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A capset in the vector space ${{\bf F}_3^n}$ over the finite field ${{\bf F}_3}$ of three elements is a subset ${A}$ of ${{\bf F}_3^n}$ that does not contain any lines ${\{ x,x+r,x+2r\}}$, where ${x,r \in {\bf F}_3^n}$ and ${r \neq 0}$. A basic problem in additive combinatorics (discussed in one of the very first posts on this blog) is to obtain good upper and lower bounds for the maximal size of a capset in ${{\bf F}_3^n}$.

Trivially, one has ${|A| \leq 3^n}$. Using Fourier methods (and the density increment argument of Roth), the bound of ${|A| \leq O( 3^n / n )}$ was obtained by Meshulam, and improved only as late as 2012 to ${O( 3^n /n^{1+c})}$ for some absolute constant ${c>0}$ by Bateman and Katz. But in a very recent breakthrough, Ellenberg (and independently Gijswijt) obtained the exponentially superior bound ${|A| \leq O( 2.756^n )}$, using a version of the polynomial method recently introduced by Croot, Lev, and Pach. (In the converse direction, a construction of Edel gives capsets as large as ${(2.2174)^n}$.) Given the success of the polynomial method in superficially similar problems such as the finite field Kakeya problem (discussed in this previous post), it was natural to wonder that this method could be applicable to the cap set problem (see for instance this MathOverflow comment of mine on this from 2010), but it took a surprisingly long time before Croot, Lev, and Pach were able to identify the precise variant of the polynomial method that would actually work here.

The proof of the capset bound is very short (Ellenberg’s and Gijswijt’s preprints are both 3 pages long, and Croot-Lev-Pach is 6 pages), but I thought I would present a slight reformulation of the argument which treats the three points on a line in ${{\bf F}_3}$ symmetrically (as opposed to treating the third point differently from the first two, as is done in the Ellenberg and Gijswijt papers; Croot-Lev-Pach also treat the middle point of a three-term arithmetic progression differently from the two endpoints, although this is a very natural thing to do in their context of ${({\bf Z}/4{\bf Z})^n}$). The basic starting point is this: if ${A}$ is a capset, then one has the identity

$\displaystyle \delta_{0^n}( x+y+z ) = \sum_{a \in A} \delta_a(x) \delta_a(y) \delta_a(z) \ \ \ \ \ (1)$

for all ${(x,y,z) \in A^3}$, where ${\delta_a(x) := 1_{a=x}}$ is the Kronecker delta function, which we view as taking values in ${{\bf F}_3}$. Indeed, (1) reflects the fact that the equation ${x+y+z=0}$ has solutions precisely when ${x,y,z}$ are either all equal, or form a line, and the latter is ruled out precisely when ${A}$ is a capset.

To exploit (1), we will show that the left-hand side of (1) is “low rank” in some sense, while the right-hand side is “high rank”. Recall that a function ${F: A \times A \rightarrow {\bf F}}$ taking values in a field ${{\bf F}}$ is of rank one if it is non-zero and of the form ${(x,y) \mapsto f(x) g(y)}$ for some ${f,g: A \rightarrow {\bf F}}$, and that the rank of a general function ${F: A \times A \rightarrow {\bf F}}$ is the least number of rank one functions needed to express ${F}$ as a linear combination. More generally, if ${k \geq 2}$, we define the rank of a function ${F: A^k \rightarrow {\bf F}}$ to be the least number of “rank one” functions of the form

$\displaystyle (x_1,\dots,x_k) \mapsto f(x_i) g(x_1,\dots,x_{i-1},x_{i+1},\dots,x_k)$

for some ${i=1,\dots,k}$ and some functions ${f: A \rightarrow {\bf F}}$, ${g: A^{k-1} \rightarrow {\bf F}}$, that are needed to generate ${F}$ as a linear combination. For instance, when ${k=3}$, the rank one functions take the form ${(x,y,z) \mapsto f(x) g(y,z)}$, ${(x,y,z) \mapsto f(y) g(x,z)}$, ${(x,y,z) \mapsto f(z) g(x,y)}$, and linear combinations of ${r}$ such rank one functions will give a function of rank at most ${r}$.

It is a standard fact in linear algebra that the rank of a diagonal matrix is equal to the number of non-zero entries. This phenomenon extends to higher dimensions:

Lemma 1 (Rank of diagonal hypermatrices) Let ${k \geq 2}$, let ${A}$ be a finite set, let ${{\bf F}}$ be a field, and for each ${a \in A}$, let ${c_a \in {\bf F}}$ be a coefficient. Then the rank of the function

$\displaystyle (x_1,\dots,x_k) \mapsto \sum_{a \in A} c_a \delta_a(x_1) \dots \delta_a(x_k) \ \ \ \ \ (2)$

is equal to the number of non-zero coefficients ${c_a}$.

Proof: We induct on ${k}$. As mentioned above, the case ${k=2}$ follows from standard linear algebra, so suppose now that ${k>2}$ and the claim has already been proven for ${k-1}$.

It is clear that the function (2) has rank at most equal to the number of non-zero ${c_a}$ (since the summands on the right-hand side are rank one functions), so it suffices to establish the lower bound. By deleting from ${A}$ those elements ${a \in A}$ with ${c_a=0}$ (which cannot increase the rank), we may assume without loss of generality that all the ${c_a}$ are non-zero. Now suppose for contradiction that (2) has rank at most ${|A|-1}$, then we obtain a representation

$\displaystyle \sum_{a \in A} c_a \delta_a(x_1) \dots \delta_a(x_k) = \sum_{i=1}^k \sum_{\alpha \in I_i} f_{i,\alpha}(x_i) g_{i,\alpha}( x_1,\dots,x_{i-1},x_{i+1},\dots,x_k) \ \ \ \ \ (3)$

for some sets ${I_1,\dots,I_k}$ of cardinalities adding up to at most ${|A|-1}$, and some functions ${f_{i,\alpha}: A \rightarrow {\bf F}}$ and ${g_{i,\alpha}: A^{k-1} \rightarrow {\bf R}}$.

Consider the space of functions ${h: A \rightarrow {\bf F}}$ that are orthogonal to all the ${f_{k,\alpha}}$, ${\alpha \in I_k}$ in the sense that

$\displaystyle \sum_{x \in A} f_{k,\alpha}(x) h(x) = 0$

for all ${\alpha \in I_k}$. This space is a vector space whose dimension ${d}$ is at least ${|A| - |I_k|}$. A basis of this space generates a ${d \times |A|}$ coordinate matrix of full rank, which implies that there is at least one non-singular ${d \times d}$ minor. This implies that there exists a function ${h: A \rightarrow {\bf F}}$ in this space which is non-zero on some subset ${A'}$ of ${A}$ of cardinality at least ${|A|-|I_k|}$.

If we multiply (3) by ${h(x_k)}$ and sum in ${x_k}$, we conclude that

$\displaystyle \sum_{a \in A} c_a h(a) \delta_a(x_1) \dots \delta_a(x_{k-1})$

$\displaystyle = \sum_{i=1}^{k-1} \sum_{\alpha \in I_i} f_{i,\alpha}(x_i)\tilde g_{i,\alpha}( x_1,\dots,x_{i-1},x_{i+1},\dots,x_{k-1})$

where

$\displaystyle \tilde g_{i,\alpha}(x_1,\dots,x_{i-1},x_{i+1},\dots,x_{k-1})$

$\displaystyle := \sum_{x_k \in A} g_{i,\alpha}(x_1,\dots,x_{i-1},x_{i+1},\dots,x_k) h(x_k).$

The right-hand side has rank at most ${|A|-1-|I_k|}$, since the summands are rank one functions. On the other hand, from induction hypothesis the left-hand side has rank at least ${|A|-|I_k|}$, giving the required contradiction. $\Box$

On the other hand, we have the following (symmetrised version of a) beautifully simple observation of Croot, Lev, and Pach:

Lemma 2 On ${({\bf F}_3^n)^3}$, the rank of the function ${(x,y,z) \mapsto \delta_{0^n}(x+y+z)}$ is at most ${3N}$, where

$\displaystyle N := \sum_{a,b,c \geq 0: a+b+c=n, b+2c \leq 2n/3} \frac{n!}{a!b!c!}.$

Proof: Using the identity ${\delta(x) = 1 - x^2}$ for ${x \in {\bf F}}$, we have

$\displaystyle \delta_{0^n}(x+y+z) = \prod_{i=1}^n (1 - (x_i+y_i+z_i)^2).$

The right-hand side is clearly a polynomial of degree ${2n}$ in ${x,y,z}$, which is then a linear combination of monomials

$\displaystyle x_1^{i_1} \dots x_n^{i_n} y_1^{j_1} \dots y_n^{j_n} z_1^{k_1} \dots z_n^{k_n}$

with ${i_1,\dots,i_n,j_1,\dots,j_n,k_1,\dots,k_n \in \{0,1,2\}}$ with

$\displaystyle i_1 + \dots + i_n + j_1 + \dots + j_n + k_1 + \dots + k_n \leq 2n.$

In particular, from the pigeonhole principle, at least one of ${i_1 + \dots + i_n, j_1 + \dots + j_n, k_1 + \dots + k_n}$ is at most ${2n/3}$.

Consider the contribution of the monomials for which ${i_1 + \dots + i_n \leq 2n/3}$. We can regroup this contribution as

$\displaystyle \sum_\alpha f_\alpha(x) g_\alpha(y,z)$

where ${\alpha}$ ranges over those ${(i_1,\dots,i_n) \in \{0,1,2\}^n}$ with ${i_1 + \dots + i_n \leq 2n/3}$, ${f_\alpha}$ is the monomial

$\displaystyle f_\alpha(x_1,\dots,x_n) := x_1^{i_1} \dots x_n^{i_n}$

and ${g_\alpha: {\bf F}_3^n \times {\bf F}_3^n \rightarrow {\bf F}_3}$ is some explicitly computable function whose exact form will not be of relevance to our argument. The number of such ${\alpha}$ is equal to ${N}$, so this contribution has rank at most ${N}$. The remaining contributions arising from the cases ${j_1 + \dots + j_n \leq 2n/3}$ and ${k_1 + \dots + k_n \leq 2n/3}$ similarly have rank at most ${N}$ (grouping the monomials so that each monomial is only counted once), so the claim follows.

Upon restricting from ${({\bf F}_3^n)^3}$ to ${A^3}$, the rank of ${(x,y,z) \mapsto \delta_{0^n}(x+y+z)}$ is still at most ${3N}$. The two lemmas then combine to give the Ellenberg-Gijswijt bound

$\displaystyle |A| \leq 3N.$

All that remains is to compute the asymptotic behaviour of ${N}$. This can be done using the general tool of Cramer’s theorem, but can also be derived from Stirling’s formula (discussed in this previous post). Indeed, if ${a = (\alpha+o(1)) n}$, ${b = (\beta+o(1)) n}$, ${c = (\gamma+o(1)) n}$ for some ${\alpha,\beta,\gamma \geq 0}$ summing to ${1}$, Stirling’s formula gives

$\displaystyle \frac{n!}{a!b!c!} = \exp( n (h(\alpha,\beta,\gamma) + o(1)) )$

where ${h}$ is the entropy function

$\displaystyle h(\alpha,\beta,\gamma) = \alpha \log \frac{1}{\alpha} + \beta \log \frac{1}{\beta} + \gamma \log \frac{1}{\gamma}.$

We then have

$\displaystyle N = \exp( n (X + o(1))$

where ${X}$ is the maximum entropy ${h(\alpha,\beta,\gamma)}$ subject to the constraints

$\displaystyle \alpha,\beta,\gamma \geq 0; \alpha+\beta+\gamma=1; \beta+2\gamma \leq 2/3.$

A routine Lagrange multiplier computation shows that the maximum occurs when

$\displaystyle \alpha = \frac{32}{3(15 + \sqrt{33})}$

$\displaystyle \beta = \frac{4(\sqrt{33}-1)}{3(15+\sqrt{33})}$

$\displaystyle \gamma = \frac{(\sqrt{33}-1)^2}{6(15+\sqrt{33})}$

and ${h(\alpha,\beta,\gamma)}$ is approximately ${1.013455}$, giving rise to the claimed bound of ${O( 2.756^n )}$.

Remark 3 As noted in the Ellenberg and Gijswijt papers, the above argument extends readily to other fields than ${{\bf F}_3}$ to control the maximal size of aubset of ${{\bf F}^n}$ that has no non-trivial solutions to the equation ${ax+by+cz=0}$, where ${a,b,c \in {\bf F}}$ are non-zero constants that sum to zero. Of course one replaces the function ${(x,y,z) \mapsto \delta_{0^n}(x+y+z)}$ in Lemma 2 by ${(x,y,z) \mapsto \delta(ax+by+cz)}$ in this case.

Remark 4 This symmetrised formulation suggests that one possible way to improve slightly on the numerical quantity ${2.756}$ by finding a more efficient way to decompose ${\delta_{0^n}(x+y+z)}$ into rank one functions, however I was not able to do so (though such improvements are reminiscent of the Strassen type algorithms for fast matrix multiplication).

Remark 5 It is tempting to see if this method can get non-trivial upper bounds for sets ${A}$ with no length ${4}$ progressions, in (say) ${{\bf F}_5^n}$. One can run the above arguments, replacing the function

$\displaystyle (x,y,z) \mapsto \delta_{0^n}(x+y+z)$

with

$\displaystyle (x,y,z,w) \mapsto \delta_{0^n}(x-2y+z) \delta_{0^n}(y-2z+w);$

this leads to the bound ${|A| \leq 4N}$ where

$\displaystyle N := \sum_{a,b,c,d,e \geq 0: a+b+c+d+e=n, b+2c+3d+4e \leq 2n} \frac{n!}{a!b!c!d!e!}.$

Unfortunately, ${N}$ is asymptotic to ${\frac{1}{2} 5^n}$ and so this bound is in fact slightly worse than the trivial bound ${|A| \leq 5^n}$! However, there is a slim chance that there is a more efficient way to decompose ${\delta(x-2y+z) \delta(y-2z+w)}$ into rank one functions that would give a non-trivial bound on ${A}$. I experimented with a few possible such decompositions but unfortunately without success.

Remark 6 Return now to the capset problem. Since Lemma 1 is valid for any field ${{\bf F}}$, one could perhaps hope to get better bounds by viewing the Kronecker delta function ${\delta}$ as taking values in another field than ${{\bf F}_3}$, such as the complex numbers ${{\bf C}}$. However, as soon as one works in a field of characteristic other than ${3}$, one can adjoin a cube root ${\omega}$ of unity, and one now has the Fourier decomposition

$\displaystyle \delta_{0^n}(x+y+z) = \frac{1}{3^n} \sum_{\xi \in {\bf F}_3^n} \omega^{\xi \cdot x} \omega^{\xi \cdot y} \omega^{\xi \cdot z}.$

Moving to the Fourier basis, we conclude from Lemma 1 that the function ${(x,y,z) \mapsto \delta_{0^n}(x+y+z)}$ on ${{\bf F}_3^n}$ now has rank exactly ${3^n}$, and so one cannot improve upon the trivial bound of ${|A| \leq 3^n}$ by this method using fields of characteristic other than three as the range field. So it seems one has to stick with ${{\bf F}_3}$ (or the algebraic completion thereof).

Thanks to Jordan Ellenberg and Ben Green for helpful discussions.

When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology.  However, there are some nice things one can do in an electronic medium, such as this blog.  Here, I would like to experiment with the ability to animate images, which I think can convey some mathematical concepts in ways that cannot be easily replicated by traditional static text and images. Given that many readers may find these animations annoying, I am placing the rest of the post below the fold.

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold ${M}$ can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space ${{\bf R}^d}$. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let ${M}$ be a compact manifold. Then there exists an embedding ${u: M \rightarrow {\bf R}^d}$ from ${M}$ to a Euclidean space ${{\bf R}^d}$.

In fact, if ${M}$ is ${n}$-dimensional, one can take ${d}$ to equal ${2n}$, which is often best possible (easy examples include the circle ${{\bf R}/{\bf Z}}$ which embeds into ${{\bf R}^2}$ but not ${{\bf R}^1}$, or the Klein bottle that embeds into ${{\bf R}^4}$ but not ${{\bf R}^3}$). One can also relax the compactness hypothesis on ${M}$ to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take ${d}$ equal to ${2n+1}$.

A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:

Theorem 2 (Nash embedding theorem) Let ${(M,g)}$ be a compact Riemannian manifold. Then there exists a isometric embedding ${u: M \rightarrow {\bf R}^d}$ from ${M}$ to a Euclidean space ${{\bf R}^d}$.

In order to obtain the isometric embedding, the dimension ${d}$ has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound

$\displaystyle d = \max( n(n+5)/2, n(n+3)/2 + 5) \ \ \ \ \ (1)$

is attained, which I believe is still the record for large ${n}$. (In the converse direction, one cannot do better than ${d = \frac{n(n+1)}{2}}$, basically because this is the number of degrees of freedom in the Riemannian metric ${g}$.) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).

I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of ${d}$, but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in ${d}$ is that it allows one to freely exploit the very useful tool of pairing together two maps ${u_1: M \rightarrow {\bf R}^{d_1}}$, ${u_2: M \rightarrow {\bf R}^{d_2}}$ to form a combined map ${(u_1,u_2): M \rightarrow {\bf R}^{d_1+d_2}}$ that can be closer to an embedding or an isometric embedding than the original maps ${u_1,u_2}$. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of ${M}$ and then pairs them together to form a “better” embedding.

In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.

In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number of useful topologies on them, including the operator norm topology, strong operator topology, and the weak operator topology. For function spaces, one can use topologies associated to various modes of convergence, such as uniform convergence, pointwise convergence, locally uniform convergence, or convergence in the sense of distributions. (A small minority of such modes are not topologisable, though, the most common of which is pointwise almost everywhere convergence; see Exercise 8 of this previous post).

Some of these topologies are much stronger than others (in that they contain many more open sets, or equivalently that they have many fewer convergent sequences and nets). However, even the weakest topologies used in analysis (e.g. convergence in distributions) tend to be Hausdorff, since this at least ensures the uniqueness of limits of sequences and nets, which is a fundamentally useful feature for analysis. On the other hand, some Hausdorff topologies used are “better” than others in that many more analysis tools are available for those topologies. In particular, topologies that come from Banach space norms are particularly valued, as such topologies (and their attendant norm and metric structures) grant access to many convenient additional results such as the Baire category theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.

Of course, most topologies placed on a vector space will not come from Banach space norms. For instance, if one takes the space ${C_0({\bf R})}$ of continuous functions on ${{\bf R}}$ that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm ${\| \|_{L^\infty}}$), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. ${L^1}$ convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.

I recently realised (while teaching a graduate class in real analysis) that the closed graph theorem provides a quick explanation for why Banach space topologies are so rare:

Proposition 1 Let ${V = (V, {\mathcal F})}$ be a Hausdorff topological vector space. Then, up to equivalence of norms, there is at most one norm ${\| \|}$ one can place on ${V}$ so that ${(V,\| \|)}$ is a Banach space whose topology is at least as strong as ${{\mathcal F}}$. In particular, there is at most one topology stronger than ${{\mathcal F}}$ that comes from a Banach space norm.

Proof: Suppose one had two norms ${\| \|_1, \| \|_2}$ on ${V}$ such that ${(V, \| \|_1)}$ and ${(V, \| \|_2)}$ were both Banach spaces with topologies stronger than ${{\mathcal F}}$. Now consider the graph of the identity function ${\hbox{id}: V \rightarrow V}$ from the Banach space ${(V, \| \|_1)}$ to the Banach space ${(V, \| \|_2)}$. This graph is closed; indeed, if ${(x_n,x_n)}$ is a sequence in this graph that converged in the product topology to ${(x,y)}$, then ${x_n}$ converges to ${x}$ in ${\| \|_1}$ norm and hence in ${{\mathcal F}}$, and similarly ${x_n}$ converges to ${y}$ in ${\| \|_2}$ norm and hence in ${{\mathcal F}}$. But limits are unique in the Hausdorff topology ${{\mathcal F}}$, so ${x=y}$. Applying the closed graph theorem (see also previous discussions on this theorem), we see that the identity map is continuous from ${(V, \| \|_1)}$ to ${(V, \| \|_2)}$; similarly for the inverse. Thus the norms ${\| \|_1, \| \|_2}$ are equivalent as claimed. $\Box$

By using various generalisations of the closed graph theorem, one can generalise the above proposition to Fréchet spaces, or even to F-spaces. The proposition can fail if one drops the requirement that the norms be stronger than a specified Hausdorff topology; indeed, if ${V}$ is infinite dimensional, one can use a Hamel basis of ${V}$ to construct a linear bijection on ${V}$ that is unbounded with respect to a given Banach space norm ${\| \|}$, and which can then be used to give an inequivalent Banach space structure on ${V}$.

One can interpret Proposition 1 as follows: once one equips a vector space with some “weak” (but still Hausdorff) topology, there is a canonical choice of “strong” topology one can place on that space that is stronger than the “weak” topology but arises from a Banach space structure (or at least a Fréchet or F-space structure), provided that at least one such structure exists. In the case of function spaces, one can usually use the topology of convergence in distribution as the “weak” Hausdorff topology for this purpose, since this topology is weaker than almost all of the other topologies used in analysis. This helps justify the common practice of describing a Banach or Fréchet function space just by giving the set of functions that belong to that space (e.g. ${{\mathcal S}({\bf R}^n)}$ is the space of Schwartz functions on ${{\bf R}^n}$) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on ${{\mathcal S}({\bf R}^n)}$ given by the usual Schwartz space seminorms).

Of course, there are still some topological vector spaces which have no “strong topology” arising from a Banach space at all. Consider for instance the space ${c_c({\bf N})}$ of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm ${\| \|}$ stronger than this topology that makes this space a Banach space. For, if there were, then letting ${e_1,e_2,e_3,\dots}$ be the standard basis of ${c_c({\bf N})}$, the series ${\sum_{n=1}^\infty 2^{-n} e_n / \| e_n \|}$ would have to converge in ${\| \|}$, and hence pointwise, to an element of ${c_c({\bf N})}$, but the only available pointwise limit for this series lies outside of ${c_c({\bf N})}$. But I do not know if there is an easily checkable criterion to test whether a given vector space (equipped with a Hausdorff “weak” toplogy) can be equipped with a stronger Banach space (or Fréchet space or ${F}$-space) topology.

There is a very nice recent paper by Lemke Oliver and Soundararajan (complete with a popular science article about it by the consistently excellent Erica Klarreich for Quanta) about a surprising (but now satisfactorily explained) bias in the distribution of pairs of consecutive primes ${p_n, p_{n+1}}$ when reduced to a small modulus ${q}$.

This phenomenon is superficially similar to the more well known Chebyshev bias concerning the reduction of a single prime ${p_n}$ to a small modulus ${q}$, but is in fact a rather different (and much stronger) bias than the Chebyshev bias, and seems to arise from a completely different source. The Chebyshev bias asserts, roughly speaking, that a randomly selected prime ${p}$ of a large magnitude ${x}$ will typically (though not always) be slightly more likely to be a quadratic non-residue modulo ${q}$ than a quadratic residue, but the bias is small (the difference in probabilities is only about ${O(1/\sqrt{x})}$ for typical choices of ${x}$), and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${\Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${\Lambda}$ is quite unbiased modulo ${q}$. The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}$. (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias. (See this article of Rubinstein and Sarnak for a more technical discussion of the Chebyshev bias, and this survey of Granville and Martin for an accessible introduction. The story of the Chebyshev bias is also related to Skewes’ number, once considered the largest explicit constant to naturally appear in a mathematical argument.)

The paper of Lemke Oliver and Soundararajan considers instead the distribution of the pairs ${(p_n \hbox{ mod } q, p_{n+1} \hbox{ mod } q)}$ for small ${q}$ and for large consecutive primes ${p_n, p_{n+1}}$, say drawn at random from the primes comparable to some large ${x}$. For sake of discussion let us just take ${q=3}$. Then all primes ${p_n}$ larger than ${3}$ are either ${1 \hbox{ mod } 3}$ or ${2 \hbox{ mod } 3}$; Chebyshev’s bias gives a very slight preference to the latter (of order ${O(1/\sqrt{x})}$, as discussed above), but apart from this, we expect the primes to be more or less equally distributed in both classes. For instance, assuming GRH, the probability that ${p_n}$ lands in ${1 \hbox{ mod } 3}$ would be ${1/2 + O( x^{-1/2+o(1)} )}$, and similarly for ${2 \hbox{ mod } 3}$.

In view of this, one would expect that up to errors of ${O(x^{-1/2+o(1)})}$ or so, the pair ${(p_n \hbox{ mod } 3, p_{n+1} \hbox{ mod } 3)}$ should be equally distributed amongst the four options ${(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$, ${(1 \hbox{ mod } 3, 2 \hbox{ mod } 3)}$, ${(2 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$, ${(2 \hbox{ mod } 3, 2 \hbox{ mod } 3)}$, thus for instance the probability that this pair is ${(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$ would naively be expected to be ${1/4 + O(x^{-1/2+o(1)})}$, and similarly for the other three tuples. These assertions are not yet proven (although some non-trivial upper and lower bounds for such probabilities can be obtained from recent work of Maynard).

However, Lemke Oliver and Soundararajan argue (backed by both plausible heuristic arguments (based ultimately on the Hardy-Littlewood prime tuples conjecture), as well as substantial numerical evidence) that there is a significant bias away from the tuples ${(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$ and ${(2 \hbox{ mod } 3, 2 \hbox{ mod } 3)}$ – informally, adjacent primes don’t like being in the same residue class! For instance, they predict that the probability of attaining ${(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$ is in fact

$\displaystyle \frac{1}{4} - \frac{1}{8} \frac{\log\log x}{\log x} + O( \frac{1}{\log x} )$

with similar predictions for the other three pairs (in fact they give a somewhat more precise prediction than this). The magnitude of this bias, being comparable to ${\log\log x / \log x}$, is significantly stronger than the Chebyshev bias of ${O(1/\sqrt{x})}$.

One consequence of this prediction is that the prime gaps ${p_{n+1}-p_n}$ are slightly less likely to be divisible by ${3}$ than naive random models of the primes would predict. Indeed, if the four options ${(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$, ${(1 \hbox{ mod } 3, 2 \hbox{ mod } 3)}$, ${(2 \hbox{ mod } 3, 1 \hbox{ mod } 3)}$, ${(2 \hbox{ mod } 3, 2 \hbox{ mod } 3)}$ all occurred with equal probability ${1/4}$, then ${p_{n+1}-p_n}$ should equal ${0 \hbox{ mod } 3}$ with probability ${1/2}$, and ${1 \hbox{ mod } 3}$ and ${2 \hbox{ mod } 3}$ with probability ${1/4}$ each (as would be the case when taking the difference of two random numbers drawn from those integers not divisible by ${3}$); but the Lemke Oliver-Soundararajan bias predicts that the probability of ${p_{n+1}-p_n}$ being divisible by three should be slightly lower, being approximately ${1/2 - \frac{1}{4} \frac{\log\log x}{\log x}}$.

Below the fold we will give a somewhat informal justification of (a simplified version of) this phenomenon, based on the Lemke Oliver-Soundararajan calculation using the prime tuples conjecture.

I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup. (I recently spoke on this work in the recent conference in Princeton in honour of Sergiu Klainerman; my slides for that talk are here.)

One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity ${\omega = \nabla \times u}$, in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become

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

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

(Throughout this post we will be working in three spatial dimensions ${{\bf R}^3}$.) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.

Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case ${\nu=0}$ of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity (which, as explained in the above slides, heuristically means that the effects of viscosity are negligible when compared against the nonlinear components of the equation). In vorticity-stream formulation, the Euler equations can be written as

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

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

As discussed in this previous blog post, a natural generalisation of this system of equations is the system

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

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

where ${T}$ is a linear operator on divergence-free vector fields that is “zeroth order” in some sense; ideally it should also be invertible, self-adjoint, and positive definite (in order to have a Hamiltonian that is comparable to the kinetic energy ${\frac{1}{2} \int_{{\bf R}^3} |u|^2}$). (In the previous blog post, it was observed that the surface quasi-geostrophic (SQG) equation could be embedded in a system of the form (1).) The system (1) has many features in common with the Euler equations; for instance vortex lines are transported by the velocity field ${u}$, and Kelvin’s circulation theorem is still valid.

So far, I have not been able to fully achieve this goal. However, I have the following partial result, stated somewhat informally:

Theorem 1 There is a “zeroth order” linear operator ${T}$ (which, unfortunately, is not invertible, self-adjoint, or positive definite) for which the system (1) exhibits smooth solutions that blowup in finite time.

The operator ${T}$ constructed is not quite a zeroth-order pseudodifferential operator; it is instead merely in the “forbidden” symbol class ${S^0_{1,1}}$, and more precisely it takes the form

$\displaystyle T v = \sum_{j \in {\bf Z}} 2^{3j} \langle v, \phi_j \rangle \psi_j \ \ \ \ \ (2)$

for some compactly supported divergence-free ${\phi,\psi}$ of mean zero with

$\displaystyle \phi_j(x) := \phi(2^j x); \quad \psi_j(x) := \psi(2^j x)$

being ${L^2}$ rescalings of ${\phi,\psi}$. This operator is still bounded on all ${L^p({\bf R}^3)}$ spaces ${1 < p < \infty}$, and so is arguably still a zeroth order operator, though not as convincingly as I would like. Another, less significant, issue with the result is that the solution constructed does not have good spatial decay properties, but this is mostly for convenience and it is likely that the construction can be localised to give solutions that have reasonable decay in space. But the biggest drawback of this theorem is the fact that ${T}$ is not invertible, self-adjoint, or positive definite, so in particular there is no non-negative Hamiltonian for this equation. It may be that some modification of the arguments below can fix these issues, but I have so far been unable to do so. Still, the construction does show that the circulation theorem is insufficient by itself to prevent blowup.

We sketch the proof of the above theorem as follows. We use the barrier method, introducing the time-varying hyperboloid domains

$\displaystyle \Omega(t) := \{ (r,\theta,z): r^2 \leq 1-t + z^2 \}$

for ${t>0}$ (expressed in cylindrical coordinates ${(r,\theta,z)}$). We will select initial data ${\omega(0)}$ to be ${\omega(0,r,\theta,z) = (0,0,\eta(r))}$ for some non-negative even bump function ${\eta}$ supported on ${[-1,1]}$, normalised so that

$\displaystyle \int\int \eta(r)\ r dr d\theta = 1;$

in particular ${\omega(0)}$ is divergence-free supported in ${\Omega(0)}$, with vortex lines connecting ${z=-\infty}$ to ${z=+\infty}$. Suppose for contradiction that we have a smooth solution ${\omega}$ to (1) with this initial data; to simplify the discussion we assume that the solution behaves well at spatial infinity (this can be justified with the choice (2) of vorticity-stream operator, but we will not do so here). Since the domains ${\Omega(t)}$ disconnect ${z=-\infty}$ from ${z=+\infty}$ at time ${t=1}$, there must exist a time ${0 < T_* < 1}$ which is the first time where the support of ${\omega(T_*)}$ touches the boundary of ${\Omega(T_*)}$, with ${\omega(t)}$ supported in ${\Omega(t)}$.

From (1) we see that the support of ${\omega(t)}$ is transported by the velocity field ${u(t)}$. Thus, at the point of contact of the support of ${\omega(T_*)}$ with the boundary of ${\Omega(T_*)}$, the inward component of the velocity field ${u(T_*)}$ cannot exceed the inward velocity of ${\Omega(T_*)}$. We will construct the functions ${\phi,\psi}$ so that this is not the case, leading to the desired contradiction. (Geometrically, what is going on here is that the operator ${T}$ is pinching the flow to pass through the narrow cylinder ${\{ z, r = O( \sqrt{1-t} )\}}$, leading to a singularity by time ${t=1}$ at the latest.)

First we observe from conservation of circulation, and from the fact that ${\omega(t)}$ is supported in ${\Omega(t)}$, that the integrals

$\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta$

are constant in both space and time for ${0 \leq t \leq T_*}$. From the choice of initial data we thus have

$\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta = 1$

for all ${t \leq T_*}$ and all ${z}$. On the other hand, if ${T}$ is of the form (2) with ${\phi = \nabla \times \eta}$ for some bump function ${\eta = (0,0,\eta_z)}$ that only has ${z}$-components, then ${\phi}$ is divergence-free with mean zero, and

$\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = 2^{-j} \langle (-\Delta) (\nabla \times \omega), \nabla \times \eta_j \rangle$

$\displaystyle = 2^{-j} \langle \omega, \eta_j \rangle$

$\displaystyle = 2^{-j} \int\int\int \omega_z(t,r,\theta,z) \eta_z(2^j r, \theta, 2^j z)\ r dr d\theta dz,$

where ${\eta_j(x) := \eta(2^j x)}$. We choose ${\eta_z}$ to be supported in the slab ${\{ C \leq z \leq 2C\}}$ for some large constant ${C}$, and to equal a function ${f(z)}$ depending only on ${z}$ on the cylinder ${\{ C \leq z \leq 2C; r \leq 10C \}}$, normalised so that ${\int f(z)\ dz = 1}$. If ${C/2^j \geq (1-t)^{1/2}}$, then ${\Omega(t)}$ passes through this cylinder, and we conclude that

$\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = -2^{-j} \int f(2^j z)\ dz$

$\displaystyle = 2^{-2j}.$

Inserting ths into (2), (1) we conclude that

$\displaystyle u = \sum_{j: C/2^j \geq (1-t)^{1/2}} 2^j \psi_j + \sum_{j: C/2^j < (1-t)^{1/2}} c_j(t) \psi_j$

for some coefficients ${c_j(t)}$. We will not be able to control these coefficients ${c_j(t)}$, but fortunately we only need to understand ${u}$ on the boundary ${\partial \Omega(t)}$, for which ${r+|z| \gg (1-t)^{1/2}}$. So, if ${\psi}$ happens to be supported on an annulus ${1 \ll r+|z| \ll 1}$, then ${\psi_j}$ vanishes on ${\partial \Omega(t)}$ if ${C}$ is large enough. We then have

$\displaystyle u = \sum_j 2^j \psi_j$

on the boundary of ${\partial \Omega(t)}$.

Let ${\Phi(r,\theta,z)}$ be a function of the form

$\displaystyle \Phi(r,\theta,z) = C z \varphi(z/r)$

where ${\varphi}$ is a bump function supported on ${[-2,2]}$ that equals ${1}$ on ${[-1,1]}$. We can perform a dyadic decomposition ${\Phi = \sum_j \Psi_j}$ where

$\displaystyle \Psi_j(r,\theta,z) = \Phi(r,\theta,z) a(2^j r)$

where ${a}$ is a bump function supported on ${[1/2,2]}$ with ${\sum_j a(2^j r) = 1}$. If we then set

$\displaystyle \psi_j = \frac{2^{-j}}{r} (-\partial_z \Psi_j, 0, \partial_r \Psi_j)$

then one can check that ${\psi_j(x) = \psi(2^j x)}$ for a function ${\psi}$ that is divergence-free and mean zero, and supported on the annulus ${1 \ll r+|z| \ll 1}$, and

$\displaystyle \sum_j 2^j \psi_j = \frac{1}{r} (-\partial_z \Phi, 0, \partial_r \Phi)$

so on ${\partial \Omega(t)}$ (where ${|z| \leq r}$) we have

$\displaystyle u = (-\frac{C}{r}, 0, 0 ).$

One can manually check that the inward velocity of this vector on ${\partial \Omega(t)}$ exceeds the inward velocity of ${\Omega(t)}$ if ${C}$ is large enough, and the claim follows.

Remark 2 The type of blowup suggested by this construction, where a unit amount of circulation is squeezed into a narrow cylinder, is of “Type II” with respect to the Navier-Stokes scaling, because Navier-Stokes-critical norms such ${L^3({\bf R}^3)}$ (or at least ${L^{3,\infty}({\bf R}^3)}$) look like they stay bounded during this squeezing procedure (the velocity field is of size about ${2^j}$ in cylinders of radius and length about ${2^j}$). So even if the various issues with ${T}$ are repaired, it does not seem likely that this construction can be directly adapted to obtain a corresponding blowup for a Navier-Stokes type equation. To get a “Type I” blowup that is consistent with Kelvin’s circulation theorem, it seems that one needs to coil the vortex lines around a loop multiple times in order to get increased circulation in a small space. This seems possible to pull off to me – there don’t appear to be any unavoidable obstructions coming from topology, scaling, or conservation laws – but would require a more complicated construction than the one given above.

In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely

Theorem 1 (Two-dimensional Vinogradov main conjecture) One has

$\displaystyle \int_{[0,1]^2} |\sum_{j=0}^N e( j x + j^2 y)|^6\ dx dy \ll N^{3+o(1)}$

as ${N \rightarrow \infty}$.

This particular case of the main conjecture has a classical proof using some elementary number theory. Indeed, the left-hand side can be viewed as the number of solutions to the system of equations

$\displaystyle j_1 + j_2 + j_3 = k_1 + k_2 + k_3$

$\displaystyle j_1^2 + j_2^2 + j_3^2 = k_1^2 + k_2^2 + k_3^2$

with ${j_1,j_2,j_3,k_1,k_2,k_3 \in \{0,\dots,N\}}$. These two equations can combine (using the algebraic identity ${(a+b-c)^2 - (a^2+b^2-c^2) = 2 (a-c)(b-c)}$ applied to ${(a,b,c) = (j_1,j_2,k_3), (k_1,k_2,j_3)}$) to imply the further equation

$\displaystyle (j_1 - k_3) (j_2 - k_3) = (k_1 - j_3) (k_2 - j_3)$

which, when combined with the divisor bound, shows that each ${k_1,k_2,j_3}$ is associated to ${O(N^{o(1)})}$ choices of ${j_1,j_2,k_3}$ excluding diagonal cases when two of the ${j_1,j_2,j_3,k_1,k_2,k_3}$ collide, and this easily yields Theorem 1. However, the Bourgain-Demeter-Guth argument (which, in the two dimensional case, is essentially contained in a previous paper of Bourgain and Demeter) does not require the divisor bound, and extends for instance to the the more general case where ${j}$ ranges in a ${1}$-separated set of reals between ${0}$ to ${N}$.

In this special case, the Bourgain-Demeter argument simplifies, as the lower dimensional inductive hypothesis becomes a simple ${L^2}$ almost orthogonality claim, and the multilinear Kakeya estimate needed is also easy (collapsing to just Fubini’s theorem). Also one can work entirely in the context of the Vinogradov main conjecture, and not turn to the increased generality of decoupling inequalities (though this additional generality is convenient in higher dimensions). As such, I am presenting this special case as an introduction to the Bourgain-Demeter-Guth machinery.

We now give the specialisation of the Bourgain-Demeter argument to Theorem 1. It will suffice to establish the bound

$\displaystyle \int_{[0,1]^2} |\sum_{j=0}^N e( j x + j^2 y)|^p\ dx dy \ll N^{p/2+o(1)}$

for all ${4, (where we keep ${p}$ fixed and send ${N}$ to infinity), as the ${L^6}$ bound then follows by combining the above bound with the trivial bound ${|\sum_{j=0}^N e( j x + j^2 x^2)| \ll N}$. Accordingly, for any ${\eta > 0}$ and ${4, we let ${P(p,\eta)}$ denote the claim that

$\displaystyle \int_{[0,1]^2} |\sum_{j=0}^N e( j x + j^2 y)|^p\ dx dy \ll N^{p/2+\eta+o(1)}$

as ${N \rightarrow \infty}$. Clearly, for any fixed ${p}$, ${P(p,\eta)}$ holds for some large ${\eta}$, and it will suffice to establish

Proposition 2 Let ${4, and let ${\eta>0}$ be such that ${P(p,\eta)}$ holds. Then there exists ${0 < \eta' < \eta}$ such that ${P(p,\eta')}$ holds.

Indeed, this proposition shows that for ${4, the infimum of the ${\eta}$ for which ${P(p,\eta)}$ holds is zero.

We prove the proposition below the fold, using a simplified form of the methods discussed in the previous blog post. To simplify the exposition we will be a bit cavalier with the uncertainty principle, for instance by essentially ignoring the tails of rapidly decreasing functions.

Given any finite collection of elements ${(f_i)_{i \in I}}$ in some Banach space ${X}$, the triangle inequality tells us that

$\displaystyle \| \sum_{i \in I} f_i \|_X \leq \sum_{i \in I} \|f_i\|_X.$

However, when the ${f_i}$ all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. For instance, if ${X}$ is a Hilbert space and the ${f_i}$ are mutually orthogonal, we have the Pythagorean theorem

$\displaystyle \| \sum_{i \in I} f_i \|_X = (\sum_{i \in I} \|f_i\|_X^2)^{1/2}.$

For sake of comparison, from the triangle inequality and Cauchy-Schwarz one has the general inequality

$\displaystyle \| \sum_{i \in I} f_i \|_X \leq (\# I)^{1/2} (\sum_{i \in I} \|f_i\|_X^2)^{1/2} \ \ \ \ \ (1)$

for any finite collection ${(f_i)_{i \in I}}$ in any Banach space ${X}$, where ${\# I}$ denotes the cardinality of ${I}$. Thus orthogonality in a Hilbert space yields “square root cancellation”, saving a factor of ${(\# I)^{1/2}}$ or so over the trivial bound coming from the triangle inequality.

More generally, let us somewhat informally say that a collection ${(f_i)_{i \in I}}$ exhibits decoupling in ${X}$ if one has the Pythagorean-like inequality

$\displaystyle \| \sum_{i \in I} f_i \|_X \ll_\varepsilon (\# I)^\varepsilon (\sum_{i \in I} \|f_i\|_X^2)^{1/2}$

for any ${\varepsilon>0}$, thus one obtains almost the full square root cancellation in the ${X}$ norm. The theory of almost orthogonality can then be viewed as the theory of decoupling in Hilbert spaces such as ${L^2({\bf R}^n)}$. In ${L^p}$ spaces for ${p < 2}$ one usually does not expect this sort of decoupling; for instance, if the ${f_i}$ are disjointly supported one has

$\displaystyle \| \sum_{i \in I} f_i \|_{L^p} = (\sum_{i \in I} \|f_i\|_{L^p}^p)^{1/p}$

and the right-hand side can be much larger than ${(\sum_{i \in I} \|f_i\|_{L^p}^2)^{1/2}}$ when ${p < 2}$. At the opposite extreme, one usually does not expect to get decoupling in ${L^\infty}$, since one could conceivably align the ${f_i}$ to all attain a maximum magnitude at the same location with the same phase, at which point the triangle inequality in ${L^\infty}$ becomes sharp.

However, in some cases one can get decoupling for certain ${2 < p < \infty}$. For instance, suppose we are in ${L^4}$, and that ${f_1,\dots,f_N}$ are bi-orthogonal in the sense that the products ${f_i f_j}$ for ${1 \leq i < j \leq N}$ are pairwise orthogonal in ${L^2}$. Then we have

$\displaystyle \| \sum_{i = 1}^N f_i \|_{L^4}^2 = \| (\sum_{i=1}^N f_i)^2 \|_{L^2}$

$\displaystyle = \| \sum_{1 \leq i,j \leq N} f_i f_j \|_{L^2}$

$\displaystyle \ll (\sum_{1 \leq i,j \leq N} \|f_i f_j \|_{L^2}^2)^{1/2}$

$\displaystyle = \| (\sum_{1 \leq i,j \leq N} |f_i f_j|^2)^{1/2} \|_{L^2}$

$\displaystyle = \| \sum_{i=1}^N |f_i|^2 \|_{L^2}$

$\displaystyle \leq \sum_{i=1}^N \| |f_i|^2 \|_{L^2}$

$\displaystyle = \sum_{i=1}^N \|f_i\|_{L^4}^2$

giving decoupling in ${L^4}$. (Similarly if each of the ${f_i f_j}$ is orthogonal to all but ${O_\varepsilon( N^\varepsilon )}$ of the other ${f_{i'} f_{j'}}$.) A similar argument also gives ${L^6}$ decoupling when one has tri-orthogonality (with the ${f_i f_j f_k}$ mostly orthogonal to each other), and so forth. As a slight variant, Khintchine’s inequality also indicates that decoupling should occur for any fixed ${2 < p < \infty}$ if one multiplies each of the ${f_i}$ by an independent random sign ${\epsilon_i \in \{-1,+1\}}$.

In recent years, Bourgain and Demeter have been establishing decoupling theorems in ${L^p({\bf R}^n)}$ spaces for various key exponents of ${2 < p < \infty}$, in the “restriction theory” setting in which the ${f_i}$ are Fourier transforms of measures supported on different portions of a given surface or curve; this builds upon the earlier decoupling theorems of Wolff. In a recent paper with Guth, they established the following decoupling theorem for the curve ${\gamma({\bf R}) \subset {\bf R}^n}$ parameterised by the polynomial curve

$\displaystyle \gamma: t \mapsto (t, t^2, \dots, t^n).$

For any ball ${B = B(x_0,r)}$ in ${{\bf R}^n}$, let ${w_B: {\bf R}^n \rightarrow {\bf R}^+}$ denote the weight

$\displaystyle w_B(x) := \frac{1}{(1 + \frac{|x-x_0|}{r})^{100n}},$

which should be viewed as a smoothed out version of the indicator function ${1_B}$ of ${B}$. In particular, the space ${L^p(w_B) = L^p({\bf R}^n, w_B(x)\ dx)}$ can be viewed as a smoothed out version of the space ${L^p(B)}$. For future reference we observe a fundamental self-similarity of the curve ${\gamma({\bf R})}$: any arc ${\gamma(I)}$ in this curve, with ${I}$ a compact interval, is affinely equivalent to the standard arc ${\gamma([0,1])}$.

Theorem 1 (Decoupling theorem) Let ${n \geq 1}$. Subdivide the unit interval ${[0,1]}$ into ${N}$ equal subintervals ${I_i}$ of length ${1/N}$, and for each such ${I_i}$, let ${f_i: {\bf R}^n \rightarrow {\bf R}}$ be the Fourier transform

$\displaystyle f_i(x) = \int_{\gamma(I_i)} e(x \cdot \xi)\ d\mu_i(\xi)$

of a finite Borel measure ${\mu_i}$ on the arc ${\gamma(I_i)}$, where ${e(\theta) := e^{2\pi i \theta}}$. Then the ${f_i}$ exhibit decoupling in ${L^{n(n+1)}(w_B)}$ for any ball ${B}$ of radius ${N^n}$.

Orthogonality gives the ${n=1}$ case of this theorem. The bi-orthogonality type arguments sketched earlier only give decoupling in ${L^p}$ up to the range ${2 \leq p \leq 2n}$; the point here is that we can now get a much larger value of ${n}$. The ${n=2}$ case of this theorem was previously established by Bourgain and Demeter (who obtained in fact an analogous theorem for any curved hypersurface). The exponent ${n(n+1)}$ (and the radius ${N^n}$) is best possible, as can be seen by the following basic example. If

$\displaystyle f_i(x) := \int_{I_i} e(x \cdot \gamma(\xi)) g_i(\xi)\ d\xi$

where ${g_i}$ is a bump function adapted to ${I_i}$, then standard Fourier-analytic computations show that ${f_i}$ will be comparable to ${1/N}$ on a rectangular box of dimensions ${N \times N^2 \times \dots \times N^n}$ (and thus volume ${N^{n(n+1)/2}}$) centred at the origin, and exhibit decay away from this box, with ${\|f_i\|_{L^{n(n+1)}(w_B)}}$ comparable to

$\displaystyle 1/N \times (N^{n(n+1)/2})^{1/(n(n+1))} = 1/\sqrt{N}.$

On the other hand, ${\sum_{i=1}^N f_i}$ is comparable to ${1}$ on a ball of radius comparable to ${1}$ centred at the origin, so ${\|\sum_{i=1}^N f_i\|_{L^{n(n+1)}(w_B)}}$ is ${\gg 1}$, which is just barely consistent with decoupling. This calculation shows that decoupling will fail if ${n(n+1)}$ is replaced by any larger exponent, and also if the radius of the ball ${B}$ is reduced to be significantly smaller than ${N^n}$.

This theorem has the following consequence of importance in analytic number theory:

Corollary 2 (Vinogradov main conjecture) Let ${s, n, N \geq 1}$ be integers, and let ${\varepsilon > 0}$. Then

$\displaystyle \int_{[0,1]^n} |\sum_{j=1}^N e( j x_1 + j^2 x_2 + \dots + j^n x_n)|^{2s}\ dx_1 \dots dx_n$

$\displaystyle \ll_{\varepsilon,s,n} N^{s+\varepsilon} + N^{2s - \frac{n(n+1)}{2}+\varepsilon}.$

Proof: By the Hölder inequality (and the trivial bound of ${N}$ for the exponential sum), it suffices to treat the critical case ${s = n(n+1)/2}$, that is to say to show that

$\displaystyle \int_{[0,1]^n} |\sum_{j=1}^N e( j x_1 + j^2 x_2 + \dots + j^n x_n)|^{n(n+1)}\ dx_1 \dots dx_n \ll_{\varepsilon,n} N^{\frac{n(n+1)}{2}+\varepsilon}.$

We can rescale this as

$\displaystyle \int_{[0,N] \times [0,N^2] \times \dots \times [0,N^n]} |\sum_{j=1}^N e( x \cdot \gamma(j/N) )|^{n(n+1)}\ dx \ll_{\varepsilon,n} N^{3\frac{n(n+1)}{2}+\varepsilon}.$

As the integrand is periodic along the lattice ${N{\bf Z} \times N^2 {\bf Z} \times \dots \times N^n {\bf Z}}$, this is equivalent to

$\displaystyle \int_{[0,N^n]^n} |\sum_{j=1}^N e( x \cdot \gamma(j/N) )|^{n(n+1)}\ dx \ll_{\varepsilon,n} N^{\frac{n(n+1)}{2}+n^2+\varepsilon}.$

The left-hand side may be bounded by ${\ll \| \sum_{j=1}^N f_j \|_{L^{n(n+1)}(w_B)}^{n(n+1)}}$, where ${B := B(0,N^n)}$ and ${f_j(x) := e(x \cdot \gamma(j/N))}$. Since

$\displaystyle \| f_j \|_{L^{n(n+1)}(w_B)} \ll (N^{n^2})^{\frac{1}{n(n+1)}},$

the claim now follows from the decoupling theorem and a brief calculation. $\Box$

Using the Plancherel formula, one may equivalently (when ${s}$ is an integer) write the Vinogradov main conjecture in terms of solutions ${j_1,\dots,j_s,k_1,\dots,k_s \in \{1,\dots,N\}}$ to the system of equations

$\displaystyle j_1^i + \dots + j_s^i = k_1^i + \dots + k_s^i \forall i=1,\dots,n,$

but we will not use this formulation here.

A history of the Vinogradov main conjecture may be found in this survey of Wooley; prior to the Bourgain-Demeter-Guth theorem, the conjecture was solved completely for ${n \leq 3}$, or for ${n > 3}$ and ${s}$ either below ${n(n+1)/2 - n/3 + O(n^{2/3})}$ or above ${n(n-1)}$, with the bulk of recent progress coming from the efficient congruencing technique of Wooley. It has numerous applications to exponential sums, Waring’s problem, and the zeta function; to give just one application, the main conjecture implies the predicted asymptotic for the number of ways to express a large number as the sum of ${23}$ fifth powers (the previous best result required ${28}$ fifth powers). The Bourgain-Demeter-Guth approach to the Vinogradov main conjecture, based on decoupling, is ostensibly very different from the efficient congruencing technique, which relies heavily on the arithmetic structure of the program, but it appears (as I have been told from second-hand sources) that the two methods are actually closely related, with the former being a sort of “Archimedean” version of the latter (with the intervals ${I_i}$ in the decoupling theorem being analogous to congruence classes in the efficient congruencing method); hopefully there will be some future work making this connection more precise. One advantage of the decoupling approach is that it generalises to non-arithmetic settings in which the set ${\{1,\dots,N\}}$ that ${j}$ is drawn from is replaced by some other similarly separated set of real numbers. (A random thought – could this allow the Vinogradov-Korobov bounds on the zeta function to extend to Beurling zeta functions?)

Below the fold we sketch the Bourgain-Demeter-Guth argument proving Theorem 1.

I thank Jean Bourgain and Andrew Granville for helpful discussions.

Let ${\lambda}$ denote the Liouville function. The prime number theorem is equivalent to the estimate

$\displaystyle \sum_{n \leq x} \lambda(n) = o(x)$

as ${x \rightarrow \infty}$, that is to say that ${\lambda}$ exhibits cancellation on large intervals such as ${[1,x]}$. This result can be improved to give cancellation on shorter intervals. For instance, using the known zero density estimates for the Riemann zeta function, one can establish that

$\displaystyle \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n)|\ dx = o( HX ) \ \ \ \ \ (1)$

as ${X \rightarrow \infty}$ if ${X^{1/6+\varepsilon} \leq H \leq X}$ for some fixed ${\varepsilon>0}$; I believe this result is due to Ramachandra (see also Exercise 21 of this previous blog post), and in fact one could obtain a better error term on the right-hand side that for instance gained an arbitrary power of ${\log X}$. On the Riemann hypothesis (or the weaker density hypothesis), it was known that the ${X^{1/6+\varepsilon}}$ could be lowered to ${X^\varepsilon}$.

Early this year, there was a major breakthrough by Matomaki and Radziwill, who (among other things) showed that the asymptotic (1) was in fact valid for any ${H = H(X)}$ with ${H \leq X}$ that went to infinity as ${X \rightarrow \infty}$, thus yielding cancellation on extremely short intervals. This has many further applications; for instance, this estimate, or more precisely its extension to other “non-pretentious” bounded multiplicative functions, was a key ingredient in my recent solution of the Erdös discrepancy problem, as well as in obtaining logarithmically averaged cases of Chowla’s conjecture, such as

$\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1)}{n} = o(\log x). \ \ \ \ \ (2)$

It is of interest to twist the above estimates by phases such as the linear phase ${n \mapsto e(\alpha n) := e^{2\pi i \alpha n}}$. In 1937, Davenport showed that

$\displaystyle \sup_\alpha |\sum_{n \leq x} \lambda(n) e(\alpha n)| \ll_A x \log^{-A} x$

which of course improves the prime number theorem. Recently with Matomaki and Radziwill, we obtained a common generalisation of this estimate with (1), showing that

$\displaystyle \sup_\alpha \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(HX) \ \ \ \ \ (3)$

as ${X \rightarrow \infty}$, for any ${H = H(X) \leq X}$ that went to infinity as ${X \rightarrow \infty}$. We were able to use this estimate to obtain an averaged form of Chowla’s conjecture.

In that paper, we asked whether one could improve this estimate further by moving the supremum inside the integral, that is to say to establish the bound

$\displaystyle \int_X^{2X} \sup_\alpha |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(HX) \ \ \ \ \ (4)$

as ${X \rightarrow \infty}$, for any ${H = H(X) \leq X}$ that went to infinity as ${X \rightarrow \infty}$. This bound is asserting that ${\lambda}$ is locally Fourier-uniform on most short intervals; it can be written equivalently in terms of the “local Gowers ${U^2}$ norm” as

$\displaystyle \int_X^{2X} \sum_{1 \leq a \leq H} |\sum_{x \leq n \leq x+H} \lambda(n) \lambda(n+a)|^2\ dx = o( H^3 X )$

from which one can see that this is another averaged form of Chowla’s conjecture (stronger than the one I was able to prove with Matomaki and Radziwill, but a consequence of the unaveraged Chowla conjecture). If one inserted such a bound into the machinery I used to solve the Erdös discrepancy problem, it should lead to further averaged cases of Chowla’s conjecture, such as

$\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1) \lambda(n+2)}{n} = o(\log x), \ \ \ \ \ (5)$

though I have not fully checked the details of this implication. It should also have a number of new implications for sign patterns of the Liouville function, though we have not explored these in detail yet.

One can write (4) equivalently in the form

$\displaystyle \int_X^{2X} \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )\ dx = o(HX) \ \ \ \ \ (6)$

uniformly for all ${x}$-dependent phases ${\alpha(x), \beta(x)}$. In contrast, (3) is equivalent to the subcase of (6) when the linear phase coefficient ${\alpha(x)}$ is independent of ${x}$. This dependency of ${\alpha(x)}$ on ${x}$ seems to necessitate some highly nontrivial additive combinatorial analysis of the function ${x \mapsto \alpha(x)}$ in order to establish (4) when ${H}$ is small. To date, this analysis has proven to be elusive, but I would like to record what one can do with more classical methods like Vaughan’s identity, namely:

Proposition 1 The estimate (4) (or equivalently (6)) holds in the range ${X^{2/3+\varepsilon} \leq H \leq X}$ for any fixed ${\varepsilon>0}$. (In fact one can improve the right-hand side by an arbitrary power of ${\log X}$ in this case.)

The values of ${H}$ in this range are far too large to yield implications such as new cases of the Chowla conjecture, but it appears that the ${2/3}$ exponent is the limit of “classical” methods (at least as far as I was able to apply them), in the sense that one does not do any combinatorial analysis on the function ${x \mapsto \alpha(x)}$, nor does one use modern equidistribution results on “Type III sums” that require deep estimates on Kloosterman-type sums. The latter may shave a little bit off of the ${2/3}$ exponent, but I don’t see how one would ever hope to go below ${1/2}$ without doing some non-trivial combinatorics on the function ${x \mapsto \alpha(x)}$. UPDATE: I have come across this paper of Zhan which uses mean-value theorems for L-functions to lower the ${2/3}$ exponent to ${5/8}$.

Let me now sketch the proof of the proposition, omitting many of the technical details. We first remark that known estimates on sums of the Liouville function (or similar functions such as the von Mangoldt function) in short arithmetic progressions, based on zero-density estimates for Dirichlet ${L}$-functions, can handle the “major arc” case of (4) (or (6)) where ${\alpha}$ is restricted to be of the form ${\alpha = \frac{a}{q} + O( X^{-1/6-\varepsilon} )}$ for ${q = O(\log^{O(1)} X)}$ (the exponent here being of the same numerology as the ${X^{1/6+\varepsilon}}$ exponent in the classical result of Ramachandra, tied to the best zero density estimates currently available); for instance a modification of the arguments in this recent paper of Koukoulopoulos would suffice. Thus we can restrict attention to “minor arc” values of ${\alpha}$ (or ${\alpha(x)}$, using the interpretation of (6)).

Next, one breaks up ${\lambda}$ (or the closely related Möbius function) into Dirichlet convolutions using one of the standard identities (e.g. Vaughan’s identity or Heath-Brown’s identity), as discussed for instance in this previous post (which is focused more on the von Mangoldt function, but analogous identities exist for the Liouville and Möbius functions). The exact choice of identity is not terribly important, but the upshot is that ${\lambda(n)}$ can be decomposed into ${\log^{O(1)} X}$ terms, each of which is either of the “Type I” form

$\displaystyle \sum_{d \sim D; m \sim M: dm=n} a_d$

for some coefficients ${a_d}$ that are roughly of logarithmic size on the average, and scales ${D, M}$ with ${D \ll X^{2/3}}$ and ${DM \sim X}$, or else of the “Type II” form

$\displaystyle \sum_{d \sim D; m \sim M: dm=n} a_d b_m$

for some coefficients ${a_d, b_m}$ that are roughly of logarithmic size on the average, and scales ${D,M}$ with ${X^{1/3} \ll D,M \ll X^{2/3}}$ and ${DM \sim X}$. As discussed in the previous post, the ${2/3}$ exponent is a natural barrier in these identities if one is unwilling to also consider “Type III” type terms which are roughly of the shape of the third divisor function ${\tau_3(n) := \sum_{d_1d_2d_3=1} 1}$.

A Type I sum makes a contribution to ${ \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )}$ that can be bounded (via Cauchy-Schwarz) in terms of an expression such as

$\displaystyle \sum_{d \sim D} | \sum_{x/d \leq m \leq x/d+H/d} e(\alpha(x) dm )|^2.$

The inner sum exhibits a lot of cancellation unless ${\alpha(x) d}$ is within ${O(D/H)}$ of an integer. (Here, “a lot” should be loosely interpreted as “gaining many powers of ${\log X}$ over the trivial bound”.) Since ${H}$ is significantly larger than ${D}$, standard Vinogradov-type manipulations (see e.g. Lemma 13 of these previous notes) show that this bad case occurs for many ${d}$ only when ${\alpha}$ is “major arc”, which is the case we have specifically excluded. This lets us dispose of the Type I contributions.

A Type II sum makes a contribution to ${ \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )}$ roughly of the form

$\displaystyle \sum_{d \sim D} | \sum_{x/d \leq m \leq x/d+H/d} b_m e(\alpha(x) dm)|.$

We can break this up into a number of sums roughly of the form

$\displaystyle \sum_{d = d_0 + O( H / M )} | \sum_{x/d_0 \leq m \leq x/d_0 + H/D} b_m e(\alpha(x) dm)|$

for ${d_0 \sim D}$; note that the ${d}$ range is non-trivial because ${H}$ is much larger than ${M}$. Applying the usual bilinear sum Cauchy-Schwarz methods (e.g. Theorem 14 of these notes) we conclude that there is a lot of cancellation unless one has ${\alpha(x) = a/q + O( \frac{X \log^{O(1)} X}{H^2} )}$ for some ${q = O(\log^{O(1)} X)}$. But with ${H \geq X^{2/3+\varepsilon}}$, ${X \log^{O(1)} X/H^2}$ is well below the threshold ${X^{-1/6-\varepsilon}}$ for the definition of major arc, so we can exclude this case and obtain the required cancellation.

A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum).

Theorem 1 (Halasz inequality) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a multiplicative function bounded in magnitude by ${1}$, and suppose that ${x \geq 3}$, ${T \geq 1}$, and ${ M \geq 0}$ are such that

$\displaystyle \sum_{p \leq x} \frac{1 - \hbox{Re}(f(p) p^{-it})}{p} \geq M \ \ \ \ \ (1)$

for all real numbers ${t}$ with ${|t| \leq T}$. Then

$\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{\sqrt{T}}.$

As a qualitative corollary, we conclude (by standard compactness arguments) that if

$\displaystyle \sum_{p} \frac{1 - \hbox{Re}(f(p) p^{-it})}{p} = +\infty$

for all real ${t}$, then

$\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) = o(1) \ \ \ \ \ (2)$

as ${x \rightarrow \infty}$. In the more recent work of this paper of Granville and Soundararajan, the sharper bound

$\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{T} + \frac{\log\log x}{\log x}$

is obtained (with a more precise description of the ${(1+M) e^{-M}}$ term).

The usual proofs of Halasz’s theorem are somewhat lengthy (though there has been a recent simplification, in forthcoming work of Granville, Harper, and Soundarajan). Below the fold I would like to give a relatively short proof of the following “cheap” version of the inequality, which has slightly weaker quantitative bounds, but still suffices to give qualitative conclusions such as (2).

Theorem 2 (Cheap Halasz inequality) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a multiplicative function bounded in magnitude by ${1}$. Let ${T \geq 1}$ and ${M \geq 0}$, and suppose that ${x}$ is sufficiently large depending on ${T,M}$. If (1) holds for all ${|t| \leq T}$, then

$\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M/2} + \frac{1}{T}.$

The non-optimal exponent ${1/2}$ can probably be improved a bit by being more careful with the exponents, but I did not try to optimise it here. A similar bound appears in the first paper of Halasz on this topic.

The idea of the argument is to split ${f}$ as a Dirichlet convolution ${f = f_1 * f_2 * f_3}$ where ${f_1,f_2,f_3}$ is the portion of ${f}$ coming from “small”, “medium”, and “large” primes respectively (with the dividing line between the three types of primes being given by various powers of ${x}$). Using a Perron-type formula, one can express this convolution in terms of the product of the Dirichlet series of ${f_1,f_2,f_3}$ respectively at various complex numbers ${1+it}$ with ${|t| \leq T}$. One can use ${L^2}$ based estimates to control the Dirichlet series of ${f_2,f_3}$, while using the hypothesis (1) one can get ${L^\infty}$ estimates on the Dirichlet series of ${f_1}$. (This is similar to the Fourier-analytic approach to ternary additive problems, such as Vinogradov’s theorem on representing large odd numbers as the sum of three primes.) This idea was inspired by a similar device used in the work of Granville, Harper, and Soundarajan. A variant of this argument also appears in unpublished work of Adam Harper.

I thank Andrew Granville for helpful comments which led to significant simplifications of the argument.