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A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:

Theorem 1 (Discrete inverse theorem for Gowers norms) Let {N \geq 1} and {s \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z} \rightarrow [-1,1]} is a function supported on {[N] := \{1,\dots,N\}} such that

\displaystyle  \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle  \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.

For the definitions of “filtered nilmanifold”, “degree”, “complexity”, and “polynomial sequence”, see the paper of Ben, Tammy, and myself. (I should caution the reader that this blog post will presume a fair amount of familiarity with this subfield of additive combinatorics.) This result has a number of applications, for instance to establishing asymptotics for linear equations in the primes, but this will not be the focus of discussion here.

The purpose of this post is to record the observation that this “discrete” inverse theorem, together with an equidistribution theorem for nilsequences that Ben and I worked out in a separate paper, implies a continuous version:

Theorem 2 (Continuous inverse theorem for Gowers norms) Let {s \geq 1} be an integer, and let {\delta>0}. Suppose that {f: {\bf R} \rightarrow [-1,1]} is a measurable function supported on {[0,1]} such that

\displaystyle  \int_{{\bf R}^{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(t+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1})\ dt dh_1 \dots dh_{s+1} \geq \delta. \ \ \ \ \ (1)

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a (smooth) polynomial sequence {g: {\bf R} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle  \int_{\bf R} f(t) F(g(t) \Gamma)\ dt \gg_{s,\delta} 1.

The interval {[0,1]} can be easily replaced with any other fixed interval by a change of variables. A key point here is that the bounds are completely uniform in the choice of {f}. Note though that the coefficients of {g} can be arbitrarily large (and this is necessary, as can be seen just by considering functions of the form {f(t) = \cos( \xi t)} for some arbitrarily large frequency {\xi}).

It is likely that one could prove Theorem 2 by carefully going through the proof of Theorem 1 and replacing all instances of {{\bf Z}} with {{\bf R}} (and making appropriate modifications to the argument to accommodate this). However, the proof of Theorem 1 is quite lengthy. Here, we shall proceed by the usual limiting process of viewing the continuous interval {[0,1]} as a limit of the discrete interval {\frac{1}{N} \cdot [N]} as {N \rightarrow \infty}. However there will be some problems taking the limit due to a failure of compactness, and specifically with regards to the coefficients of the polynomial sequence {g: {\bf N} \rightarrow G} produced by Theorem 1, after normalising these coefficients by {N}. Fortunately, a factorisation theorem from a paper of Ben Green and myself resolves this problem by splitting {g} into a “smooth” part which does enjoy good compactness properties, as well as “totally equidistributed” and “periodic” parts which can be eliminated using the measurability (and thus, approximate smoothness), of {f}.

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Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:

Theorem 1 (Szemerédi’s theorem) Let {N} be a positive integer, and let {f: {\bf Z}/N{\bf Z} \rightarrow [0,1]} be a function with {{\bf E}_{x \in {\bf Z}/N{\bf Z}} f(x) \geq \delta} for some {\delta>0}, where we use the averaging notation {{\bf E}_{x \in A} f(x) := \frac{1}{|A|} \sum_{x \in A} f(x)}, {{\bf E}_{x,r \in A} f(x) := \frac{1}{|A|^2} \sum_{x, r \in A} f(x)}, etc.. Then for {k \geq 3} we have

\displaystyle  {\bf E}_{x,r \in {\bf Z}/N{\bf Z}} f(x) f(x+r) \dots f(x+(k-1)r) \geq c(k,\delta)

for some {c(k,\delta)>0} depending only on {k,\delta}.

The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases {k=1,2} as they are trivial and somewhat degenerate.

There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.

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In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel lists of facts (stated somewhat informally). Firstly, some facts about the prime factorisation of large integers:

  • Every positive integer {m} has a prime factorisation

    \displaystyle  m = p_1 p_2 \dots p_r

    into (not necessarily distinct) primes {p_1,\dots,p_r}, which is unique up to rearrangement. Taking logarithms, we obtain a partition

    \displaystyle  \log m = \log p_1 + \log p_2 + \dots + \log p_r

    of {\log m}.

  • (Prime number theorem) A randomly selected integer {m} of size {m \sim N} will be prime with probability {\approx \frac{1}{\log N}} when {N} is large.
  • If {m \sim N} is a randomly selected large integer of size {N}, and {p = p_i} is a randomly selected prime factor of {m = p_1 \dots p_r} (with each index {i} being chosen with probability {\frac{\log p_i}{\log m}}), then {\log p_i} is approximately uniformly distributed between {0} and {\log N}. (See Proposition 9 of this previous blog post.)
  • The set of real numbers {\{ \frac{\log p_i}{\log m}: i=1,\dots,r \}} arising from the prime factorisation {m = p_1 \dots p_r} of a large random number {m \sim N} converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit {N \rightarrow \infty}. (See the previously mentioned blog post for a definition of the Poisson-Dirichlet process, and a proof of this claim.)

Now for the facts about the cycle decomposition of large permutations:

  • Every permutation {\sigma \in S_n} has a cycle decomposition

    \displaystyle  \sigma = C_1 \dots C_r

    into disjoint cycles {C_1,\dots,C_r}, which is unique up to rearrangement, and where we count each fixed point of {\sigma} as a cycle of length {1}. If {|C_i|} is the length of the cycle {C_i}, we obtain a partition

    \displaystyle  n = |C_1| + \dots + |C_r|

    of {n}.

  • (Prime number theorem for permutations) A randomly selected permutation of {S_n} will be an {n}-cycle with probability exactly {1/n}. (This was noted in this previous blog post.)
  • If {\sigma} is a random permutation in {S_n}, and {C_i} is a randomly selected cycle of {\sigma} (with each {i} being selected with probability {|C_i|/n}), then {|C_i|} is exactly uniformly distributed on {\{1,\dots,n\}}. (See Proposition 8 of this blog post.)
  • The set of real numbers {\{ \frac{|C_i|}{n} \}} arising from the cycle decomposition {\sigma = C_1 \dots C_r} of a random permutation {\sigma \in S_n} converges (in a suitable sense) to the Poisson-Dirichlet process in the limit {n \rightarrow \infty}. (Again, see this previous blog post for details.)

See this previous blog post (or the aforementioned article of Granville, or the Notices article of Arratia, Barbour, and Tavaré) for further exploration of the analogy between prime factorisation of integers and cycle decomposition of permutations.

There is however something unsatisfying about the analogy, in that it is not clear why there should be such a kinship between integer prime factorisation and permutation cycle decomposition. It turns out that the situation is clarified if one uses another fundamental analogy in number theory, namely the analogy between integers and polynomials {P \in {\mathbf F}_q[T]} over a finite field {{\mathbf F}_q}, discussed for instance in this previous post; this is the simplest case of the more general function field analogy between number fields and function fields. Just as we restrict attention to positive integers when talking about prime factorisation, it will be reasonable to restrict attention to monic polynomials {P}. We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:

  • Every monic polynomial {f \in {\mathbf F}_q[T]} has a factorisation

    \displaystyle  f = P_1 \dots P_r

    into irreducible monic polynomials {P_1,\dots,P_r \in {\mathbf F}_q[T]}, which is unique up to rearrangement. Taking degrees, we obtain a partition

    \displaystyle  \hbox{deg} f = \hbox{deg} P_1 + \dots + \hbox{deg} P_r

    of {\hbox{deg} f}.

  • (Prime number theorem for polynomials) A randomly selected monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} will be irreducible with probability {\approx \frac{1}{n}} when {q} is fixed and {n} is large.
  • If {f \in {\mathbf F}_q[T]} is a random monic polynomial of degree {n}, and {P_i} is a random irreducible factor of {f = P_1 \dots P_r} (with each {i} selected with probability {\hbox{deg} P_i / n}), then {\hbox{deg} P_i} is approximately uniformly distributed in {\{1,\dots,n\}} when {q} is fixed and {n} is large.
  • The set of real numbers {\{ \hbox{deg} P_i / n \}} arising from the factorisation {f = P_1 \dots P_r} of a randomly selected polynomial {f \in {\mathbf F}_q[T]} of degree {n} converges (in a suitable sense) to the Poisson-Dirichlet process when {q} is fixed and {n} is large.

The above list of facts addressed the large {n} limit of the polynomial ring {{\mathbf F}_q[T]}, where the order {q} of the field is held fixed, but the degrees of the polynomials go to infinity. This is the limit that is most closely analogous to the integers {{\bf Z}}. However, there is another interesting asymptotic limit of polynomial rings to consider, namely the large {q} limit where it is now the degree {n} that is held fixed, but the order {q} of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the characteristic {p} of the field goes to infinity (again keeping the degree {n} fixed), although all of the results proven below for the large {p} limit turn out to be true as well in the large {q} limit.

The large {q} (or large {p}) limit is technically a different limit than the large {n} limit, but in practice the asymptotic statistics of the two limits often agree quite closely. For instance, here is the prime number theorem in the large {q} limit:

Theorem 1 (Prime number theorem) The probability that a random monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} is irreducible is {\frac{1}{n}+o(1)} in the limit where {n} is fixed and the characteristic {p} goes to infinity.

Proof: There are {q^n} monic polynomials {f \in {\mathbf F}_q[T]} of degree {n}. If {f} is irreducible, then the {n} zeroes of {f} are distinct and lie in the finite field {{\mathbf F}_{q^n}}, but do not lie in any proper subfield of that field. Conversely, every element {\alpha} of {{\mathbf F}_{q^n}} that does not lie in a proper subfield is the root of a unique monic polynomial in {{\mathbf F}_q[T]} of degree {f} (the minimal polynomial of {\alpha}). Since the union of all the proper subfields of {{\mathbf F}_{q^n}} has size {o(q^n)}, the total number of irreducible polynomials of degree {n} is thus {\frac{q^n - o(q^n)}{n}}, and the claim follows. \Box

Remark 2 The above argument and inclusion-exclusion in fact gives the well known exact formula {\frac{1}{n} \sum_{d|n} \mu(\frac{n}{d}) q^d} for the number of irreducible monic polynomials of degree {n}.

Now we can give a precise connection between the cycle distribution of a random permutation, and (the large {p} limit of) the irreducible factorisation of a polynomial, giving a (somewhat indirect, but still connected) link between permutation cycle decomposition and integer factorisation:

Theorem 3 The partition {\{ \hbox{deg}(P_1), \dots, \hbox{deg}(P_r) \}} of a random monic polynomial {f= P_1 \dots P_r\in {\mathbf F}_q[T]} of degree {n} converges in distribution to the partition {\{ |C_1|, \dots, |C_r|\}} of a random permutation {\sigma = C_1 \dots C_r \in S_n} of length {n}, in the limit where {n} is fixed and the characteristic {p} goes to infinity.

We can quickly prove this theorem as follows. We first need a basic fact:

Lemma 4 (Most polynomials square-free in large {q} limit) A random monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} will be square-free with probability {1-o(1)} when {n} is fixed and {q} (or {p}) goes to infinity. In a similar spirit, two randomly selected monic polynomials {f,g} of degree {n,m} will be coprime with probability {1-o(1)} if {n,m} are fixed and {q} or {p} goes to infinity.

Proof: For any polynomial {g} of degree {m}, the probability that {f} is divisible by {g^2} is at most {1/q^{2m}}. Summing over all polynomials of degree {1 \leq m \leq n/2}, and using the union bound, we see that the probability that {f} is not squarefree is at most {\sum_{1 \leq m \leq n/2} \frac{q^m}{q^{2m}} = o(1)}, giving the first claim. For the second, observe from the first claim (and the fact that {fg} has only a bounded number of factors) that {fg} is squarefree with probability {1-o(1)}, giving the claim. \Box

Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation {\sigma \in S_n} consisting of {c_k} cycles of length {k} for {k=1,\dots,r}, where {c_k} are nonnegative integers with {\sum_{k=1}^r k c_k = n}, is precisely

\displaystyle  \frac{1}{\prod_{k=1}^r c_k! k^{c_k}},

since there are {\prod_{k=1}^r c_k! k^{c_k}} ways to write a given tuple of cycles {C_1,\dots,C_r} in cycle notation in nondecreasing order of length, and {n!} ways to select the labels for the cycle notation. On the other hand, by Theorem 1 (and using Lemma 4 to isolate the small number of cases involving repeated factors) the number of monic polynomials of degree {n} that are the product of {c_k} irreducible polynomials of degree {k} is

\displaystyle  \frac{1}{\prod_{k=1}^r c_k!} \prod_{k=1}^r ( (\frac{1}{k}+o(1)) q^k )^{c_k} + o( q^n )

which simplifies to

\displaystyle  \frac{1+o(1)}{\prod_{k=1}^r c_k! k^{c_k}} q^n,

and the claim follows.

This was a fairly short calculation, but it still doesn’t quite explain why there is such a link between the cycle decomposition {\sigma = C_1 \dots C_r} of permutations and the factorisation {f = P_1 \dots P_r} of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in {S_n} with the multiplication structure of polynomials; however, these structures are too dissimilar to set up a convincing analogy. For instance, the multiplication law on polynomials is abelian and non-invertible, whilst the multiplication law on {S_n} is (extremely) non-abelian but invertible. Also, the multiplication of a degree {n} and a degree {m} polynomial is a degree {n+m} polynomial, whereas the group multiplication law on permutations does not take a permutation in {S_n} and a permutation in {S_m} and return a permutation in {S_{n+m}}.

I recently found (after some discussions with Ben Green) what I feel to be a satisfying conceptual (as opposed to computational) explanation of this link, which I will place below the fold.

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Suppose that {A \subset B} are two subgroups of some ambient group {G}, with the index {K := [B:A]} of {A} in {B} being finite. Then {B} is the union of {K} left cosets of {A}, thus {B = SA} for some set {S \subset B} of cardinality {K}. The elements {s} of {S} are not entirely arbitrary with regards to {A}. For instance, if {A} is a normal subgroup of {B}, then for each {s \in S}, the conjugation map {g \mapsto s^{-1} g s} preserves {A}. In particular, if we write {A^s := s^{-1} A s} for the conjugate of {A} by {s}, then

\displaystyle  A = A^s.

Even if {A} is not normal in {B}, it turns out that the conjugation map {g \mapsto s^{-1} g s} approximately preserves {A}, if {K} is bounded. To quantify this, let us call two subgroups {A,B} {K}-commensurate for some {K \geq 1} if one has

\displaystyle  [A : A \cap B], [B : A \cap B] \leq K.

Proposition 1 Let {A \subset B} be groups, with finite index {K = [B:A]}. Then for every {s \in B}, the groups {A} and {A^s} are {K}-commensurate, in fact

\displaystyle  [A : A \cap A^s ] = [A^s : A \cap A^s ] \leq K.

Proof: One can partition {B} into {K} left translates {xA} of {A}, as well as {K} left translates {yA^s} of {A^s}. Combining the partitions, we see that {B} can be partitioned into at most {K^2} non-empty sets of the form {xA \cap yA^s}. Each of these sets is easily seen to be a left translate of the subgroup {A \cap A^s}, thus {[B: A \cap A^s] \leq K^2}. Since

\displaystyle [B: A \cap A^s] = [B:A] [A: A \cap A^s] = [B:A^s] [A^s: A \cap A^s]

and {[B:A] = [B:A^s]=K}, we obtain the claim. \Box

One can replace the inclusion {A \subset B} by commensurability, at the cost of some worsening of the constants:

Corollary 2 Let {A, B} be {K}-commensurate subgroups of {G}. Then for every {s \in B}, the groups {A} and {A^s} are {K^2}-commensurate.

Proof: Applying the previous proposition with {A} replaced by {A \cap B}, we see that for every {s \in B}, {A \cap B} and {(A \cap B)^s} are {K}-commensurate. Since {A \cap B} and {(A \cap B)^s} have index at most {K} in {A} and {A^s} respectively, the claim follows. \Box

It turns out that a similar phenomenon holds for the more general concept of an approximate group, and gives a “classification” of all the approximate groups {B} containing a given approximate group {A} as a “bounded index approximate subgroup”. Recall that a {K}-approximate group {A} in a group {G} for some {K \geq 1} is a symmetric subset of {G} containing the identity, such that the product set {A^2 := \{ a_1 a_2: a_1,a_2 \in A\}} can be covered by at most {K} left translates of {A} (and thus also {K} right translates, by the symmetry of {A}). For simplicity we will restrict attention to finite approximate groups {A} so that we can use their cardinality {A} as a measure of size. We call finite two approximate groups {A,B} {K}-commensurate if one has

\displaystyle  |A^2 \cap B^2| \geq \frac{1}{K} |A|, \frac{1}{K} |B|;

note that this is consistent with the previous notion of commensurability for genuine groups.

Theorem 3 Let {G} be a group, and let {K_1,K_2,K_3 \geq 1} be real numbers. Let {A} be a finite {K_1}-approximate group, and let {B} be a symmetric subset of {G} that contains {A}.

  • (i) If {B} is a {K_2}-approximate group with {|B| \leq K_3 |A|}, then one has {B \subset SA} for some set {S} of cardinality at most {K_1 K_2 K_3}. Furthermore, for each {s \in S}, the approximate groups {A} and {A^s} are {K_1 K_2^5 K_3}-commensurate.
  • (ii) Conversely, if {B \subset SA} for some set {S} of cardinality at most {K_3}, and {A} and {A^s} are {K_2}-commensurate for all {s \in S}, then {|B| \leq K_3 |A|}, and {B} is a {K_1^6 K_2 K_3^2}-approximate group.

Informally, the assertion that {B} is an approximate group containing {A} as a “bounded index approximate subgroup” is equivalent to {B} being covered by a bounded number of shifts {sA} of {A}, where {s} approximately normalises {A^2} in the sense that {A^2} and {(A^2)^s} have large intersection. Thus, to classify all such {B}, the problem essentially reduces to that of classifying those {s} that approximately normalise {A^2}.

To prove the theorem, we recall some standard lemmas from arithmetic combinatorics, which are the foundation stones of the “Ruzsa calculus” that we will use to establish our results:

Lemma 4 (Ruzsa covering lemma) If {A} and {B} are finite non-empty subsets of {G}, then one has {B \subset SAA^{-1}} for some set {S \subset B} with cardinality {|S| \leq \frac{|BA|}{|A|}}.

Proof: We take {S} to be a subset of {B} with the property that the translates {sA, s \in S} are disjoint in {BA}, and such that {S} is maximal with respect to set inclusion. The required properties of {S} are then easily verified. \Box

Lemma 5 (Ruzsa triangle inequality) If {A,B,C} are finite non-empty subsets of {G}, then

\displaystyle  |A C^{-1}| \leq |A B^{-1}| |B C^{-1}| / |B|.

Proof: If {ac^{-1}} is an element of {AC^{-1}} with {a \in A} and {c \in C}, then from the identity {ac^{-1} = (ab^{-1}) (bc^{-1})} we see that {ac^{-1}} can be written as the product of an element of {AB^{-1}} and an element of {BC^{-1}} in at least {|B|} distinct ways. The claim follows. \Box

Now we can prove (i). By the Ruzsa covering lemma, {B} can be covered by at most

\displaystyle \frac{|BA|}{|A|} \leq \frac{|B^2|}{|A|} \leq \frac{K_2 |B|}{|A|} \leq K_2 K_3

left-translates of {A^2}, and hence by at most {K_1 K_2 K_3} left-translates of {A}, thus {B \subset SA} for some {|S| \leq K_1 K_2 K_3}. Since {sA} only intersects {B} if {s \in BA}, we may assume that

\displaystyle  S \subset BA \subset B^2

and hence for any {s \in S}

\displaystyle  |A^s A| \leq |B^2 A B^2 A| \leq |B^6|

\displaystyle  \leq K_2^5 |B| \leq K_2^5 K_3 |A|.

By the Ruzsa covering lemma again, this implies that {A^s} can be covered by at most {K_2^5 K_3} left-translates of {A^2}, and hence by at most {K_1 K_2^5 K_3} left-translates of {A}. By the pigeonhole principle, there thus exists a group element {g} with

\displaystyle  |A^s \cap gA| \geq \frac{1}{K_1 K_2^5 K_3} |A|.


\displaystyle  |A^s \cap gA| \leq | (A^s \cap gA)^{-1} (A^s \cap gA)|


\displaystyle  (A^s \cap gA)^{-1} (A^s \cap gA) \subset A^2 \cap (A^s)^2

the claim follows.

Now we prove (ii). Clearly

\displaystyle  |B| \leq |S| |A| \leq K_3 |A|.

Now we control the size of {B^2 A}. We have

\displaystyle  |B^2 A| \leq |SA SA^2| \leq K_3^2 \sup_{s \in S} |A s A^2| = K_3^2 \sup_{s \in S} |A^s A^2|.

From the Ruzsa triangle inequality and symmetry we have

\displaystyle  |A^s A^2| \leq \frac{ |A^s (A^2 \cap (A^2)^s)| |(A^2 \cap (A^2)^s) A^2|}{|A^2 \cap (A^2)^s|}

\displaystyle  \leq \frac{ |(A^3)^s| |A^4| }{|A|/K_2}

\displaystyle  \leq K_2 \frac{ |A^3| |A^4| }{|A|}

\displaystyle  \leq K_1^5 K_2 |A|

and thus

\displaystyle  |B^2 A| \leq K_1^5 K_2 K_3^2 |A|.

By the Ruzsa covering lemma, this implies that {B^2} is covered by at most {K_1^5 K_2 K_3^2} left-translates of {A^2}, hence by at most {K_1^6 K_2 K_3^2} left-translates of {A}. Since {A \subset B}, the claim follows.

We now establish some auxiliary propositions about commensurability of approximate groups. The first claim is that commensurability is approximately transitive:

Proposition 6 Let {A} be a {K_1}-approximate group, {B} be a {K_2}-approximate group, and {C} be a {K_3}-approximate group. If {A} and {B} are {K_4}-commensurate, and {B} and {C} are {K_5}-commensurate, then {A} and {C} are {K_1^2 K_2^3 K_2^3 K_4 K_5 \max(K_1,K_3)}-commensurate.

Proof: From two applications of the Ruzsa triangle inequality we have

\displaystyle  |AC| \leq \frac{|A (A^2 \cap B^2)| |(A^2 \cap B^2) (B^2 \cap C^2)| |(B^2 \cap C^2) C|}{|A^2 \cap B^2| |B^2 \cap C^2|}

\displaystyle  \leq \frac{|A^3| |B^4| |C^3|}{ (|A|/K_4) (|B|/K_5)}

\displaystyle  \leq K_4 K_5 \frac{K_1^2 |A| K_2^3 |B| K_3^2 |C|}{ |A| |B| }

\displaystyle  = K_1^2 K_2^3 K_3^2 K_4 K_5 |C|.

By the Ruzsa covering lemma, we may thus cover {A} by at most {K_1^2 K_2^3 K_3^2 K_4 K_5} left-translates of {C^2}, and hence by {K_1^2 K_2^3 K_3^3 K_4 K_5} left-translates of {C}. By the pigeonhole principle, there thus exists a group element {g} such that

\displaystyle  |A \cap gC| \geq \frac{1}{K_1^2 K_2^3 K_3^3 K_4 K_5} |A|,

and so by arguing as in the proof of part (i) of the theorem we have

\displaystyle  |A^2 \cap C^2| \geq \frac{1}{K_1^2 K_2^3 K_3^3 K_4 K_5} |A|

and similarly

\displaystyle  |A^2 \cap C^2| \geq \frac{1}{K_1^3 K_2^3 K_3^2 K_4 K_5} |C|

and the claim follows. \Box

The next proposition asserts that the union and (modified) intersection of two commensurate approximate groups is again an approximate group:

Proposition 7 Let {A} be a {K_1}-approximate group, {B} be a {K_2}-approximate group, and suppose that {A} and {B} are {K_3}-commensurate. Then {A \cup B} is a {K_1 + K_2 + K_1^2 K_2^4 K_3 + K_1^4 K_2^2 K_3}-approximate subgroup, and {A^2 \cap B^2} is a {K_1^6 K_2^3 K_3}-approximate subgroup.

Using this proposition, one may obtain a variant of the previous theorem where the containment {A \subset B} is replaced by commensurability; we leave the details to the interested reader.

Proof: We begin with {A \cup B}. Clearly {A \cup B} is symmetric and contains the identity. We have {(A \cup B)^2 = A^2 \cup AB \cup BA \cup B^2}. The set {A^2} is already covered by {K_1} left translates of {A}, and hence of {A \cup B}; similarly {B^2} is covered by {K_2} left translates of {A \cup B}. As for {AB}, we observe from the Ruzsa triangle inequality that

\displaystyle  |AB^2| \leq \frac{|A (A^2 \cap B^2)| |(A^2 \cap B^2) B^2|}{|A^2 \cap B^2|}

\displaystyle  \leq \frac{|A^3| |B^4|}{|A|/K_3}

\displaystyle  \leq K_1^2 K_2^3 K_3 |B|

and hence by the Ruzsa covering lemma, {AB} is covered by at most {K_1^2 K_2^3 K_3} left translates of {B^2}, and hence by {K_1^2 K_2^4 K_3} left translates of {B}, and hence of {A \cup B}. Similarly {BA} is covered by at most {K_1^4 K_2^2 K_3} left translates of {B}. The claim follows.

Now we consider {A^2 \cap B^2}. Again, this is clearly symmetric and contains the identity. Repeating the previous arguments, we see that {A} is covered by at most {K_1^2 K_2^3 K_3} left-translates of {B}, and hence there exists a group element {g} with

\displaystyle  |A \cap gB| \geq \frac{1}{K_1^2 K_2^3 K_3} |A|.

Now observe that

\displaystyle  |(A^2 \cap B^2)^2 (A \cap gB)| \leq |A^5| \leq K_1^4 |A|

and so by the Ruzsa covering lemma, {(A^2 \cap B^2)^2} can be covered by at most {K_1^6 K_2^3 K_3} left-translates of {(A \cap gB) (A \cap gB)^{-1}}. But this latter set is (as observed previously) contained in {A^2 \cap B^2}, and the claim follows. \Box

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)


\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)


\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


\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


\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


\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


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}}.

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In 1946, Ulam, in response to a theorem of Anning and Erdös, posed the following problem:

Problem 1 (Erdös-Ulam problem) Let {S \subset {\bf R}^2} be a set such that the distance between any two points in {S} is rational. Is it true that {S} cannot be (topologically) dense in {{\bf R}^2}?

The paper of Anning and Erdös addressed the case that all the distances between two points in {S} were integer rather than rational in the affirmative.

The Erdös-Ulam problem remains open; it was discussed recently over at Gödel’s lost letter. It is in fact likely (as we shall see below) that the set {S} in the above problem is not only forbidden to be topologically dense, but also cannot be Zariski dense either. If so, then the structure of {S} is quite restricted; it was shown by Solymosi and de Zeeuw that if {S} fails to be Zariski dense, then all but finitely many of the points of {S} must lie on a single line, or a single circle. (Conversely, it is easy to construct examples of dense subsets of a line or circle in which all distances are rational, though in the latter case the square of the radius of the circle must also be rational.)

The main tool of the Solymosi-de Zeeuw analysis was Faltings’ celebrated theorem that every algebraic curve of genus at least two contains only finitely many rational points. The purpose of this post is to observe that an affirmative answer to the full Erdös-Ulam problem similarly follows from the conjectured analogue of Falting’s theorem for surfaces, namely the following conjecture of Bombieri and Lang:

Conjecture 2 (Bombieri-Lang conjecture) Let {X} be a smooth projective irreducible algebraic surface defined over the rationals {{\bf Q}} which is of general type. Then the set {X({\bf Q})} of rational points of {X} is not Zariski dense in {X}.

In fact, the Bombieri-Lang conjecture has been made for varieties of arbitrary dimension, and for more general number fields than the rationals, but the above special case of the conjecture is the only one needed for this application. We will review what “general type” means (for smooth projective complex varieties, at least) below the fold.

The Bombieri-Lang conjecture is considered to be extremely difficult, in particular being substantially harder than Faltings’ theorem, which is itself a highly non-trivial result. So this implication should not be viewed as a practical route to resolving the Erdös-Ulam problem unconditionally; rather, it is a demonstration of the power of the Bombieri-Lang conjecture. Still, it was an instructive algebraic geometry exercise for me to carry out the details of this implication, which quickly boils down to verifying that a certain quite explicit algebraic surface is of general type (Theorem 4 below). As I am not an expert in the subject, my computations here will be rather tedious and pedestrian; it is likely that they could be made much slicker by exploiting more of the machinery of modern algebraic geometry, and I would welcome any such streamlining by actual experts in this area. (For similar reasons, there may be more typos and errors than usual in this post; corrections are welcome as always.) My calculations here are based on a similar calculation of van Luijk, who used analogous arguments to show (assuming Bombieri-Lang) that the set of perfect cuboids is not Zariski-dense in its projective parameter space.

We also remark that in a recent paper of Makhul and Shaffaf, the Bombieri-Lang conjecture (or more precisely, a weaker consequence of that conjecture) was used to show that if {S} is a subset of {{\bf R}^2} with rational distances which intersects any line in only finitely many points, then there is a uniform bound on the cardinality of the intersection of {S} with any line. I have also recently learned (private communication) that an unpublished work of Shaffaf has obtained a result similar to the one in this post, namely that the Erdös-Ulam conjecture follows from the Bombieri-Lang conjecture, plus an additional conjecture about the rational curves in a specific surface.

Let us now give the elementary reductions to the claim that a certain variety is of general type. For sake of contradiction, let {S} be a dense set such that the distance between any two points is rational. Then {S} certainly contains two points that are a rational distance apart. By applying a translation, rotation, and a (rational) dilation, we may assume that these two points are {(0,0)} and {(1,0)}. As {S} is dense, there is a third point of {S} not on the {x} axis, which after a reflection we can place in the upper half-plane; we will write it as {(a,\sqrt{b})} with {b>0}.

Given any two points {P, Q} in {S}, the quantities {|P|^2, |Q|^2, |P-Q|^2} are rational, and so by the cosine rule the dot product {P \cdot Q} is rational as well. Since {(1,0) \in S}, this implies that the {x}-component of every point {P} in {S} is rational; this in turn implies that the product of the {y}-coordinates of any two points {P,Q} in {S} is rational as well (since this differs from {P \cdot Q} by a rational number). In particular, {a} and {b} are rational, and all of the points in {S} now lie in the lattice {\{ ( x, y\sqrt{b}): x, y \in {\bf Q} \}}. (This fact appears to have first been observed in the 1988 habilitationschrift of Kemnitz.)

Now take four points {(x_j,y_j \sqrt{b})}, {j=1,\dots,4} in {S} in general position (so that the octuplet {(x_1,y_1\sqrt{b},\dots,x_4,y_4\sqrt{b})} avoids any pre-specified hypersurface in {{\bf C}^8}); this can be done if {S} is dense. (If one wished, one could re-use the three previous points {(0,0), (1,0), (a,\sqrt{b})} to be three of these four points, although this ultimately makes little difference to the analysis.) If {(x,y\sqrt{b})} is any point in {S}, then the distances {r_j} from {(x,y\sqrt{b})} to {(x_j,y_j\sqrt{b})} are rationals that obey the equations

\displaystyle (x - x_j)^2 + b (y-y_j)^2 = r_j^2

for {j=1,\dots,4}, and thus determine a rational point in the affine complex variety {V = V_{b,x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4} \subset {\bf C}^5} defined as

\displaystyle V := \{ (x,y,r_1,r_2,r_3,r_4) \in {\bf C}^6:

\displaystyle (x - x_j)^2 + b (y-y_j)^2 = r_j^2 \hbox{ for } j=1,\dots,4 \}.

By inspecting the projection {(x,y,r_1,r_2,r_3,r_4) \rightarrow (x,y)} from {V} to {{\bf C}^2}, we see that {V} is a branched cover of {{\bf C}^2}, with the generic cover having {2^4=16} points (coming from the different ways to form the square roots {r_1,r_2,r_3,r_4}); in particular, {V} is a complex affine algebraic surface, defined over the rationals. By inspecting the monodromy around the four singular base points {(x,y) = (x_i,y_i)} (which switch the sign of one of the roots {r_i}, while keeping the other three roots unchanged), we see that the variety {V} is connected away from its singular set, and thus irreducible. As {S} is topologically dense in {{\bf R}^2}, it is Zariski-dense in {{\bf C}^2}, and so {S} generates a Zariski-dense set of rational points in {V}. To solve the Erdös-Ulam problem, it thus suffices to show that

Claim 3 For any non-zero rational {b} and for rationals {x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4} in general position, the rational points of the affine surface {V = V_{b,x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4}} is not Zariski dense in {V}.

This is already very close to a claim that can be directly resolved by the Bombieri-Lang conjecture, but {V} is affine rather than projective, and also contains some singularities. The first issue is easy to deal with, by working with the projectivisation

\displaystyle \overline{V} := \{ [X,Y,Z,R_1,R_2,R_3,R_4] \in {\bf CP}^6: Q(X,Y,Z,R_1,R_2,R_3,R_4) = 0 \} \ \ \ \ \ (1)


of {V}, where {Q: {\bf C}^7 \rightarrow {\bf C}^4} is the homogeneous quadratic polynomial

\displaystyle (X,Y,Z,R_1,R_2,R_3,R_4) := (Q_j(X,Y,Z,R_1,R_2,R_3,R_4) )_{j=1}^4


\displaystyle Q_j(X,Y,Z,R_1,R_2,R_3,R_4) := (X-x_j Z)^2 + b (Y-y_jZ)^2 - R_j^2

and the projective complex space {{\bf CP}^6} is the space of all equivalence classes {[X,Y,Z,R_1,R_2,R_3,R_4]} of tuples {(X,Y,Z,R_1,R_2,R_3,R_4) \in {\bf C}^7 \backslash \{0\}} up to projective equivalence {(\lambda X, \lambda Y, \lambda Z, \lambda R_1, \lambda R_2, \lambda R_3, \lambda R_4) \sim (X,Y,Z,R_1,R_2,R_3,R_4)}. By identifying the affine point {(x,y,r_1,r_2,r_3,r_4)} with the projective point {(X,Y,1,R_1,R_2,R_3,R_4)}, we see that {\overline{V}} consists of the affine variety {V} together with the set {\{ [X,Y,0,R_1,R_2,R_3,R_4]: X^2+bY^2=R^2; R_j = \pm R_1 \hbox{ for } j=2,3,4\}}, which is the union of eight curves, each of which lies in the closure of {V}. Thus {\overline{V}} is the projective closure of {V}, and is thus a complex irreducible projective surface, defined over the rationals. As {\overline{V}} is cut out by four quadric equations in {{\bf CP}^6} and has degree sixteen (as can be seen for instance by inspecting the intersection of {\overline{V}} with a generic perturbation of a fibre over the generically defined projection {[X,Y,Z,R_1,R_2,R_3,R_4] \mapsto [X,Y,Z]}), it is also a complete intersection. To show (3), it then suffices to show that the rational points in {\overline{V}} are not Zariski dense in {\overline{V}}.

Heuristically, the reason why we expect few rational points in {\overline{V}} is as follows. First observe from the projective nature of (1) that every rational point is equivalent to an integer point. But for a septuple {(X,Y,Z,R_1,R_2,R_3,R_4)} of integers of size {O(N)}, the quantity {Q(X,Y,Z,R_1,R_2,R_3,R_4)} is an integer point of {{\bf Z}^4} of size {O(N^2)}, and so should only vanish about {O(N^{-8})} of the time. Hence the number of integer points {(X,Y,Z,R_1,R_2,R_3,R_4) \in {\bf Z}^7} of height comparable to {N} should be about

\displaystyle O(N)^7 \times O(N^{-8}) = O(N^{-1});

this is a convergent sum if {N} ranges over (say) powers of two, and so from standard probabilistic heuristics (see this previous post) we in fact expect only finitely many solutions, in the absence of any special algebraic structure (e.g. the structure of an abelian variety, or a birational reduction to a simpler variety) that could produce an unusually large number of solutions.

The Bombieri-Lang conjecture, Conjecture 2, can be viewed as a formalisation of the above heuristics (roughly speaking, it is one of the most optimistic natural conjectures one could make that is compatible with these heuristics while also being invariant under birational equivalence).

Unfortunately, {\overline{V}} contains some singular points. Being a complete intersection, this occurs when the Jacobian matrix of the map {Q: {\bf C}^7 \rightarrow {\bf C}^4} has less than full rank, or equivalently that the gradient vectors

\displaystyle \nabla Q_j = (2(X-x_j Z), 2(Y-y_j Z), -2x_j (X-x_j Z) - 2y_j (Y-y_j Z), \ \ \ \ \ (2)


\displaystyle 0, \dots, 0, -2R_j, 0, \dots, 0)

for {j=1,\dots,4} are linearly dependent, where the {-2R_j} is in the coordinate position associated to {R_j}. One way in which this can occur is if one of the gradient vectors {\nabla Q_j} vanish identically. This occurs at precisely {4 \times 2^3 = 32} points, when {[X,Y,Z]} is equal to {[x_j,y_j,1]} for some {j=1,\dots,4}, and one has {R_k = \pm ( (x_j - x_k)^2 + b (y_j - y_k)^2 )^{1/2}} for all {k=1,\dots,4} (so in particular {R_j=0}). Let us refer to these as the obvious singularities; they arise from the geometrically evident fact that the distance function {(x,y\sqrt{b}) \mapsto \sqrt{(x-x_j)^2 + b(y-y_j)^2}} is singular at {(x_j,y_j\sqrt{b})}.

The other way in which could occur is if a non-trivial linear combination of at least two of the gradient vectors vanishes. From (2), this can only occur if {R_j=R_k=0} for some distinct {j,k}, which from (1) implies that

\displaystyle (X - x_j Z) = \pm \sqrt{b} i (Y - y_j Z) \ \ \ \ \ (3)



\displaystyle (X - x_k Z) = \pm \sqrt{b} i (Y - y_k Z) \ \ \ \ \ (4)


for two choices of sign {\pm}. If the signs are equal, then (as {x_j, y_j, x_k, y_k} are in general position) this implies that {Z=0}, and then we have the singular point

\displaystyle [X,Y,Z,R_1,R_2,R_3,R_4] = [\pm \sqrt{b} i, 1, 0, 0, 0, 0, 0]. \ \ \ \ \ (5)


If the non-trivial linear combination involved three or more gradient vectors, then by the pigeonhole principle at least two of the signs involved must be equal, and so the only singular points are (5). So the only remaining possibility is when we have two gradient vectors {\nabla Q_j, \nabla Q_k} that are parallel but non-zero, with the signs in (3), (4) opposing. But then (as {x_j,y_j,x_k,y_k} are in general position) the vectors {(X-x_j Z, Y-y_j Z), (X-x_k Z, Y-y_k Z)} are non-zero and non-parallel to each other, a contradiction. Thus, outside of the {32} obvious singular points mentioned earlier, the only other singular points are the two points (5).

We will shortly show that the {32} obvious singularities are ordinary double points; the surface {\overline{V}} near any of these points is analytically equivalent to an ordinary cone {\{ (x,y,z) \in {\bf C}^3: z^2 = x^2 + y^2 \}} near the origin, which is a cone over a smooth conic curve {\{ (x,y) \in {\bf C}^2: x^2+y^2=1\}}. The two non-obvious singularities (5) are slightly more complicated than ordinary double points, they are elliptic singularities, which approximately resemble a cone over an elliptic curve. (As far as I can tell, this resemblance is exact in the category of real smooth manifolds, but not in the category of algebraic varieties.) If one blows up each of the point singularities of {\overline{V}} separately, no further singularities are created, and one obtains a smooth projective surface {X} (using the Segre embedding as necessary to embed {X} back into projective space, rather than in a product of projective spaces). Away from the singularities, the rational points of {\overline{V}} lift up to rational points of {X}. Assuming the Bombieri-Lang conjecture, we thus are able to answer the Erdös-Ulam problem in the affirmative once we establish

Theorem 4 The blowup {X} of {\overline{V}} is of general type.

This will be done below the fold, by the pedestrian device of explicitly constructing global differential forms on {X}; I will also be working from a complex analysis viewpoint rather than an algebraic geometry viewpoint as I am more comfortable with the former approach. (As mentioned above, though, there may well be a quicker way to establish this result by using more sophisticated machinery.)

I thank Mark Green and David Gieseker for helpful conversations (and a crash course in varieties of general type!).

Remark 5 The above argument shows in fact (assuming Bombieri-Lang) that sets {S \subset {\bf R}^2} with all distances rational cannot be Zariski-dense, and thus (by Solymosi-de Zeeuw) must lie on a single line or circle with only finitely many exceptions. Assuming a stronger version of Bombieri-Lang involving a general number field {K}, we obtain a similar conclusion with “rational” replaced by “lying in {K}” (one has to extend the Solymosi-de Zeeuw analysis to more general number fields, but this should be routine, using the analogue of Faltings’ theorem for such number fields).

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