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There are a number of ways to construct the real numbers {{\bf R}}, for instance

  • as the metric completion of {{\bf Q}} (thus, {{\bf R}} is defined as the set of Cauchy sequences of rationals, modulo Cauchy equivalence);
  • as the space of Dedekind cuts on the rationals {{\bf Q}};
  • as the space of quasimorphisms {\phi: {\bf Z} \rightarrow {\bf Z}} on the integers, quotiented by bounded functions. (I believe this construction first appears in this paper of Street, who credits the idea to Schanuel, though the germ of this construction arguably goes all the way back to Eudoxus.)

There is also a fourth family of constructions that proceeds via nonstandard analysis, as a special case of what is known as the nonstandard hull construction. (Here I will assume some basic familiarity with nonstandard analysis and ultraproducts, as covered for instance in this previous blog post.) Given an unbounded nonstandard natural number {N \in {}^* {\bf N} \backslash {\bf N}}, one can define two external additive subgroups of the nonstandard integers {{}^* {\bf Z}}:

  • The group {O(N) := \{ n \in {}^* {\bf Z}: |n| \leq CN \hbox{ for some } C \in {\bf N} \}} of all nonstandard integers of magnitude less than or comparable to {N}; and
  • The group {o(N) := \{ n \in {}^* {\bf Z}: |n| \leq C^{-1} N \hbox{ for all } C \in {\bf N} \}} of nonstandard integers of magnitude infinitesimally smaller than {N}.

The group {o(N)} is a subgroup of {O(N)}, so we may form the quotient group {O(N)/o(N)}. This space is isomorphic to the reals {{\bf R}}, and can in fact be used to construct the reals:

Proposition 1 For any coset {n + o(N)} of {O(N)/o(N)}, there is a unique real number {\hbox{st} \frac{n}{N}} with the property that {\frac{n}{N} = \hbox{st} \frac{n}{N} + o(1)}. The map {n + o(N) \mapsto \hbox{st} \frac{n}{N}} is then an isomorphism between the additive groups {O(N)/o(N)} and {{\bf R}}.

Proof: Uniqueness is clear. For existence, observe that the set {\{ x \in {\bf R}: Nx \leq n + o(N) \}} is a Dedekind cut, and its supremum can be verified to have the required properties for {\hbox{st} \frac{n}{N}}. \Box

In a similar vein, we can view the unit interval {[0,1]} in the reals as the quotient

\displaystyle  [0,1] \equiv [N] / o(N) \ \ \ \ \ (1)

where {[N]} is the nonstandard (i.e. internal) set {\{ n \in {\bf N}: n \leq N \}}; of course, {[N]} is not a group, so one should interpret {[N]/o(N)} as the image of {[N]} under the quotient map {{}^* {\bf Z} \rightarrow {}^* {\bf Z} / o(N)} (or {O(N) \rightarrow O(N)/o(N)}, if one prefers). Or to put it another way, (1) asserts that {[0,1]} is the image of {[N]} with respect to the map {\pi: n \mapsto \hbox{st} \frac{n}{N}}.

In this post I would like to record a nice measure-theoretic version of the equivalence (1), which essentially appears already in standard texts on Loeb measure (see e.g. this text of Cutland). To describe the results, we must first quickly recall the construction of Loeb measure on {[N]}. Given an internal subset {A} of {[N]}, we may define the elementary measure {\mu_0(A)} of {A} by the formula

\displaystyle  \mu_0(A) := \hbox{st} \frac{|A|}{N}.

This is a finitely additive probability measure on the Boolean algebra of internal subsets of {[N]}. We can then construct the Loeb outer measure {\mu^*(A)} of any subset {A \subset [N]} in complete analogy with Lebesgue outer measure by the formula

\displaystyle  \mu^*(A) := \inf \sum_{n=1}^\infty \mu_0(A_n)

where {(A_n)_{n=1}^\infty} ranges over all sequences of internal subsets of {[N]} that cover {A}. We say that a subset {A} of {[N]} is Loeb measurable if, for any (standard) {\epsilon>0}, one can find an internal subset {B} of {[N]} which differs from {A} by a set of Loeb outer measure at most {\epsilon}, and in that case we define the Loeb measure {\mu(A)} of {A} to be {\mu^*(A)}. It is a routine matter to show (e.g. using the Carathéodory extension theorem) that the space {{\mathcal L}} of Loeb measurable sets is a {\sigma}-algebra, and that {\mu} is a countably additive probability measure on this space that extends the elementary measure {\mu_0}. Thus {[N]} now has the structure of a probability space {([N], {\mathcal L}, \mu)}.

Now, the group {o(N)} acts (Loeb-almost everywhere) on the probability space {[N]} by the addition map, thus {T^h n := n+h} for {n \in [N]} and {h \in o(N)} (excluding a set of Loeb measure zero where {n+h} exits {[N]}). This action is clearly seen to be measure-preserving. As such, we can form the invariant factor {Z^0_{o(N)}([N]) = ([N], {\mathcal L}^{o(N)}, \mu\downharpoonright_{{\mathcal L}^{o(N)}})}, defined by restricting attention to those Loeb measurable sets {A \subset [N]} with the property that {T^h A} is equal {\mu}-almost everywhere to {A} for each {h \in o(N)}.

The claim is then that this invariant factor is equivalent (up to almost everywhere equivalence) to the unit interval {[0,1]} with Lebesgue measure {m} (and the trivial action of {o(N)}), by the same factor map {\pi: n \mapsto \hbox{st} \frac{n}{N}} used in (1). More precisely:

Theorem 2 Given a set {A \in {\mathcal L}^{o(N)}}, there exists a Lebesgue measurable set {B \subset [0,1]}, unique up to {m}-a.e. equivalence, such that {A} is {\mu}-a.e. equivalent to the set {\pi^{-1}(B) := \{ n \in [N]: \hbox{st} \frac{n}{N} \in B \}}. Conversely, if {B \in [0,1]} is Lebesgue measurable, then {\pi^{-1}(B)} is in {{\mathcal L}^{o(N)}}, and {\mu( \pi^{-1}(B) ) = m( B )}.

More informally, we have the measure-theoretic version

\displaystyle  [0,1] \equiv Z^0_{o(N)}( [N] )

of (1).

Proof: We first prove the converse. It is clear that {\pi^{-1}(B)} is {o(N)}-invariant, so it suffices to show that {\pi^{-1}(B)} is Loeb measurable with Loeb measure {m(B)}. This is easily verified when {B} is an elementary set (a finite union of intervals). By countable subadditivity of outer measure, this implies that Loeb outer measure of {\pi^{-1}(E)} is bounded by the Lebesgue outer measure of {E} for any set {E \subset [0,1]}; since every Lebesgue measurable set differs from an elementary set by a set of arbitrarily small Lebesgue outer measure, the claim follows.

Now we establish the forward claim. Uniqueness is clear from the converse claim, so it suffices to show existence. Let {A \in {\mathcal L}^{o(N)}}. Let {\epsilon>0} be an arbitrary standard real number, then we can find an internal set {A_\epsilon \subset [N]} which differs from {A} by a set of Loeb measure at most {\epsilon}. As {A} is {o(N)}-invariant, we conclude that for every {h \in o(N)}, {A_\epsilon} and {T^h A_\epsilon} differ by a set of Loeb measure (and hence elementary measure) at most {2\epsilon}. By the (contrapositive of the) underspill principle, there must exist a standard {\delta>0} such that {A_\epsilon} and {T^h A_\epsilon} differ by a set of elementary measure at most {2\epsilon} for all {|h| \leq \delta N}. If we then define the nonstandard function {f_\epsilon: [N] \rightarrow {}^* {\bf R}} by the formula

\displaystyle  f(n) := \hbox{st} \frac{1}{\delta N} \sum_{m \in [N]: m \leq n \leq m+\delta N} 1_{A_\epsilon}(m),

then from the (nonstandard) triangle inequality we have

\displaystyle  \frac{1}{N} \sum_{n \in [N]} |f(n) - 1_{A_\epsilon}(n)| \leq 3\epsilon

(say). On the other hand, {f} has the Lipschitz continuity property

\displaystyle  |f(n)-f(m)| \leq \frac{2|n-m|}{\delta N}

and so in particular we see that

\displaystyle  \hbox{st} f(n) = \tilde f( \hbox{st} \frac{n}{N} )

for some Lipschitz continuous function {\tilde f: [0,1] \rightarrow [0,1]}. If we then let {E_\epsilon} be the set where {\tilde f \geq 1 - \sqrt{\epsilon}}, one can check that {A_\epsilon} differs from {\pi^{-1}(E_\epsilon)} by a set of Loeb outer measure {O(\sqrt{\epsilon})}, and hence {A} does so also. Sending {\epsilon} to zero, we see (from the converse claim) that {1_{E_\epsilon}} is a Cauchy sequence in {L^1} and thus converges in {L^1} for some Lebesgue measurable {E}. The sets {A_\epsilon} then converge in Loeb outer measure to {\pi^{-1}(E)}, giving the claim. \Box

Thanks to the Lebesgue differentiation theorem, the conditional expectation {{\bf E}( f | Z^0_{o(N)}([N]))} of a bounded Loeb-measurable function {f: [N] \rightarrow {\bf R}} can be expressed (as a function on {[0,1]}, defined {m}-a.e.) as

\displaystyle  {\bf E}( f | Z^0_{o(N)}([N]))(x) := \lim_{\epsilon \rightarrow 0} \frac{1}{2\epsilon} \int_{[x-\epsilon N,x+\epsilon N]} f\ d\mu.

By the abstract ergodic theorem from the previous post, one can also view this conditional expectation as the element in the closed convex hull of the shifts {T^h f}, {h = o(N)} of minimal {L^2} norm. In particular, we obtain a form of the von Neumann ergodic theorem in this context: the averages {\frac{1}{H} \sum_{h=1}^H T^h f} for {H=O(N)} converge (as a net, rather than a sequence) in {L^2} to {{\bf E}( f | Z^0_{o(N)}([N]))}.

If {f: [N] \rightarrow [-1,1]} is (the standard part of) an internal function, that is to say the ultralimit of a sequence {f_n: [N_n] \rightarrow [-1,1]} of finitary bounded functions, one can view the measurable function {F := {\bf E}( f | Z^0_{o(N)}([N]))} as a limit of the {f_n} that is analogous to the “graphons” that emerge as limits of graphs (see e.g. the recent text of Lovasz on graph limits). Indeed, the measurable function {F: [0,1] \rightarrow [-1,1]} is related to the discrete functions {f_n: [N_n] \rightarrow [-1,1]} by the formula

\displaystyle  \int_a^b F(x)\ dx = \hbox{st} \lim_{n \rightarrow p} \frac{1}{N_n} \sum_{a N_n \leq m \leq b N_n} f_n(m)

for all {0 \leq a < b \leq 1}, where {p} is the nonprincipal ultrafilter used to define the nonstandard universe. In particular, from the Arzela-Ascoli diagonalisation argument there is a subsequence {n_j} such that

\displaystyle  \int_a^b F(x)\ dx = \lim_{j \rightarrow \infty} \frac{1}{N_{n_j}} \sum_{a N_{n_j} \leq m \leq b N_{n_j}} f_n(m),

thus {F} is the asymptotic density function of the {f_n}. For instance, if {f_n} is the indicator function of a randomly chosen subset of {[N_n]}, then the asymptotic density function would equal {1/2} (almost everywhere, at least).

I’m continuing to look into understanding the ergodic theory of {o(N)} actions, as I believe this may allow one to apply ergodic theory methods to the “single-scale” or “non-asymptotic” setting (in which one averages only over scales comparable to a large parameter {N}, rather than the traditional asymptotic approach of letting the scale go to infinity). I’m planning some further posts in this direction, though this is still a work in progress.

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 {g: H \rightarrow H}, and {v \in H} is a vector in that Hilbert space, then one has

\displaystyle  \lim_{N \rightarrow \infty} \frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv = \pi_{H^G} v \ \ \ \ \ (1)

for any Folner sequence {\Phi_N} of {G}, where {H^G := \{ w \in H: gw = w \hbox{ for all }g \in G \}} is the {G}-invariant subspace. Thus one can interpret {\pi_{H^G} v} as a certain average of elements of the orbit {Gv := \{ gv: g \in G \}} of {v}.

I recently discovered that there is a simple variant of this ergodic theorem 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 := \{ gv: g \in G \}} of minimal norm, and is also the unique element of {H^G} in this closed convex hull.

Proof: As the closed convex hull of {Gv} is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element {F} of minimal norm. If {T_g F \neq F} for some {g}, then the midpoint of {T_g F} and {F} would be in the closed convex hull and be of smaller norm, a contradiction; thus {F} is {G}-invariant. To finish the first claim, it suffices to show that {v-F} is orthogonal to every element {h} of {H^G}. But if this were not the case for some such {h}, we would have {\langle T_g v - F, h \rangle = \langle v-F,h\rangle \neq 0} for all {g \in G}, and thus on taking convex hulls {\langle F-F,h\rangle = \langle f-F,f\rangle \neq 0}, a contradiction.

Finally, since {T_g v - F} is orthogonal to {H^G}, the same is true for {F'-F} for any {F'} in the closed convex hull of {Gv}, and this gives the second claim. \Box

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any {\epsilon>0}, Theorem 1 implies that there is a finite convex combination {v_\epsilon} of shifts {gv} of {v} which lies within {\epsilon} (in the {H} norm) to {\pi_{H^G} v}. By the triangle inequality, all the averages {\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv_\epsilon} also lie within {\epsilon} of {\pi_{H^G} v}, but by the Folner property this implies that the averages {\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv} are eventually within {2\epsilon} (say) of {\pi_{H^G} v}, giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality) Let {G} be a group acting unitarily on a Hilbert space {H}, and let {V} be a {G}-invariant subspace of {H}. Then {V} and {H^G} are relatively orthogonal over their common subspace {V^G}, that is to say the restrictions of {V} and {H^G} to the orthogonal complement of {V^G} are orthogonal to each other.

Proof: By Theorem 1, we have {\pi_{H^G} v = \pi_{V^G} v} for all {v \in V}, and the claim follows. (Thanks to Gergely Harcos for this short argument.) \Box

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups {G}. Define a {G}-system {(X, {\mathcal X}, \mu, (T_g)_{g \in G})} to be a probability space {X = (X, {\mathcal X}, \mu)} together with a measure-preserving action {(T_g)_{g \in G}} of {G} on {X}; this gives an action of {G} on {L^2(X) = L^2(X,{\mathcal X},\mu)}, which by abuse of notation we also call {T_g}:

\displaystyle  T_g f := f \circ T_{g^{-1}}.

(In this post we follow the usual convention of defining the {L^p} spaces by quotienting out by almost everywhere equivalence.) We say that a {G}-system is ergodic if {L^2(X)^G} consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space {X} and quotienting {{\mathcal X}} by the {\sigma}-ideal of null sets, and considering maps such as {T_g} only on this quotient {\sigma}-algebra (or on the associated von Neumann algebra {L^\infty(X)} or Hilbert space {L^2(X)}). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A factor {Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})} of a {G}-system {X = (X,{\mathcal X},\mu, (T_g)_{g \in G})} is another {G}-system together with a factor map {\pi: X \rightarrow Y} which commutes with the {G}-action (thus {T_g \pi = \pi S_g} for all {g \in G}) and respects the measure in the sense that {\mu(\pi^{-1}(E)) = \nu(E)} for all {E \in {\mathcal Y}}. For instance, the {G}-invariant factor {Z^0_G(X) := (X, {\mathcal X}^G, \mu\downharpoonright_{{\mathcal X}^G}, (T_g)_{g \in G})}, formed by restricting {X} to the invariant algebra {{\mathcal X}^G := \{ E \in {\mathcal X}: T_g E = E \hbox{ a.e. for all } g \in G \}}, is a factor of {X}. (This factor is the first factor in an important hierachy, the next element of which is the Kronecker factor {Z^1_G(X)}, but we will not discuss higher elements of this hierarchy further here.) If {Y} is a factor of {X}, we refer to {X} as an extension of {Y}.

From Corollary 2 we have

Corollary 3 (Relative independence) Let {X} be a {G}-system for a group {G}, and let {Y} be a factor of {X}. Then {Y} and {Z^0_G(X)} are relatively independent over their common factor {Z^0_G(Y)}, in the sense that the spaces {L^2(Y)} and {L^2(Z^0_G(X))} are relatively orthogonal over {L^2(Z^0_G(Y))} when all these spaces are embedded into {L^2(X)}.

This has a simple consequence regarding the product {X \times Y = (X \times Y, {\mathcal X} \times {\mathcal Y}, \mu \times \nu, (T_g \oplus S_g)_{g \in G})} of two {G}-systems {X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})} and {Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}, in the case when the {Y} action is trivial:

Lemma 4 If {X,Y} are two {G}-systems, with the action of {G} on {Y} trivial, then {Z^0_G(X \times Y)} is isomorphic to {Z^0_G(X) \times Y} in the obvious fashion.

This lemma is immediate for countable {G}, since for a {G}-invariant function {f}, one can ensure that {T_g f = f} holds simultaneously for all {g \in G} outside of a null set, but is a little trickier for uncountable {G}.

Proof: It is clear that {Z^0_G(X) \times Y} is a factor of {Z^0_G(X \times Y)}. To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero {f \in L^2(Z^0_G(X \times Y))} which is orthogonal to {L^2(Z^0_G(X) \times Y)}. In particular, we have {fg} orthogonal to {L^2(Z^0_G(X))} for any {g \in L^\infty(Y)}. Since {fg} lies in {L^2(Z^0_G(X \times Y))}, we conclude from Corollary 3 (viewing {X} as a factor of {X \times Y}) that {fg} is also orthogonal to {L^2(X)}. Since {g} is an arbitrary element of {L^\infty(Y)}, we conclude that {f} is orthogonal to {L^2(X \times Y)} and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.) \Box

Now we discuss the notion of a group extension.

Definition 5 (Group extension) Let {G} be an arbitrary group, let {Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})} be a {G}-system, and let {K} be a compact metrisable group. A {K}-extension of {Y} is an extension {X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})} whose underlying space is {X = Y \times K} (with {{\mathcal X}} the product of {{\mathcal Y}} and the Borel {\sigma}-algebra on {K}), the factor map is {\pi: (y,k) \mapsto y}, and the shift maps {T_g} are given by

\displaystyle  T_g ( y, k ) = (S_g y, \rho_g(y) k )

where for each {g \in G}, {\rho_g: Y \rightarrow K} is a measurable map (known as the cocycle associated to the {K}-extension {X}).

An important special case of a {K}-extension arises when the measure {\mu} is the product of {\nu} with the Haar measure {dk} on {K}. In this case, {X} also has a {K}-action {k': (y,k) \mapsto (y,k(k')^{-1})} that commutes with the {G}-action, making {X} a {G \times K}-system. More generally, {\mu} could be the product of {\nu} with the Haar measure {dh} of some closed subgroup {H} of {K}, with {\rho_g} taking values in {H}; then {X} is now a {G \times H} system. In this latter case we will call {X} {H}-uniform.

If {X} is a {K}-extension of {Y} and {U: Y \rightarrow K} is a measurable map, we can define the gauge transform {X_U} of {X} to be the {K}-extension of {Y} whose measure {\mu_U} is the pushforward of {\mu} under the map {(y,k) \mapsto (y, U(y) k)}, and whose cocycles {\rho_{g,U}: Y \rightarrow K} are given by the formula

\displaystyle  \rho_{g,U}(y) := U(gy) \rho_g(y) U(y)^{-1}.

It is easy to see that {X_U} is a {K}-extension that is isomorphic to {X} as a {K}-extension of {Y}; we will refer to {X_U} and {X} as equivalent systems, and {\rho_{g,U}} as cohomologous to {\rho_g}. We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem) Let {G} be an arbitrary group, let {Y} be an ergodic {G}-system, and let {K} be a compact metrisable group. Then every ergodic {K}-extension {X} of {Y} is equivalent to an {H}-uniform extension of {Y} for some closed subgroup {H} of {K}.

This theorem is usually stated for amenable groups {G}, but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, {X} and {Y} are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions via the ergodic decomposition, but the result becomes more complicated to state, and we will not do so here.

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This should be the final thread (for now, at least) for the Polymath8 project (encompassing the original Polymath8a paper, the nearly finished Polymath8b paper, and the retrospective paper), superseding the previous Polymath8b thread (which was quite full) and the Polymath8a/retrospective thread (which was more or less inactive).

On Polymath8a: I talked briefly with Andrew Granville, who is handling the paper for Algebra & Number Theory, and he said that a referee report should be coming in soon.  Apparently length of the paper is a bit of an issue (not surprising, as it is 163 pages long) and there will be some suggestions to trim the size down a bit.

In view of the length issue at A&NT, I’m now leaning towards taking up Ken Ono’s offer to submit the Polymath8b paper to the new open access journal “Research in the Mathematical Sciences“.  I think the paper is almost ready to be submitted (after the current participants sign off on it, of course), but it might be worth waiting on the Polymath8a referee report in case the changes suggested impact the 8b paper.

Finally, it is perhaps time to start working on the retrospective article, and collect some impressions from participants.  I wrote up a quick draft of my own experiences, and also pasted in Pace Nielsen’s thoughts, as well as a contribution from an undergraduate following the project (Andrew Gibson).  Hopefully we can collect a few more (either through comments on this blog, through email, or through Dropbox), and then start working on editing them together and finding some suitable concluding points to make about the Polymath8 project, and what lessons we can take from it for future projects of this type.

Given two unit vectors {v,w} in a real inner product space, one can define the correlation between these vectors to be their inner product {\langle v, w \rangle}, or in more geometric terms, the cosine of the angle {\angle(v,w)} subtended by {v} and {w}. By the Cauchy-Schwarz inequality, this is a quantity between {-1} and {+1}, with the extreme positive correlation {+1} occurring when {v,w} are identical, the extreme negative correlation {-1} occurring when {v,w} are diametrically opposite, and the zero correlation {0} occurring when {v,w} are orthogonal. This notion is closely related to the notion of correlation between two non-constant square-integrable real-valued random variables {X,Y}, which is the same as the correlation between two unit vectors {v,w} lying in the Hilbert space {L^2(\Omega)} of square-integrable random variables, with {v} being the normalisation of {X} defined by subtracting off the mean {\mathbf{E} X} and then dividing by the standard deviation of {X}, and similarly for {w} and {Y}.

One can also define correlation for complex (Hermitian) inner product spaces by taking the real part {\hbox{Re} \langle , \rangle} of the complex inner product to recover a real inner product.

While reading the (highly recommended) recent popular maths book “How not to be wrong“, by my friend and co-author Jordan Ellenberg, I came across the (important) point that correlation is not necessarily transitive: if {X} correlates with {Y}, and {Y} correlates with {Z}, then this does not imply that {X} correlates with {Z}. A simple geometric example is provided by the three unit vectors

\displaystyle  u := (1,0); v := (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}); w := (0,1)

in the Euclidean plane {{\bf R}^2}: {u} and {v} have a positive correlation of {\frac{1}{\sqrt{2}}}, as does {v} and {w}, but {u} and {w} are not correlated with each other. Or: for a typical undergraduate course, it is generally true that good exam scores are correlated with a deep understanding of the course material, and memorising from flash cards are correlated with good exam scores, but this does not imply that memorising flash cards is correlated with deep understanding of the course material.

However, there are at least two situations in which some partial version of transitivity of correlation can be recovered. The first is in the “99%” regime in which the correlations are very close to {1}: if {u,v,w} are unit vectors such that {u} is very highly correlated with {v}, and {v} is very highly correlated with {w}, then this does imply that {u} is very highly correlated with {w}. Indeed, from the identity

\displaystyle  \| u-v \| = 2^{1/2} (1 - \langle u,v\rangle)^{1/2}

(and similarly for {v-w} and {u-w}) and the triangle inequality

\displaystyle  \|u-w\| \leq \|u-v\| + \|v-w\|,

we see that

\displaystyle  (1 - \langle u,w \rangle)^{1/2} \leq (1 - \langle u,v\rangle)^{1/2} + (1 - \langle v,w\rangle)^{1/2}. \ \ \ \ \ (1)

Thus, for instance, if {\langle u, v \rangle \geq 1-\epsilon} and {\langle v,w \rangle \geq 1-\epsilon}, then {\langle u,w \rangle \geq 1-4\epsilon}. This is of course closely related to (though slightly weaker than) the triangle inequality for angles:

\displaystyle  \angle(u,w) \leq \angle(u,v) + \angle(v,w).

Remark 1 (Thanks to Andrew Granville for conversations leading to this observation.) The inequality (1) also holds for sub-unit vectors, i.e. vectors {u,v,w} with {\|u\|, \|v\|, \|w\| \leq 1}. This comes by extending {u,v,w} in directions orthogonal to all three original vectors and to each other in order to make them unit vectors, enlarging the ambient Hilbert space {H} if necessary. More concretely, one can apply (1) to the unit vectors

\displaystyle  (u, \sqrt{1-\|u\|^2}, 0, 0), (v, 0, \sqrt{1-\|v\|^2}, 0), (w, 0, 0, \sqrt{1-\|w\|^2})

in {H \times {\bf R}^3}.

But even in the “{1\%}” regime in which correlations are very weak, there is still a version of transitivity of correlation, known as the van der Corput lemma, which basically asserts that if a unit vector {v} is correlated with many unit vectors {u_1,\dots,u_n}, then many of the pairs {u_i,u_j} will then be correlated with each other. Indeed, from the Cauchy-Schwarz inequality

\displaystyle  |\langle v, \sum_{i=1}^n u_i \rangle|^2 \leq \|v\|^2 \| \sum_{i=1}^n u_i \|^2

we see that

\displaystyle  (\sum_{i=1}^n \langle v, u_i \rangle)^2 \leq \sum_{1 \leq i,j \leq n} \langle u_i, u_j \rangle. \ \ \ \ \ (2)

Thus, for instance, if {\langle v, u_i \rangle \geq \epsilon} for at least {\epsilon n} values of {i=1,\dots,n}, then {\sum_{1 \leq i,j \leq n} \langle u_i, u_j \rangle} must be at least {\epsilon^3 n^2}, which implies that {\langle u_i, u_j \rangle \geq \epsilon^3/2} for at least {\epsilon^3 n^2/2} pairs {(i,j)}. Or as another example: if a random variable {X} exhibits at least {1\%} positive correlation with {n} other random variables {Y_1,\dots,Y_n}, then if {n > 10,000}, at least two distinct {Y_i,Y_j} must have positive correlation with each other (although this argument does not tell you which pair {Y_i,Y_j} are so correlated). Thus one can view this inequality as a sort of `pigeonhole principle” for correlation.

A similar argument (multiplying each {u_i} by an appropriate sign {\pm 1}) shows the related van der Corput inequality

\displaystyle  (\sum_{i=1}^n |\langle v, u_i \rangle|)^2 \leq \sum_{1 \leq i,j \leq n} |\langle u_i, u_j \rangle|, \ \ \ \ \ (3)

and this inequality is also true for complex inner product spaces. (Also, the {u_i} do not need to be unit vectors for this inequality to hold.)

Geometrically, the picture is this: if {v} positively correlates with all of the {u_1,\dots,u_n}, then the {u_1,\dots,u_n} are all squashed into a somewhat narrow cone centred at {v}. The cone is still wide enough to allow a few pairs {u_i, u_j} to be orthogonal (or even negatively correlated) with each other, but (when {n} is large enough) it is not wide enough to allow all of the {u_i,u_j} to be so widely separated. Remarkably, the bound here does not depend on the dimension of the ambient inner product space; while increasing the number of dimensions should in principle add more “room” to the cone, this effect is counteracted by the fact that in high dimensions, almost all pairs of vectors are close to orthogonal, and the exceptional pairs that are even weakly correlated to each other become exponentially rare. (See this previous blog post for some related discussion; in particular, Lemma 2 from that post is closely related to the van der Corput inequality presented here.)

A particularly common special case of the van der Corput inequality arises when {v} is a unit vector fixed by some unitary operator {T}, and the {u_i} are shifts {u_i = T^i u} of a single unit vector {u}. In this case, the inner products {\langle v, u_i \rangle} are all equal, and we arrive at the useful van der Corput inequality

\displaystyle  |\langle v, u \rangle|^2 \leq \frac{1}{n^2} \sum_{1 \leq i,j \leq n} |\langle T^i u, T^j u \rangle|. \ \ \ \ \ (4)

(In fact, one can even remove the absolute values from the right-hand side, by using (2) instead of (4).) Thus, to show that {v} has negligible correlation with {u}, it suffices to show that the shifts of {u} have negligible correlation with each other.

Here is a basic application of the van der Corput inequality:

Proposition 1 (Weyl equidistribution estimate) Let {P: {\bf Z} \rightarrow {\bf R}/{\bf Z}} be a polynomial with at least one non-constant coefficient irrational. Then one has

\displaystyle  \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e( P(n) ) = 0,

where {e(x) := e^{2\pi i x}}.

Note that this assertion implies the more general assertion

\displaystyle  \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e( kP(n) ) = 0

for any non-zero integer {k} (simply by replacing {P} by {kP}), which by the Weyl equidistribution criterion is equivalent to the sequence {P(1), P(2),\dots} being asymptotically equidistributed in {{\bf R}/{\bf Z}}.

Proof: We induct on the degree {d} of the polynomial {P}, which must be at least one. If {d} is equal to one, the claim is easily established from the geometric series formula, so suppose that {d>1} and that the claim has already been proven for {d-1}. If the top coefficient {a_d} of {P(n) = a_d n^d + \dots + a_0} is rational, say {a_d = \frac{p}{q}}, then by partitioning the natural numbers into residue classes modulo {q}, we see that the claim follows from the induction hypothesis; so we may assume that the top coefficient {a_d} is irrational.

In order to use the van der Corput inequality as stated above (i.e. in the formalism of inner product spaces) we will need a non-principal ultrafilter {p} (see e.g this previous blog post for basic theory of ultrafilters); we leave it as an exercise to the reader to figure out how to present the argument below without the use of ultrafilters (or similar devices, such as Banach limits). The ultrafilter {p} defines an inner product {\langle, \rangle_p} on bounded complex sequences {z = (z_1,z_2,z_3,\dots)} by setting

\displaystyle  \langle z, w \rangle_p := \hbox{st} \lim_{N \rightarrow p} \frac{1}{N} \sum_{n=1}^N z_n \overline{w_n}.

Strictly speaking, this inner product is only positive semi-definite rather than positive definite, but one can quotient out by the null vectors to obtain a positive-definite inner product. To establish the claim, it will suffice to show that

\displaystyle  \langle 1, e(P) \rangle_p = 0

for every non-principal ultrafilter {p}.

Note that the space of bounded sequences (modulo null vectors) admits a shift {T}, defined by

\displaystyle  T (z_1,z_2,\dots) := (z_2,z_3,\dots).

This shift becomes unitary once we quotient out by null vectors, and the constant sequence {1} is clearly a unit vector that is invariant with respect to the shift. So by the van der Corput inequality, we have

\displaystyle  |\langle 1, e(P) \rangle_p| \leq \frac{1}{n^2} \sum_{1 \leq i,j \leq n} |\langle T^i e(P), T^j e(P) \rangle_p|

for any {n \geq 1}. But we may rewrite {\langle T^i e(P), T^j e(P) \rangle = \langle 1, e(T^j P - T^i P) \rangle_p}. Then observe that if {i \neq j}, {T^j P - T^i P} is a polynomial of degree {d-1} whose {d-1} coefficient is irrational, so by induction hypothesis we have {\langle T^i e(P), T^j e(P) \rangle_p = 0} for {i \neq j}. For {i=j} we of course have {\langle T^i e(P), T^j e(P) \rangle_p = 1}, and so

\displaystyle  |\langle 1, e(P) \rangle_p| \leq \frac{1}{n^2} \times n

for any {n}. Letting {n \rightarrow \infty}, we obtain the claim. \Box

This is the eleventh thread for the Polymath8b project to obtain new bounds for the quantity

H_m := \liminf_{n \to\infty} p_{n+m} - p_n;

the previous thread may be found here.

The main focus is now on writing up the results, with a draft paper close to completion here (with the directory of source files here).    Most of the sections are now written up more or less completely, with the exception of the appendix on narrow admissible tuples, which was awaiting the bounds on such tuples to stabilise.  There is now also an acknowledgments section (linking to the corresponding page on the wiki, which participants should check to see if their affiliations etc. are posted correctly), and in the final remarks section there is now also some discussion about potential improvements to the H_m bounds.  I’ve also added some mention of a recent paper of Banks, Freiberg and Maynard which makes use of some of our results (in particular, that M_{50,1/25} > 4).  On the other hand, the portions of the writeup relating to potential improvements to the MPZ estimates have been commented out, as it appears that one cannot easily obtain the exponential sum estimates required to make those go through.  (Perhaps, if there are significant new developments, one could incorporate them into a putative Polymath8c project, although at present I think there’s not much urgency to start over once again.)

Regarding the numerics in Section 7 of the paper, one thing which is missing at present is some links to code in case future readers wish to verify the results; alternatively one could include such code and data into the arXiv submission.

It’s about time to discuss possible journals to submit the paper to.  Ken Ono has invited us to submit to his new journal, “Research in the Mathematical Sciences“.  Another option would be to submit to the same journal “Algebra & Number Theory” that is currently handling our Polymath8a paper (no news on the submission there, but it is a very long paper), although I think the papers are independent enough that it is not absolutely necessary to place them in the same journal.  A third natural choice is “Mathematics of Computation“, though I should note that when the Polymath4 paper was submitted there, the editors required us to use our real names instead of the D.H.J. Polymath pseudonym as it would have messed up their metadata system otherwise.  (But I can check with the editor there before submitting to see if there is some workaround now, perhaps their policies have changed.)  At present I have no strong preferences regarding journal selection, and would welcome further thoughts and proposals.  (It is perhaps best to avoid the journals that I am editor or associate editor of, namely Amer. J. Math, Forum of Mathematics, Analysis & PDE, and Dynamics and PDE, due to conflict of interest (and in the latter two cases, specialisation to a different area of mathematics)).

 

 

 

 

Many fluid equations are expected to exhibit turbulence in their solutions, in which a significant portion of their energy ends up in high frequency modes. A typical example arises from the three-dimensional periodic Navier-Stokes equations

\displaystyle  \partial_t u + u \cdot \nabla u = \nu \Delta u + \nabla p + f

\displaystyle  \nabla \cdot u = 0

where {u: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3} is the velocity field, {f: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3} is a forcing term, {p: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}} is a pressure field, and {\nu > 0} is the viscosity. To study the dynamics of energy for this system, we first pass to the Fourier transform

\displaystyle  \hat u(t,k) := \int_{{\bf R}^3/{\bf Z}^3} u(t,x) e^{-2\pi i k \cdot x}

so that the system becomes

\displaystyle  \partial_t \hat u(t,k) + 2\pi \sum_{k = k_1 + k_2} (\hat u(t,k_1) \cdot ik_2) \hat u(t,k_2) =

\displaystyle  - 4\pi^2 \nu |k|^2 \hat u(t,k) + 2\pi ik \hat p(t,k) + \hat f(t,k) \ \ \ \ \ (1)

\displaystyle  k \cdot \hat u(t,k) = 0.

We may normalise {u} (and {f}) to have mean zero, so that {\hat u(t,0)=0}. Then we introduce the dyadic energies

\displaystyle  E_N(t) := \sum_{|k| \sim N} |\hat u(t,k)|^2

where {N \geq 1} ranges over the powers of two, and {|k| \sim N} is shorthand for {N \leq |k| < 2N}. Taking the inner product of (1) with {\hat u(t,k)}, we obtain the energy flow equation

\displaystyle  \partial_t E_N = \sum_{N_1,N_2} \Pi_{N,N_1,N_2} - D_N + F_N \ \ \ \ \ (2)

where {N_1,N_2} range over powers of two, {\Pi_{N,N_1,N_2}} is the energy flow rate

\displaystyle  \Pi_{N,N_1,N_2} := -2\pi \sum_{k=k_1+k_2: |k| \sim N, |k_1| \sim N_1, |k_2| \sim N_2}

\displaystyle  (\hat u(t,k_1) \cdot ik_2) (\hat u(t,k) \cdot \hat u(t,k_2)),

{D_N} is the energy dissipation rate

\displaystyle  D_N := 4\pi^2 \nu \sum_{|k| \sim N} |k|^2 |\hat u(t,k)|^2

and {F_N} is the energy injection rate

\displaystyle  F_N := \sum_{|k| \sim N} \hat u(t,k) \cdot \hat f(t,k).

The Navier-Stokes equations are notoriously difficult to solve in general. Despite this, Kolmogorov in 1941 was able to give a convincing heuristic argument for what the distribution of the dyadic energies {E_N} should become over long times, assuming that some sort of distributional steady state is reached. It is common to present this argument in the form of dimensional analysis, but one can also give a more “first principles” form Kolmogorov’s argument, which I will do here. Heuristically, one can divide the frequency scales {N} into three regimes:

  • The injection regime in which the energy injection rate {F_N} dominates the right-hand side of (2);
  • The energy flow regime in which the flow rates {\Pi_{N,N_1,N_2}} dominate the right-hand side of (2); and
  • The dissipation regime in which the dissipation {D_N} dominates the right-hand side of (2).

If we assume a fairly steady and smooth forcing term {f}, then {\hat f} will be supported on the low frequency modes {k=O(1)}, and so we heuristically expect the injection regime to consist of the low scales {N=O(1)}. Conversely, if we take the viscosity {\nu} to be small, we expect the dissipation regime to only occur for very large frequencies {N}, with the energy flow regime occupying the intermediate frequencies.

We can heuristically predict the dividing line between the energy flow regime. Of all the flow rates {\Pi_{N,N_1,N_2}}, it turns out in practice that the terms in which {N_1,N_2 = N+O(1)} (i.e., interactions between comparable scales, rather than widely separated scales) will dominate the other flow rates, so we will focus just on these terms. It is convenient to return back to physical space, decomposing the velocity field {u} into Littlewood-Paley components

\displaystyle  u_N(t,x) := \sum_{|k| \sim N} \hat u(t,k) e^{2\pi i k \cdot x}

of the velocity field {u(t,x)} at frequency {N}. By Plancherel’s theorem, this field will have an {L^2} norm of {E_N(t)^{1/2}}, and as a naive model of turbulence we expect this field to be spread out more or less uniformly on the torus, so we have the heuristic

\displaystyle  |u_N(t,x)| = O( E_N(t)^{1/2} ),

and a similar heuristic applied to {\nabla u_N} gives

\displaystyle  |\nabla u_N(t,x)| = O( N E_N(t)^{1/2} ).

(One can consider modifications of the Kolmogorov model in which {u_N} is concentrated on a lower-dimensional subset of the three-dimensional torus, leading to some changes in the numerology below, but we will not consider such variants here.) Since

\displaystyle  \Pi_{N,N_1,N_2} = - \int_{{\bf R}^3/{\bf Z}^3} u_N \cdot ( (u_{N_1} \cdot \nabla) u_{N_2} )\ dx

we thus arrive at the heuristic

\displaystyle  \Pi_{N,N_1,N_2} = O( N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} ).

Of course, there is the possibility that due to significant cancellation, the energy flow is significantly less than {O( N E_N(t)^{3/2} )}, but we will assume that cancellation effects are not that significant, so that we typically have

\displaystyle  \Pi_{N,N_1,N_2} \sim N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} \ \ \ \ \ (3)

or (assuming that {E_N} does not oscillate too much in {N}, and {N_1,N_2} are close to {N})

\displaystyle  \Pi_{N,N_1,N_2} \sim N E_N^{3/2}.

On the other hand, we clearly have

\displaystyle  D_N \sim \nu N^2 E_N.

We thus expect to be in the dissipation regime when

\displaystyle  N \gtrsim \nu^{-1} E_N^{1/2} \ \ \ \ \ (4)

and in the energy flow regime when

\displaystyle  1 \lesssim N \lesssim \nu^{-1} E_N^{1/2}. \ \ \ \ \ (5)

Now we study the energy flow regime further. We assume a “statistically scale-invariant” dynamics in this regime, in particular assuming a power law

\displaystyle  E_N \sim A N^{-\alpha} \ \ \ \ \ (6)

for some {A,\alpha > 0}. From (3), we then expect an average asymptotic of the form

\displaystyle  \Pi_{N,N_1,N_2} \approx A^{3/2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \ \ \ \ \ (7)

for some structure constants {c_{N,N_1,N_2} \sim 1} that depend on the exact nature of the turbulence; here we have replaced the factor {N_2} by the comparable term {(N N_1 N_2)^{1/3}} to make things more symmetric. In order to attain a steady state in the energy flow regime, we thus need a cancellation in the structure constants:

\displaystyle  \sum_{N_1,N_2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \approx 0. \ \ \ \ \ (8)

On the other hand, if one is assuming statistical scale invariance, we expect the structure constants to be scale-invariant (in the energy flow regime), in that

\displaystyle  c_{\lambda N, \lambda N_1, \lambda N_2} = c_{N,N_1,N_2} \ \ \ \ \ (9)

for dyadic {\lambda > 0}. Also, since the Euler equations conserve energy, the energy flows {\Pi_{N,N_1,N_2}} symmetrise to zero,

\displaystyle  \Pi_{N,N_1,N_2} + \Pi_{N,N_2,N_1} + \Pi_{N_1,N,N_2} + \Pi_{N_1,N_2,N} + \Pi_{N_2,N,N_1} + \Pi_{N_2,N_1,N} = 0,

which from (7) suggests a similar cancellation among the structure constants

\displaystyle  c_{N,N_1,N_2} + c_{N,N_2,N_1} + c_{N_1,N,N_2} + c_{N_1,N_2,N} + c_{N_2,N,N_1} + c_{N_2,N_1,N} \approx 0.

Combining this with the scale-invariance (9), we see that for fixed {N}, we may organise the structure constants {c_{N,N_1,N_2}} for dyadic {N_1,N_2} into sextuples which sum to zero (including some degenerate tuples of order less than six). This will automatically guarantee the cancellation (8) required for a steady state energy distribution, provided that

\displaystyle  \frac{1}{3} - \frac{\alpha}{2} = 0

or in other words

\displaystyle  \alpha = \frac{2}{3};

for any other value of {\alpha}, there is no particular reason to expect this cancellation (8) to hold. Thus we are led to the heuristic conclusion that the most stable power law distribution for the energies {E_N} is the {2/3} law

\displaystyle  E_N \sim A N^{-2/3} \ \ \ \ \ (10)

or in terms of shell energies, we have the famous Kolmogorov 5/3 law

\displaystyle  \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim A k_0^{-5/3}.

Given that frequency interactions tend to cascade from low frequencies to high (if only because there are so many more high frequencies than low ones), the above analysis predicts a stablising effect around this power law: scales at which a law (6) holds for some {\alpha > 2/3} are likely to lose energy in the near-term, while scales at which a law (6) hold for some {\alpha< 2/3} are conversely expected to gain energy, thus nudging the exponent of power law towards {2/3}.

We can solve for {A} in terms of energy dissipation as follows. If we let {N_*} be the frequency scale demarcating the transition from the energy flow regime (5) to the dissipation regime (4), we have

\displaystyle  N_* \sim \nu^{-1} E_{N_*}

and hence by (10)

\displaystyle  N_* \sim \nu^{-1} A N_*^{-2/3}.

On the other hand, if we let {\epsilon := D_{N_*}} be the energy dissipation at this scale {N_*} (which we expect to be the dominant scale of energy dissipation), we have

\displaystyle  \epsilon \sim \nu N_*^2 E_N \sim \nu N_*^2 A N_*^{-2/3}.

Some simple algebra then lets us solve for {A} and {N_*} as

\displaystyle  N_* \sim (\frac{\epsilon}{\nu^3})^{1/4}

and

\displaystyle  A \sim \epsilon^{2/3}.

Thus, we have the Kolmogorov prediction

\displaystyle  \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim \epsilon^{2/3} k_0^{-5/3}

for

\displaystyle  1 \lesssim k_0 \lesssim (\frac{\epsilon}{\nu^3})^{1/4}

with energy dissipation occuring at the high end {k_0 \sim (\frac{\epsilon}{\nu^3})^{1/4}} of this scale, which is counterbalanced by the energy injection at the low end {k_0 \sim 1} of the scale.

Let {V} be a quasiprojective variety defined over a finite field {{\bf F}_q}, thus for instance {V} could be an affine variety

\displaystyle  V = \{ x \in {\bf A}^d: P_1(x) = \dots = P_m(x) = 0\} \ \ \ \ \ (1)

where {{\bf A}^d} is {d}-dimensional affine space and {P_1,\dots,P_m: {\bf A}^d \rightarrow {\bf A}} are a finite collection of polynomials with coefficients in {{\bf F}_q}. Then one can define the set {V[{\bf F}_q]} of {{\bf F}_q}-rational points, and more generally the set {V[{\bf F}_{q^n}]} of {{\bf F}_{q^n}}-rational points for any {n \geq 1}, since {{\bf F}_{q^n}} can be viewed as a field extension of {{\bf F}_q}. Thus for instance in the affine case (1) we have

\displaystyle  V[{\bf F}_{q^n}] := \{ x \in {\bf F}_{q^n}^d: P_1(x) = \dots = P_m(x) = 0\}.

The Weil conjectures are concerned with understanding the number

\displaystyle  S_n := |V[{\bf F}_{q^n}]| \ \ \ \ \ (2)

of {{\bf F}_{q^n}}-rational points over a variety {V}. The first of these conjectures was proven by Dwork, and can be phrased as follows.

Theorem 1 (Rationality of the zeta function) Let {V} be a quasiprojective variety defined over a finite field {{\bf F}_q}, and let {S_n} be given by (2). Then there exist a finite number of algebraic integers {\alpha_1,\dots,\alpha_k, \beta_1,\dots,\beta_{k'} \in O_{\overline{{\bf Q}}}} (known as characteristic values of {V}), such that

\displaystyle  S_n = \alpha_1^n + \dots + \alpha_k^n - \beta_1^n - \dots - \beta_{k'}^n

for all {n \geq 1}.

After cancelling, we may of course assume that {\alpha_i \neq \beta_j} for any {i=1,\dots,k} and {j=1,\dots,k'}, and then it is easy to see (as we will see below) that the {\alpha_1,\dots,\alpha_k,\beta_1,\dots,\beta_{k'}} become uniquely determined up to permutations of the {\alpha_1,\dots,\alpha_k} and {\beta_1,\dots,\beta_{k'}}. These values are known as the characteristic values of {V}. Since {S_n} is a rational integer (i.e. an element of {{\bf Z}}) rather than merely an algebraic integer (i.e. an element of the ring of integers {O_{\overline{{\bf Q}}}} of the algebraic closure {\overline{{\bf Q}}} of {{\bf Q}}), we conclude from the above-mentioned uniqueness that the set of characteristic values are invariant with respect to the Galois group {Gal(\overline{{\bf Q}} / {\bf Q} )}. To emphasise this Galois invariance, we will not fix a specific embedding {\iota_\infty: \overline{{\bf Q}} \rightarrow {\bf C}} of the algebraic numbers into the complex field {{\bf C} = {\bf C}_\infty}, but work with all such embeddings simultaneously. (Thus, for instance, {\overline{{\bf Q}}} contains three cube roots of {2}, but which of these is assigned to the complex numbers {2^{1/3}}, {e^{2\pi i/3} 2^{1/3}}, {e^{4\pi i/3} 2^{1/3}} will depend on the choice of embedding {\iota_\infty}.)

An equivalent way of phrasing Dwork’s theorem is that the ({T}-form of the) zeta function

\displaystyle \zeta_V(T) := \exp( \sum_{n=1}^\infty \frac{S_n}{n} T^n )

associated to {V} (which is well defined as a formal power series in {T}, at least) is equal to a rational function of {T} (with the {\alpha_1,\dots,\alpha_k} and {\beta_1,\dots,\beta_{k'}} being the poles and zeroes of {\zeta_V} respectively). Here, we use the formal exponential

\displaystyle  \exp(X) := 1 + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \dots.

Equivalently, the ({s}-form of the) zeta-function {s \mapsto \zeta_V(q^{-s})} is a meromorphic function on the complex numbers {{\bf C}} which is also periodic with period {2\pi i/\log q}, and which has only finitely many poles and zeroes up to this periodicity.

Dwork’s argument relies primarily on {p}-adic analysis – an analogue of complex analysis, but over an algebraically complete (and metrically complete) extension {{\bf C}_p} of the {p}-adic field {{\bf Q}_p}, rather than over the Archimedean complex numbers {{\bf C}}. The argument is quite effective, and in particular gives explicit upper bounds for the number {k+k'} of characteristic values in terms of the complexity of the variety {V}; for instance, in the affine case (1) with {V} of degree {D}, Bombieri used Dwork’s methods (in combination with Deligne’s theorem below) to obtain the bound {k+k' \leq (4D+9)^{2d+1}}, and a subsequent paper of Hooley established the slightly weaker bound {k+k' \leq (11D+11)^{d+m+2}} purely from Dwork’s methods (a similar bound had also been pointed out in unpublished work of Dwork). In particular, one has bounds that are uniform in the field {{\bf F}_q}, which is an important fact for many analytic number theory applications.

These {p}-adic arguments stand in contrast with Deligne’s resolution of the last (and deepest) of the Weil conjectures:

Theorem 2 (Riemann hypothesis) Let {V} be a quasiprojective variety defined over a finite field {{\bf F}_q}, and let {\lambda \in \overline{{\bf Q}}} be a characteristic value of {V}. Then there exists a natural number {w} such that {|\iota_\infty(\lambda)|_\infty = q^{w/2}} for every embedding {\iota_\infty: \overline{{\bf Q}} \rightarrow {\bf C}}, where {| |_\infty} denotes the usual absolute value on the complex numbers {{\bf C} = {\bf C}_\infty}. (Informally: {\lambda} and all of its Galois conjugates have complex magnitude {q^{w/2}}.)

To put it another way that closely resembles the classical Riemann hypothesis, all the zeroes and poles of the {s}-form {s \mapsto \zeta_V(q^{-s})} lie on the critical lines {\{ s \in {\bf C}: \hbox{Re}(s) = \frac{w}{2} \}} for {w=0,1,2,\dots}. (See this previous blog post for further comparison of various instantiations of the Riemann hypothesis.) Whereas Dwork uses {p}-adic analysis, Deligne uses the essentially orthogonal technique of ell-adic cohomology to establish his theorem. However, ell-adic methods can be used (via the Grothendieck-Lefschetz trace formula) to establish rationality, and conversely, in this paper of Kedlaya p-adic methods are used to establish the Riemann hypothesis. As pointed out by Kedlaya, the ell-adic methods are tied to the intrinsic geometry of {V} (such as the structure of sheaves and covers over {V}), while the {p}-adic methods are more tied to the extrinsic geometry of {V} (how {V} sits inside its ambient affine or projective space).

In this post, I would like to record my notes on Dwork’s proof of Theorem 1, drawing heavily on the expositions of Serre, Hooley, Koblitz, and others.

The basic strategy is to control the rational integers {S_n} both in an “Archimedean” sense (embedding the rational integers inside the complex numbers {{\bf C}_\infty} with the usual norm {||_\infty}) as well as in the “{p}-adic” sense, with {p} the characteristic of {{\bf F}_q} (embedding the integers now in the “complexification” {{\bf C}_p} of the {p}-adic numbers {{\bf Q}_p}, which is equipped with a norm {||_p} that we will recall later). (This is in contrast to the methods of ell-adic cohomology, in which one primarily works over an {\ell}-adic field {{\bf Q}_\ell} with {\ell \neq p,\infty}.) The Archimedean control is trivial:

Proposition 3 (Archimedean control of {S_n}) With {S_n} as above, and any embedding {\iota_\infty: \overline{{\bf Q}} \rightarrow {\bf C}}, we have

\displaystyle  |\iota_\infty(S_n)|_\infty \leq C q^{A n}

for all {n} and some {C, A >0} independent of {n}.

Proof: Since {S_n} is a rational integer, {|\iota_\infty(S_n)|_\infty} is just {|S_n|_\infty}. By decomposing {V} into affine pieces, we may assume that {V} is of the affine form (1), then we trivially have {|S_n|_\infty \leq q^{nd}}, and the claim follows. \Box

Another way of thinking about this Archimedean control is that it guarantees that the zeta function {T \mapsto \zeta_V(T)} can be defined holomorphically on the open disk in {{\bf C}_\infty} of radius {q^{-A}} centred at the origin.

The {p}-adic control is significantly more difficult, and is the main component of Dwork’s argument:

Proposition 4 ({p}-adic control of {S_n}) With {S_n} as above, and using an embedding {\iota_p: \overline{{\bf Q}} \rightarrow {\bf C}_p} (defined later) with {p} the characteristic of {{\bf F}_q}, we can find for any real {A > 0} a finite number of elements {\alpha_1,\dots,\alpha_k,\beta_1,\dots,\beta_{k'} \in {\bf C}_p} such that

\displaystyle  |\iota_p(S_n) - (\alpha_1^n + \dots + \alpha_k^n - \beta_1^n - \dots - \beta_{k'}^n)|_p \leq q^{-An}

for all {n}.

Another way of thinking about this {p}-adic control is that it guarantees that the zeta function {T \mapsto \zeta_V(T)} can be defined meromorphically on the entire {p}-adic complex field {{\bf C}_p}.

Proposition 4 is ostensibly much weaker than Theorem 1 because of (a) the error term of {p}-adic magnitude at most {Cq^{-An}}; (b) the fact that the number {k+k'} of potential characteristic values here may go to infinity as {A \rightarrow \infty}; and (c) the potential characteristic values {\alpha_1,\dots,\alpha_k,\beta_1,\dots,\beta_{k'}} only exist inside the complexified {p}-adics {{\bf C}_p}, rather than in the algebraic integers {O_{\overline{{\bf Q}}}}. However, it turns out that by combining {p}-adic control on {S_n} in Proposition 4 with the trivial control on {S_n} in Proposition 3, one can obtain Theorem 1 by an elementary argument that does not use any further properties of {S_n} (other than the obvious fact that the {S_n} are rational integers), with the {A} in Proposition 4 chosen to exceed the {A} in Proposition 3. We give this argument (essentially due to Borel) below the fold.

The proof of Proposition 4 can be split into two pieces. The first piece, which can be viewed as the number-theoretic component of the proof, uses external descriptions of {V} such as (1) to obtain the following decomposition of {S_n}:

Proposition 5 (Decomposition of {S_n}) With {\iota_p} and {S_n} as above, we can decompose {\iota_p(S_n)} as a finite linear combination (over the integers) of sequences {S'_n \in {\bf C}_p}, such that for each such sequence {n \mapsto S'_n}, the zeta functions

\displaystyle  \zeta'(T) := \exp( \sum_{n=1}^\infty \frac{S'_n}{n} T^n ) = \sum_{n=0}^\infty c_n T^n

are entire in {{\bf C}_p}, by which we mean that

\displaystyle  |c_n|_p^{1/n} \rightarrow 0

as {n \rightarrow \infty}.

This proposition will ultimately be a consequence of the properties of the Teichmuller lifting {\tau: \overline{{\bf F}_p}^\times \rightarrow {\bf C}_p^\times}.

The second piece, which can be viewed as the “{p}-adic complex analytic” component of the proof, relates the {p}-adic entire nature of a zeta function with control on the associated sequence {S'_n}, and can be interpreted (after some manipulation) as a {p}-adic version of the Weierstrass preparation theorem:

Proposition 6 ({p}-adic Weierstrass preparation theorem) Let {S'_n} be a sequence in {{\bf C}_p}, such that the zeta function

\displaystyle  \zeta'(T) := \exp( \sum_{n=1}^\infty \frac{S'_n}{n} T^n )

is entire in {{\bf C}_p}. Then for any real {A > 0}, there exist a finite number of elements {\beta_1,\dots,\beta_{k'} \in {\bf C}_p} such that

\displaystyle  |\iota_p(S'_n) + \beta_1^n + \dots + \beta_{k'}^n|_p \leq q^{-An}

for all {n} and some {C>0}.

Clearly, the combination of Proposition 5 and Proposition 6 (and the non-Archimedean nature of the {||_p} norm) imply Proposition 4.

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This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”.

Note: the discussion here will be highly non-rigorous in nature, being extremely loose in particular with asymptotic notation and with the notion of dimension. Caveat emptor.

One of the most infamous unsolved problems at the intersection of geometric measure theory, incidence combinatorics, and real-variable harmonic analysis is the Kakeya set conjecture. We will focus on the following three-dimensional case of the conjecture, stated informally as follows:

Conjecture 1 (Kakeya conjecture) Let {E} be a subset of {{\bf R}^3} that contains a unit line segment in every direction. Then {\hbox{dim}(E) = 3}.

This conjecture is not precisely formulated here, because we have not specified exactly what type of set {E} is (e.g. measurable, Borel, compact, etc.) and what notion of dimension we are using. We will deliberately ignore these technical details in this post. It is slightly more convenient for us here to work with lines instead of unit line segments, so we work with the following slight variant of the conjecture (which is essentially equivalent):

Conjecture 2 (Kakeya conjecture, again) Let {{\cal L}} be a family of lines in {{\bf R}^3} that meet {B(0,1)} and contain a line in each direction. Let {E} be the union of the restriction {\ell \cap B(0,2)} to {B(0,2)} of every line {\ell} in {{\cal L}}. Then {\hbox{dim}(E) = 3}.

As the space of all directions in {{\bf R}^3} is two-dimensional, we thus see that {{\cal L}} is an (at least) two-dimensional subset of the four-dimensional space of lines in {{\bf R}^3} (actually, it lies in a compact subset of this space, since we have constrained the lines to meet {B(0,1)}). One could then ask if this is the only property of {{\cal L}} that is needed to establish the Kakeya conjecture, that is to say if any subset of {B(0,2)} which contains a two-dimensional family of lines (restricted to {B(0,2)}, and meeting {B(0,1)}) is necessarily three-dimensional. Here we have an easy counterexample, namely a plane in {B(0,2)} (passing through the origin), which contains a two-dimensional collection of lines. However, we can exclude this case by adding an additional axiom, leading to what one might call a “strong” Kakeya conjecture:

Conjecture 3 (Strong Kakeya conjecture) Let {{\cal L}} be a two-dimensional family of lines in {{\bf R}^3} that meet {B(0,1)}, and assume the Wolff axiom that no (affine) plane contains more than a one-dimensional family of lines in {{\cal L}}. Let {E} be the union of the restriction {\ell \cap B(0,2)} of every line {\ell} in {{\cal L}}. Then {\hbox{dim}(E) = 3}.

Actually, to make things work out we need a more quantitative version of the Wolff axiom in which we constrain the metric entropy (and not just dimension) of lines that lie close to a plane, rather than exactly on the plane. However, for the informal discussion here we will ignore these technical details. Families of lines that lie in different directions will obey the Wolff axiom, but the converse is not true in general.

In 1995, Wolff established the important lower bound {\hbox{dim}(E) \geq 5/2} (for various notions of dimension, e.g. Hausdorff dimension) for sets {E} in Conjecture 3 (and hence also for the other forms of the Kakeya problem). However, there is a key obstruction to going beyond the {5/2} barrier, coming from the possible existence of half-dimensional (approximate) subfields of the reals {{\bf R}}. To explain this problem, it easiest to first discuss the complex version of the strong Kakeya conjecture, in which all relevant (real) dimensions are doubled:

Conjecture 4 (Strong Kakeya conjecture over {{\bf C}}) Let {{\cal L}} be a four (real) dimensional family of complex lines in {{\bf C}^3} that meet the unit ball {B(0,1)} in {{\bf C}^3}, and assume the Wolff axiom that no four (real) dimensional (affine) subspace contains more than a two (real) dimensional family of complex lines in {{\cal L}}. Let {E} be the union of the restriction {\ell \cap B(0,2)} of every complex line {\ell} in {{\cal L}}. Then {E} has real dimension {6}.

The argument of Wolff can be adapted to the complex case to show that all sets {E} occuring in Conjecture 4 have real dimension at least {5}. Unfortunately, this is sharp, due to the following fundamental counterexample:

Proposition 5 (Heisenberg group counterexample) Let {H \subset {\bf C}^3} be the Heisenberg group

\displaystyle  H = \{ (z_1,z_2,z_3) \in {\bf C}^3: \hbox{Im}(z_1) = \hbox{Im}(z_2 \overline{z_3}) \}

and let {{\cal L}} be the family of complex lines

\displaystyle  \ell_{s,t,\alpha} := \{ (\overline{\alpha} z + t, z, sz + \alpha): z \in {\bf C} \}

with {s,t \in {\bf R}} and {\alpha \in {\bf C}}. Then {H} is a five (real) dimensional subset of {{\bf C}^3} that contains every line in the four (real) dimensional set {{\cal L}}; however each four real dimensional (affine) subspace contains at most a two (real) dimensional set of lines in {{\cal L}}. In particular, the strong Kakeya conjecture over the complex numbers is false.

This proposition is proven by a routine computation, which we omit here. The group structure on {H} is given by the group law

\displaystyle  (z_1,z_2,z_3) \cdot (w_1,w_2,w_3) = (z_1 + w_1 + z_2 \overline{w_3} - z_3 \overline{w_2}, z_2 +w_2, z_3+w_3),

giving {E} the structure of a {2}-step simply-connected nilpotent Lie group, isomorphic to the usual Heisenberg group over {{\bf R}^2}. Note that while the Heisenberg group is a counterexample to the complex strong Kakeya conjecture, it is not a counterexample to the complex form of the original Kakeya conjecture, because the complex lines {{\cal L}} in the Heisenberg counterexample do not point in distinct directions, but instead only point in a three (real) dimensional subset of the four (real) dimensional space of available directions for complex lines. For instance, one has the one real-dimensional family of parallel lines

\displaystyle  \ell_{0,t,0} = \{ (t, z, 0): z \in {\bf C}\}

with {t \in {\bf R}}; multiplying this family of lines on the right by a group element in {H} gives other families of parallel lines, which in fact sweep out all of {{\cal L}}.

The Heisenberg counterexample ultimately arises from the “half-dimensional” (and hence degree two) subfield {{\bf R}} of {{\bf C}}, which induces an involution {z \mapsto \overline{z}} which can then be used to define the Heisenberg group {H} through the formula

\displaystyle  H = \{ (z_1,z_2,z_3) \in {\bf C}^3: z_1 - \overline{z_1} = z_2 \overline{z_3} - z_3 \overline{z_2} \}.

Analogous Heisenberg counterexamples can also be constructed if one works over finite fields {{\bf F}_{q^2}} that contain a “half-dimensional” subfield {{\bf F}_q}; we leave the details to the interested reader. Morally speaking, if {{\bf R}} in turn contained a subfield of dimension {1/2} (or even a subring or “approximate subring”), then one ought to be able to use this field to generate a counterexample to the strong Kakeya conjecture over the reals. Fortunately, such subfields do not exist; this was a conjecture of Erdos and Volkmann that was proven by Edgar and Miller, and more quantitatively by Bourgain (answering a question of Nets Katz and myself). However, this fact is not entirely trivial to prove, being a key example of the sum-product phenomenon.

We thus see that to go beyond the {5/2} dimension bound of Wolff for the 3D Kakeya problem over the reals, one must do at least one of two things:

  • (a) Exploit the distinct directions of the lines in {{\mathcal L}} in a way that goes beyond the Wolff axiom; or
  • (b) Exploit the fact that {{\bf R}} does not contain half-dimensional subfields (or more generally, intermediate-dimensional approximate subrings).

(The situation is more complicated in higher dimensions, as there are more obstructions than the Heisenberg group; for instance, in four dimensions quadric surfaces are an important obstruction, as discussed in this paper of mine.)

Various partial or complete results on the Kakeya problem over various fields have been obtained through route (a) or route (b). For instance, in 2000, Nets Katz, Izabella Laba and myself used route (a) to improve Wolff’s lower bound of {5/2} for Kakeya sets very slightly to {5/2+10^{-10}} (for a weak notion of dimension, namely upper Minkowski dimension). In 2004, Bourgain, Katz, and myself established a sum-product estimate which (among other things) ruled out approximate intermediate-dimensional subrings of {{\bf F}_p}, and then pursued route (b) to obtain a corresponding improvement {5/2+\epsilon} to the Kakeya conjecture over finite fields of prime order. The analogous (discretised) sum-product estimate over the reals was established by Bourgain in 2003, which in principle would allow one to extend the result of Katz, Laba and myself to the strong Kakeya setting, but this has not been carried out in the literature. Finally, in 2009, Dvir used route (a) and introduced the polynomial method (as discussed previously here) to completely settle the Kakeya conjecture in finite fields.

Below the fold, I present a heuristic argument of Nets Katz and myself, which in principle would use route (b) to establish the full (strong) Kakeya conjecture. In broad terms, the strategy is as follows:

  1. Assume that the (strong) Kakeya conjecture fails, so that there are sets {E} of the form in Conjecture 3 of dimension {3-\sigma} for some {\sigma>0}. Assume that {E} is “optimal”, in the sense that {\sigma} is as large as possible.
  2. Use the optimality of {E} (and suitable non-isotropic rescalings) to establish strong forms of standard structural properties expected of such sets {E}, namely “stickiness”, “planiness”, “local graininess” and “global graininess” (we will roughly describe these properties below). Heuristically, these properties are constraining {E} to “behave like” a putative Heisenberg group counterexample.
  3. By playing all these structural properties off of each other, show that {E} can be parameterised locally by a one-dimensional set which generates a counterexample to Bourgain’s sum-product theorem. This contradiction establishes the Kakeya conjecture.

Nets and I have had an informal version of argument for many years, but were never able to make a satisfactory theorem (or even a partial Kakeya result) out of it, because we could not rigorously establish anywhere near enough of the necessary structural properties (stickiness, planiness, etc.) on the optimal set {E} for a large number of reasons (one of which being that we did not have a good notion of dimension that did everything that we wished to demand of it). However, there is beginning to be movement in these directions (e.g. in this recent result of Guth using the polynomial method obtaining a weak version of local graininess on certain Kakeya sets). In view of this (and given that neither Nets or I have been actively working in this direction for some time now, due to many other projects), we’ve decided to distribute these ideas more widely than before, and in particular on this blog.

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Let {{\bf F}_q} be a finite field of order {q = p^n}, and let {C} be an absolutely irreducible smooth projective curve defined over {{\bf F}_q} (and hence over the algebraic closure {k := \overline{{\bf F}_q}} of that field). For instance, {C} could be the projective elliptic curve

\displaystyle C = \{ [x,y,z]: y^2 z = x^3 + ax z^2 + b z^3 \}

in the projective plane {{\bf P}^2 = \{ [x,y,z]: (x,y,z) \neq (0,0,0) \}}, where {a,b \in {\bf F}_q} are coefficients whose discriminant {-16(4a^3+27b^2)} is non-vanishing, which is the projective version of the affine elliptic curve

\displaystyle \{ (x,y): y^2 = x^3 + ax + b \}.

To each such curve {C} one can associate a genus {g}, which we will define later; for instance, elliptic curves have genus {1}. We can also count the cardinality {|C({\bf F}_q)|} of the set {C({\bf F}_q)} of {{\bf F}_q}-points of {C}. The Hasse-Weil bound relates the two:

Theorem 1 (Hasse-Weil bound) {||C({\bf F}_q)| - q - 1| \leq 2g\sqrt{q}}.

The usual proofs of this bound proceed by first establishing a trace formula of the form

\displaystyle |C({\bf F}_{p^n})| = p^n - \sum_{i=1}^{2g} \alpha_i^n + 1 \ \ \ \ \ (1)

 

for some complex numbers {\alpha_1,\dots,\alpha_{2g}} independent of {n}; this is in fact a special case of the Lefschetz-Grothendieck trace formula, and can be interpreted as an assertion that the zeta function associated to the curve {C} is rational. The task is then to establish a bound {|\alpha_i| \leq \sqrt{p}} for all {i=1,\dots,2g}; this (or more precisely, the slightly stronger assertion {|\alpha_i| = \sqrt{p}}) is the Riemann hypothesis for such curves. This can be done either by passing to the Jacobian variety of {C} and using a certain duality available on the cohomology of such varieties, known as Rosati involution; alternatively, one can pass to the product surface {C \times C} and apply the Riemann-Roch theorem for that surface.

In 1969, Stepanov introduced an elementary method (a version of what is now known as the polynomial method) to count (or at least to upper bound) the quantity {|C({\bf F}_q)|}. The method was initially restricted to hyperelliptic curves, but was soon extended to general curves. In particular, Bombieri used this method to give a short proof of the following weaker version of the Hasse-Weil bound:

Theorem 2 (Weak Hasse-Weil bound) If {q} is a perfect square, and {q \geq (g+1)^4}, then {|C({\bf F}_q)| \leq q + (2g+1) \sqrt{q} + 1}.

In fact, the bound on {|C({\bf F}_q)|} can be sharpened a little bit further, as we will soon see.

Theorem 2 is only an upper bound on {|C({\bf F}_q)|}, but there is a Galois-theoretic trick to convert (a slight generalisation of) this upper bound to a matching lower bound, and if one then uses the trace formula (1) (and the “tensor power trick” of sending {n} to infinity to control the weights {\alpha_i}) one can then recover the full Hasse-Weil bound. We discuss these steps below the fold.

I’ve discussed Bombieri’s proof of Theorem 2 in this previous post (in the special case of hyperelliptic curves), but now wish to present the full proof, with some minor simplifications from Bombieri’s original presentation; it is mostly elementary, with the deepest fact from algebraic geometry needed being Riemann’s inequality (a weak form of the Riemann-Roch theorem).

The first step is to reinterpret {|C({\bf F}_q)|} as the number of points of intersection between two curves {C_1,C_2} in the surface {C \times C}. Indeed, if we define the Frobenius endomorphism {\hbox{Frob}_q} on any projective space by

\displaystyle \hbox{Frob}_q( [x_0,\dots,x_n] ) := [x_0^q, \dots, x_n^q]

then this map preserves the curve {C}, and the fixed points of this map are precisely the {{\bf F}_q} points of {C}:

\displaystyle C({\bf F}_q) = \{ z \in C: \hbox{Frob}_q(z) = z \}.

Thus one can interpret {|C({\bf F}_q)|} as the number of points of intersection between the diagonal curve

\displaystyle \{ (z,z): z \in C \}

and the Frobenius graph

\displaystyle \{ (z, \hbox{Frob}_q(z)): z \in C \}

which are copies of {C} inside {C \times C}. But we can use the additional hypothesis that {q} is a perfect square to write this more symmetrically, by taking advantage of the fact that the Frobenius map has a square root

\displaystyle \hbox{Frob}_q = \hbox{Frob}_{\sqrt{q}}^2

with {\hbox{Frob}_{\sqrt{q}}} also preserving {C}. One can then also interpret {|C({\bf F}_q)|} as the number of points of intersection between the curve

\displaystyle C_1 := \{ (z, \hbox{Frob}_{\sqrt{q}}(z)): z \in C \} \ \ \ \ \ (2)

 

and its transpose

\displaystyle C_2 := \{ (\hbox{Frob}_{\sqrt{q}}(w), w): w \in C \}.

Let {k(C \times C)} be the field of rational functions on {C \times C} (with coefficients in {k}), and define {k(C_1)}, {k(C_2)}, and {k(C_1 \cap C_2)} analogously )(although {C_1 \cap C_2} is likely to be disconnected, so {k(C_1 \cap C_2)} will just be a ring rather than a field. We then (morally) have the commuting square

\displaystyle \begin{array}{ccccc} && k(C \times C) && \\ & \swarrow & & \searrow & \\ k(C_1) & & & & k(C_2) \\ & \searrow & & \swarrow & \\ && k(C_1 \cap C_2) && \end{array},

if we ignore the issue that a rational function on, say, {C \times C}, might blow up on all of {C_1} and thus not have a well-defined restriction to {C_1}. We use {\pi_1: k(C \times C) \rightarrow k(C_1)} and {\pi_2: k(C \times C) \rightarrow k(C_2)} to denote the restriction maps. Furthermore, we have obvious isomorphisms {\iota_1: k(C_1) \rightarrow k(C)}, {\iota_2: k(C_2) \rightarrow k(C)} coming from composing with the graphing maps {z \mapsto (z, \hbox{Frob}_{\sqrt{q}}(z))} and {w \mapsto (\hbox{Frob}_{\sqrt{q}}(w), w)}.

The idea now is to find a rational function {f \in k(C \times C)} on the surface {C \times C} of controlled degree which vanishes when restricted to {C_1}, but is non-vanishing (and not blowing up) when restricted to {C_2}. On {C_2}, we thus get a non-zero rational function {f \downharpoonright_{C_2}} of controlled degree which vanishes on {C_1 \cap C_2} – which then lets us bound the cardinality of {C_1 \cap C_2} in terms of the degree of {f \downharpoonright_{C_2}}. (In Bombieri’s original argument, one required vanishing to high order on the {C_1} side, but in our presentation, we have factored out a {\hbox{Frob}_{\sqrt{q}}} term which removes this high order vanishing condition.)

To find this {f}, we will use linear algebra. Namely, we will locate a finite-dimensional subspace {V} of {k(C \times C)} (consisting of certain “controlled degree” rational functions) which projects injectively to {k(C_2)}, but whose projection to {k(C_1)} has strictly smaller dimension than {V} itself. The rank-nullity theorem then forces the existence of a non-zero element {P} of {V} whose projection to {k(C_1)} vanishes, but whose projection to {k(C_2)} is non-zero.

Now we build {V}. Pick a {{\bf F}_q} point {P_\infty} of {C}, which we will think of as being a point at infinity. (For the purposes of proving Theorem 2, we may clearly assume that {C({\bf F}_q)} is non-empty.) Thus {P_\infty} is fixed by {\hbox{Frob}_q}. To simplify the exposition, we will also assume that {P_\infty} is fixed by the square root {\hbox{Frob}_{\sqrt{q}}} of {\hbox{Frob}_q}; in the opposite case when {\hbox{Frob}_{\sqrt{q}}} has order two when acting on {P_\infty}, the argument is essentially the same, but all references to {P_\infty} in the second factor of {C \times C} need to be replaced by {\hbox{Frob}_{\sqrt{q}} P_\infty} (we leave the details to the interested reader).

For any natural number {n}, define {R_n} to be the set of rational functions {f \in k(C)} which are allowed to have a pole of order up to {n} at {P_\infty}, but have no other poles on {C}; note that as we are assuming {C} to be smooth, it is unambiguous what a pole is (and what order it will have). (In the fancier language of divisors and Cech cohomology, we have {R_n = H^0( C, {\mathcal O}_C(-n P_\infty) )}.) The space {R_n} is clearly a vector space over {k}; one can view intuitively as the space of “polynomials” on {C} of “degree” at most {n}. When {n=0}, {R_0} consists just of the constant functions. Indeed, if {f \in R_0}, then the image {f(C)} of {f} avoids {\infty} and so lies in the affine line {k = {\mathbf P}^1 \backslash \{\infty\}}; but as {C} is projective, the image {f(C)} needs to be compact (hence closed) in {{\mathbf P}^1}, and must therefore be a point, giving the claim.

For higher {n \geq 1}, we have the easy relations

\displaystyle \hbox{dim}(R_{n-1}) \leq \hbox{dim}(R_n) \leq \hbox{dim}(R_{n-1})+1. \ \ \ \ \ (3)

 

The former inequality just comes from the trivial inclusion {R_{n-1} \subset R_n}. For the latter, observe that if two functions {f, g} lie in {R_n}, so that they each have a pole of order at most {n} at {P_\infty}, then some linear combination of these functions must have a pole of order at most {n-1} at {P_\infty}; thus {R_{n-1}} has codimension at most one in {R_n}, giving the claim.

From (3) and induction we see that each of the {R_n} are finite dimensional, with the trivial upper bound

\displaystyle \hbox{dim}(R_n) \leq n+1. \ \ \ \ \ (4)

 

Riemann’s inequality complements this with the lower bound

\displaystyle \hbox{dim}(R_n) \geq n+1-g, \ \ \ \ \ (5)

 

thus one has {\hbox{dim}(R_n) = \hbox{dim}(R_{n-1})+1} for all but at most {g} exceptions (in fact, exactly {g} exceptions as it turns out). This is a consequence of the Riemann-Roch theorem; it can be proven from abstract nonsense (the snake lemma) if one defines the genus {g} in a non-standard fashion (as the dimension of the first Cech cohomology {H^1(C)} of the structure sheaf {{\mathcal O}_C} of {C}), but to obtain this inequality with a standard definition of {g} (e.g. as the dimension of the zeroth Cech cohomolgy {H^0(C, \Omega_C^1)} of the line bundle of differentials) requires the more non-trivial tool of Serre duality.

At any rate, now that we have these vector spaces {R_n}, we will define {V \subset k(C \times C)} to be a tensor product space

\displaystyle V = R_\ell \otimes R_m

for some natural numbers {\ell, m \geq 0} which we will optimise in later. That is to say, {V} is spanned by functions of the form {(z,w) \mapsto f(z) g(w)} with {f \in R_\ell} and {g \in R_m}. This is clearly a linear subspace of {k(C \times C)} of dimension {\hbox{dim}(R_\ell) \hbox{dim}(R_m)}, and hence by Rieman’s inequality we have

\displaystyle \hbox{dim}(V) \geq (\ell+1-g) (m+1-g) \ \ \ \ \ (6)

 

if

\displaystyle \ell,m \geq g-1. \ \ \ \ \ (7)

 

Observe that {\iota_1 \circ \pi_1} maps a tensor product {(z,w) \mapsto f(z) g(w)} to a function {z \mapsto f(z) g(\hbox{Frob}_{\sqrt{q}} z)}. If {f \in R_\ell} and {g \in R_m}, then we see that the function {z \mapsto f(z) g(\hbox{Frob}_{\sqrt{q}} z)} has a pole of order at most {\ell+m\sqrt{q}} at {P_\infty}. We conclude that

\displaystyle \iota_1 \circ \pi_1( V ) \subset R_{\ell + m\sqrt{q}} \ \ \ \ \ (8)

 

and in particular by (4)

\displaystyle \hbox{dim}(\pi_1(V)) \leq \ell + m \sqrt{q} + 1 \ \ \ \ \ (9)

 

and similarly

\displaystyle \hbox{dim}(\pi_2(V)) \leq \ell \sqrt{q} + m + 1. \ \ \ \ \ (10)

 

We will choose {m} to be a bit bigger than {\ell}, to make the {\pi_2} image of {V} smaller than that of {\pi_1}. From (6), (10) we see that if we have the inequality

\displaystyle (\ell+1-g) (m+1-g) > \ell \sqrt{q}+m + 1 \ \ \ \ \ (11)

 

(together with (7)) then {\pi_2} cannot be injective.

On the other hand, we have the following basic fact:

Lemma 3 (Injectivity) If

\displaystyle \ell < \sqrt{q}, \ \ \ \ \ (12)

 

then {\pi_1: V \rightarrow \pi_1(V)} is injective.

Proof: From (3), we can find a linear basis {f_1,\dots,f_a} of {R_\ell} such that each of the {f_i} has a distinct order {d_i} of pole at {P_\infty} (somewhere between {0} and {\ell} inclusive). Similarly, we may find a linear basis {g_1,\dots,g_b} of {R_m} such that each of the {g_j} has a distinct order {e_j} of pole at {P_\infty} (somewhere between {0} and {m} inclusive). The functions {z \mapsto f_i(z) g_j(\hbox{Frob}_{\sqrt{q}} z)} then span {\iota_1(\pi_1(V))}, and the order of pole at {P_\infty} is {d_i + \sqrt{q} e_j}. But since {\ell < \sqrt{q}}, these orders are all distinct, and so these functions must be linearly independent. The claim follows. \Box

This gives us the following bound:

Proposition 4 Let {\ell,m} be natural numbers such that (7), (11), (12) hold. Then {|C({\bf F}_q)| \leq \ell + m \sqrt{q}}.

Proof: As {\pi_2} is not injective, we can find {f \in V} with {\pi_2(f)} vanishing. By the above lemma, the function {\iota_1(\pi_1(f))} is then non-zero, but it must also vanish on {\iota_1(C_1 \cap C_2)}, which has cardinality {|C({\bf F}_q)|}. On the other hand, by (8), {\iota_1(\pi_1(f))} has a pole of order at most {\ell+m\sqrt{q}} at {P_\infty} and no other poles. Since the number of poles and zeroes of a rational function on a projective curve must add up to zero, the claim follows. \Box

If {q \geq (g+1)^4}, we may make the explicit choice

\displaystyle m := \sqrt{q}+2g; \quad \ell := \lfloor \frac{g}{g+1} \sqrt{q} \rfloor + g + 1

and a brief calculation then gives Theorem 2. In some cases one can optimise things a bit further. For instance, in the genus zero case {g=0} (e.g. if {C} is just the projective line {{\mathbf P}^1}) one may take {\ell=1, m = \sqrt{q}} and conclude the absolutely sharp bound {|C({\bf F}_q)| \leq q+1} in this case; in the case of the projective line {{\mathbf P}^1}, the function {f} is in fact the very concrete function {f(z,w) := z - w^{\sqrt{q}}}.

Remark 1 When {q = p^{2n+1}} is not a perfect square, one can try to run the above argument using the factorisation {\hbox{Frob}_q = \hbox{Frob}_{p^n} \hbox{Frob}_{p^{n+1}}} instead of {\hbox{Frob}_q = \hbox{Frob}_{\sqrt{q}} \hbox{Frob}_{\sqrt{q}}}. This gives a weaker version of the above bound, of the shape {|C({\bf F}_q)| \leq q + O( \sqrt{p} \sqrt{q} )}. In the hyperelliptic case at least, one can erase this loss by working with a variant of the argument in which one requires {f} to vanish to high order at {C_1}, rather than just to first order; see this survey article of mine for details.

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Roth’s theorem on arithmetic progressions asserts that every subset of the integers {{\bf Z}} of positive upper density contains infinitely many arithmetic progressions of length three. There are many versions and variants of this theorem. Here is one of them:

Theorem 1 (Roth’s theorem) Let {G = (G,+)} be a compact abelian group, with Haar probability measure {\mu}, which is {2}-divisible (i.e. the map {x \mapsto 2x} is surjective) and let {A} be a measurable subset of {G} with {\mu(A) \geq \alpha} for some {0 < \alpha < 1}. Then we have

\displaystyle  \int_G \int_G 1_A(x) 1_A(x+r) 1_A(x+2r)\ d\mu(x) d\mu(r) \gg_\alpha 1,

where {X \gg_\alpha Y} denotes the bound {X \geq c_\alpha Y} for some {c_\alpha > 0} depending only on {\alpha}.

This theorem is usually formulated in the case that {G} is a finite abelian group of odd order (in which case the result is essentially due to Meshulam) or more specifically a cyclic group {G = {\bf Z}/N{\bf Z}} of odd order (in which case it is essentially due to Varnavides), but is also valid for the more general setting of {2}-divisible compact abelian groups, as we shall shortly see. One can be more precise about the dependence of the implied constant {c_\alpha} on {\alpha}, but to keep the exposition simple we will work at the qualitative level here, without trying at all to get good quantitative bounds. The theorem is also true without the {2}-divisibility hypothesis, but the proof we will discuss runs into some technical issues due to the degeneracy of the {2r} shift in that case.

We can deduce Theorem 1 from the following more general Khintchine-type statement. Let {\hat G} denote the Pontryagin dual of a compact abelian group {G}, that is to say the set of all continuous homomorphisms {\xi: x \mapsto \xi \cdot x} from {G} to the (additive) unit circle {{\bf R}/{\bf Z}}. Thus {\hat G} is a discrete abelian group, and functions {f \in L^2(G)} have a Fourier transform {\hat f \in \ell^2(\hat G)} defined by

\displaystyle  \hat f(\xi) := \int_G f(x) e^{-2\pi i \xi \cdot x}\ d\mu(x).

If {G} is {2}-divisible, then {\hat G} is {2}-torsion-free in the sense that the map {\xi \mapsto 2 \xi} is injective. For any finite set {S \subset \hat G} and any radius {\rho>0}, define the Bohr set

\displaystyle  B(S,\rho) := \{ x \in G: \sup_{\xi \in S} \| \xi \cdot x \|_{{\bf R}/{\bf Z}} < \rho \}

where {\|\theta\|_{{\bf R}/{\bf Z}}} denotes the distance of {\theta} to the nearest integer. We refer to the cardinality {|S|} of {S} as the rank of the Bohr set. We record a simple volume bound on Bohr sets:

Lemma 2 (Volume packing bound) Let {G} be a compact abelian group with Haar probability measure {\mu}. For any Bohr set {B(S,\rho)}, we have

\displaystyle  \mu( B( S, \rho ) ) \gg_{|S|, \rho} 1.

Proof: We can cover the torus {({\bf R}/{\bf Z})^S} by {O_{|S|,\rho}(1)} translates {\theta+Q} of the cube {Q := \{ (\theta_\xi)_{\xi \in S} \in ({\bf R}/{\bf Z})^S: \sup_{\xi \in S} \|\theta_\xi\|_{{\bf R}/{\bf Z}} < \rho/2 \}}. Then the sets {\{ x \in G: (\xi \cdot x)_{\xi \in S} \in \theta + Q \}} form an cover of {G}. But all of these sets lie in a translate of {B(S,\rho)}, and the claim then follows from the translation invariance of {\mu}. \Box

Given any Bohr set {B(S,\rho)}, we define a normalised “Lipschitz” cutoff function {\nu_{B(S,\rho)}: G \rightarrow {\bf R}} by the formula

\displaystyle  \nu_{B(S,\rho)}(x) = c_{B(S,\rho)} (1 - \frac{1}{\rho} \sup_{\xi \in S} \|\xi \cdot x\|_{{\bf R}/{\bf Z}})_+ \ \ \ \ \ (1)

where {c_{B(S,\rho)}} is the constant such that

\displaystyle  \int_G \nu_{B(S,\rho)}\ d\mu = 1,

thus

\displaystyle c_{B(S,\rho)} = \left( \int_{B(S,\rho)} (1 - \frac{1}{\rho} \sup_{\xi \in S} \|\xi \cdot x\|_{{\bf R}/{\bf Z}})\ d\mu(x) \right)^{-1}.

The function {\nu_{B(S,\rho)}} should be viewed as an {L^1}-normalised “tent function” cutoff to {B(S,\rho)}. Note from Lemma 2 that

\displaystyle  1 \ll_{|S|,\rho} c_{B(S,\rho)} \ll_{|S|,\rho} 1. \ \ \ \ \ (2)

We then have the following sharper version of Theorem 1:

Theorem 3 (Roth-Khintchine theorem) Let {G = (G,+)} be a {2}-divisible compact abelian group, with Haar probability measure {\mu}, and let {\epsilon>0}. Then for any measurable function {f: G \rightarrow [0,1]}, there exists a Bohr set {B(S,\rho)} with {|S| \ll_\epsilon 1} and {\rho \gg_\epsilon 1} such that

\displaystyle  \int_G \int_G f(x) f(x+r) f(x+2r) \nu_{B(S,\rho)}*\nu_{B(S,\rho)}(r)\ d\mu(x) d\mu(r) \ \ \ \ \ (3)

\displaystyle  \geq (\int_G f\ d\mu)^3 - O(\epsilon)

where {*} denotes the convolution operation

\displaystyle  f*g(x) := \int_G f(y) g(x-y)\ d\mu(y).

A variant of this result (expressed in the language of ergodic theory) appears in this paper of Bergelson, Host, and Kra; a combinatorial version of the Bergelson-Host-Kra result that is closer to Theorem 3 subsequently appeared in this paper of Ben Green and myself, but this theorem arguably appears implicitly in a much older paper of Bourgain. To see why Theorem 3 implies Theorem 1, we apply the theorem with {f := 1_A} and {\epsilon} equal to a small multiple of {\alpha^3} to conclude that there is a Bohr set {B(S,\rho)} with {|S| \ll_\alpha 1} and {\rho \gg_\alpha 1} such that

\displaystyle  \int_G \int_G 1_A(x) 1_A(x+r) 1_A(x+2r) \nu_{B(S,\rho)}*\nu_{B(S,\rho)}(r)\ d\mu(x) d\mu(r) \gg \alpha^3.

But from (2) we have the pointwise bound {\nu_{B(S,\rho)}*\nu_{B(S,\rho)} \ll_\alpha 1}, and Theorem 1 follows.

Below the fold, we give a short proof of Theorem 3, using an “energy pigeonholing” argument that essentially dates back to the 1986 paper of Bourgain mentioned previously (not to be confused with a later 1999 paper of Bourgain on Roth’s theorem that was highly influential, for instance in emphasising the importance of Bohr sets). The idea is to use the pigeonhole principle to choose the Bohr set {B(S,\rho)} to capture all the “large Fourier coefficients” of {f}, but such that a certain “dilate” of {B(S,\rho)} does not capture much more Fourier energy of {f} than {B(S,\rho)} itself. The bound (3) may then be obtained through elementary Fourier analysis, without much need to explicitly compute things like the Fourier transform of an indicator function of a Bohr set. (However, the bound obtained by this argument is going to be quite poor – of tower-exponential type.) To do this we perform a structural decomposition of {f} into “structured”, “small”, and “highly pseudorandom” components, as is common in the subject (e.g. in this previous blog post), but even though we crucially need to retain non-negativity of one of the components in this decomposition, we can avoid recourse to conditional expectation with respect to a partition (or “factor”) of the space, using instead convolution with one of the {\nu_{B(S,\rho)}} considered above to achieve a similar effect.

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