it)\ d\sigma.$

Conclude in particular that if the Lindelöf hypothesis holds, then ${S(T) = o(\log T)}$ as ${T \rightarrow \infty}$.

A little bit more about the normalised limit profiles ${F}$ are known unconditionally, beyond (i)-(iv). For instance, from Exercise 3 of Notes 5 we have ${\zeta(1/2 + it ) = O( t^{1/6+o(1)} )}$ as ${t \rightarrow +\infty}$, which implies that any normalised limit profile ${F}$ for ${\log|\zeta|}$ is bounded by ${1/6}$ on the critical line, beating the bound of ${1/4}$ coming from convexity and (ii), (iii), and then convexity can be used to further bound ${F}$ away from the critical line also. Some further small improvements of this type are known (coming from various methods for estimating exponential sums), though they fall well short of determining ${F}$ completely at our current level of understanding. Of course, given that we believe the Riemann hypothesis (and hence the Lindelöf hypothesis) to be true, the only actual limit profile that should exist is ${\max(0,\frac{1}{2}-\sigma)}$ (in fact this assertion is equivalent to the Lindelöf hypothesis, by the arguments above).

Better control on limiting profiles is available if we do not insist on controlling ${\zeta}$ for all values of the height parameter ${T}$, but only for most such values, thanks to the existence of several mean value theorems for the zeta function, as discussed in Notes 6; we discuss this below the fold.