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Kaisa Matomäki, Maksym Radziwill, and I just uploaded to the arXiv our paper “Fourier uniformity of bounded multiplicative functions in short intervals on average“. This paper is the outcome of our attempts during the MSRI program in analytic number theory last year to attack the local Fourier uniformity conjecture for the Liouville function {\lambda}. This conjecture generalises a landmark result of Matomäki and Radziwill, who show (among other things) that one has the asymptotic

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

whenever {X \rightarrow \infty} and {H = H(X)} goes to infinity as {X \rightarrow \infty}. Informally, this says that the Liouville function has small mean for almost all short intervals {[x,x+H]}. The remarkable thing about this theorem is that there is no lower bound on how {H} goes to infinity with {X}; one can take for instance {H = \log\log\log X}. This lack of lower bound was crucial when I applied this result (or more precisely, a generalisation of this result to arbitrary non-pretentious bounded multiplicative functions) a few years ago to solve the Erdös discrepancy problem, as well as a logarithmically averaged two-point Chowla conjecture, for instance it implies that

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

The local Fourier uniformity conjecture asserts the stronger asymptotic

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

under the same hypotheses on {H} and {X}. As I worked out in a previous paper, this conjecture would imply a logarithmically averaged three-point Chowla conjecture, implying for instance that

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

This particular bound also follows from some slightly different arguments of Joni Teräväinen and myself, but the implication would also work for other non-pretentious bounded multiplicative functions, whereas the arguments of Joni and myself rely more heavily on the specific properties of the Liouville function (in particular that {\lambda(p)=-1} for all primes {p}).

There is also a higher order version of the local Fourier uniformity conjecture in which the linear phase {{}e(-\alpha n)} is replaced with a polynomial phase such as {e(-\alpha_d n^d - \dots - \alpha_1 n - \alpha_0)}, or more generally a nilsequence {\overline{F(g(n) \Gamma)}}; as shown in my previous paper, this conjecture implies (and is in fact equivalent to, after logarithmic averaging) a logarithmically averaged version of the full Chowla conjecture (not just the two-point or three-point versions), as well as a logarithmically averaged version of the Sarnak conjecture.

The main result of the current paper is to obtain some cases of the local Fourier uniformity conjecture:

Theorem 1 The asymptotic (2) is true when {H = X^\theta} for a fixed {\theta > 0}.

Previously this was known for {\theta > 5/8} by the work of Zhan (who in fact proved the stronger pointwise assertion {\sup_{\alpha \in {\bf R}} |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha n)|= o(H)} for {X \leq x \leq 2X} in this case). In a previous paper with Kaisa and Maksym, we also proved a weak version

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

of (2) for any {H} growing arbitrarily slowly with {X}; this is stronger than (1) (and is in fact proven by a variant of the method) but significantly weaker than (2), because in the latter the worst-case {\alpha} is permitted to depend on the {x} parameter, whereas in (3) {\alpha} must remain independent of {x}.

Unfortunately, the restriction {H = X^\theta} is not strong enough to give applications to Chowla-type conjectures (one would need something more like {H = \log^\theta X} for this). However, it can still be used to control some sums that had not previously been manageable. For instance, a quick application of the circle method lets one use the above theorem to derive the asymptotic

\displaystyle  \sum_{h \leq H} \sum_{n \leq X} \lambda(n) \Lambda(n+h) \Lambda(n+2h) = o( H X )

whenever {H = X^\theta} for a fixed {\theta > 0}, where {\Lambda} is the von Mangoldt function. Amusingly, the seemingly simpler question of establishing the expected asymptotic for

\displaystyle  \sum_{h \leq H} \sum_{n \leq X} \Lambda(n+h) \Lambda(n+2h)

is only known in the range {\theta \geq 1/6} (from the work of Zaccagnini). Thus we have a rare example of a number theory sum that becomes easier to control when one inserts a Liouville function!

We now give an informal description of the strategy of proof of the theorem (though for numerous technical reasons, the actual proof deviates in some respects from the description given here). If (2) failed, then for many values of {x \in [X,2X]} we would have the lower bound

\displaystyle  |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha_x n)| \gg 1

for some frequency {\alpha_x \in{\bf R}}. We informally describe this correlation between {\lambda(n)} and {e(\alpha_x n)} by writing

\displaystyle  \lambda(n) \approx e(\alpha_x n) \ \ \ \ \ (4)

for {n \in [x,x+H]} (informally, one should view this as asserting that {\lambda(n)} “behaves like” a constant multiple of {e(\alpha_x n)}). For sake of discussion, suppose we have this relationship for all {x \in [X,2X]}, not just many.

As mentioned before, the main difficulty here is to understand how {\alpha_x} varies with {x}. As it turns out, the multiplicativity properties of the Liouville function place a significant constraint on this dependence. Indeed, if we let {p} be a fairly small prime (e.g. of size {H^\varepsilon} for some {\varepsilon>0}), and use the identity {\lambda(np) = \lambda(n) \lambda(p) = - \lambda(n)} for the Liouville function to conclude (at least heuristically) from (4) that

\displaystyle  \lambda(n) \approx e(\alpha_x n p)

for {n \in [x/p, x/p + H/p]}. (In practice, we will have this sort of claim for many primes {p} rather than all primes {p}, after using tools such as the Turán-Kubilius inequality, but we ignore this distinction for this informal argument.)

Now let {x, y \in [X,2X]} and {p,q \sim P} be primes comparable to some fixed range {P = H^\varepsilon} such that

\displaystyle  x/p = y/q + O( H/P). \ \ \ \ \ (5)

Then we have both

\displaystyle  \lambda(n) \approx e(\alpha_x n p)

and

\displaystyle  \lambda(n) \approx e(\alpha_y n q)

on essentially the same range of {n} (two nearby intervals of length {\sim H/P}). This suggests that the frequencies {p \alpha_x} and {q \alpha_y} should be close to each other modulo {1}, in particular one should expect the relationship

\displaystyle  p \alpha_x = q \alpha_y + O( \frac{P}{H} ) \hbox{ mod } 1. \ \ \ \ \ (6)

Comparing this with (5) one is led to the expectation that {\alpha_x} should depend inversely on {x} in some sense (for instance one can check that

\displaystyle  \alpha_x = T/x \ \ \ \ \ (7)

would solve (6) if {T = O( X / H^2 )}; by Taylor expansion, this would correspond to a global approximation of the form {\lambda(n) \approx n^{iT}}). One now has a problem of an additive combinatorial flavour (or of a “local to global” flavour), namely to leverage the relation (6) to obtain global control on {\alpha_x} that resembles (7).

A key obstacle in solving (6) efficiently is the fact that one only knows that {p \alpha_x} and {q \alpha_y} are close modulo {1}, rather than close on the real line. One can start resolving this problem by the Chinese remainder theorem, using the fact that we have the freedom to shift (say) {\alpha_y} by an arbitrary integer. After doing so, one can arrange matters so that one in fact has the relationship

\displaystyle  p \alpha_x = q \alpha_y + O( \frac{P}{H} ) \hbox{ mod } p \ \ \ \ \ (8)

whenever {x,y \in [X,2X]} and {p,q \sim P} obey (5). (This may force {\alpha_q} to become extremely large, on the order of {\prod_{p \sim P} p}, but this will not concern us.)

Now suppose that we have {y,y' \in [X,2X]} and primes {q,q' \sim P} such that

\displaystyle  y/q = y'/q' + O(H/P). \ \ \ \ \ (9)

For every prime {p \sim P}, we can find an {x} such that {x/p} is within {O(H/P)} of both {y/q} and {y'/q'}. Applying (8) twice we obtain

\displaystyle  p \alpha_x = q \alpha_y + O( \frac{P}{H} ) \hbox{ mod } p

and

\displaystyle  p \alpha_x = q' \alpha_{y'} + O( \frac{P}{H} ) \hbox{ mod } p

and thus by the triangle inequality we have

\displaystyle  q \alpha_y = q' \alpha_{y'} + O( \frac{P}{H} ) \hbox{ mod } p

for all {p \sim P}; hence by the Chinese remainder theorem

\displaystyle  q \alpha_y = q' \alpha_{y'} + O( \frac{P}{H} ) \hbox{ mod } \prod_{p \sim P} p.

In practice, in the regime {H = X^\theta} that we are considering, the modulus {\prod_{p \sim P} p} is so huge we can effectively ignore it (in the spirit of the Lefschetz principle); so let us pretend that we in fact have

\displaystyle  q \alpha_y = q' \alpha_{y'} + O( \frac{P}{H} ) \ \ \ \ \ (10)

whenever {y,y' \in [X,2X]} and {q,q' \sim P} obey (9).

Now let {k} be an integer to be chosen later, and suppose we have primes {p_1,\dots,p_k,q_1,\dots,q_k \sim P} such that the difference

\displaystyle  q = |p_1 \dots p_k - q_1 \dots q_k|

is small but non-zero. If {k} is chosen so that

\displaystyle  P^k \approx \frac{X}{H}

(where one is somewhat loose about what {\approx} means) then one can then find real numbers {x_1,\dots,x_k \sim X} such that

\displaystyle  \frac{x_j}{p_j} = \frac{x_{j+1}}{q_j} + O( \frac{H}{P} )

for {j=1,\dots,k}, with the convention that {x_{k+1} = x_1}. We then have

\displaystyle  p_j \alpha_{x_j} = q_j \alpha_{x_{j+1}} + O( \frac{P}{H} )

which telescopes to

\displaystyle  p_1 \dots p_k \alpha_{x_1} = q_1 \dots q_k \alpha_{x_1} + O( \frac{P^k}{H} )

and thus

\displaystyle  q \alpha_{x_1} = O( \frac{P^k}{H} )

and hence

\displaystyle  \alpha_{x_1} = O( \frac{P^k}{H} ) \approx O( \frac{X}{H^2} ).

In particular, for each {x \sim X}, we expect to be able to write

\displaystyle  \alpha_x = \frac{T_x}{x} + O( \frac{1}{H} )

for some {T_x = O( \frac{X^2}{H^2} )}. This quantity {T_x} can vary with {x}; but from (10) and a short calculation we see that

\displaystyle  T_y = T_{y'} + O( \frac{X}{H} )

whenever {y, y' \in [X,2X]} obey (9) for some {q,q' \sim P}.

Now imagine a “graph” in which the vertices are elements {y} of {[X,2X]}, and two elements {y,y'} are joined by an edge if (9) holds for some {q,q' \sim P}. Because of exponential sum estimates on {\sum_{q \sim P} q^{it}}, this graph turns out to essentially be an “expander” in the sense that any two vertices {y,y' \in [X,2X]} can be connected (in multiple ways) by fairly short paths in this graph (if one allows one to modify one of {y} or {y'} by {O(H)}). As a consequence, we can assume that this quantity {T_y} is essentially constant in {y} (cf. the application of the ergodic theorem in this previous blog post), thus we now have

\displaystyle  \alpha_x = \frac{T}{x} + O(\frac{1}{H} )

for most {x \in [X,2X]} and some {T = O(X^2/H^2)}. By Taylor expansion, this implies that

\displaystyle  \lambda(n) \approx n^{iT}

on {[x,x+H]} for most {x}, thus

\displaystyle  \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n) n^{-iT}|\ dx \gg HX.

But this can be shown to contradict the Matomäki-Radziwill theorem (because the multiplicative function {n \mapsto \lambda(n) n^{-iT}} is known to be non-pretentious).

I’ve just uploaded to the arXiv my paper “Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion“, submitted to Revista Matematica Iberoamericana. This paper concerns the extent to which one can accurately embed the metric structure of the Heisenberg group

\displaystyle H := \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix}

into Euclidean space, which we can write as {\{ [x,y,z]: x,y,z \in {\bf R} \}} with the notation

\displaystyle [x,y,z] := \begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}.

Here we give {H} the right-invariant Carnot-Carathéodory metric {d} coming from the right-invariant vector fields

\displaystyle X := \frac{\partial}{\partial x} + y \frac{\partial}{\partial z}; \quad Y := \frac{\partial}{\partial y}

but not from the commutator vector field

\displaystyle Z := [Y,X] = \frac{\partial}{\partial z}.

This gives {H} the geometry of a Carnot group. As observed by Semmes, it follows from the Carnot group differentiation theory of Pansu that there is no bilipschitz map from {(H,d)} to any Euclidean space {{\bf R}^D} or even to {\ell^2}, since such a map must be differentiable almost everywhere in the sense of Carnot groups, which in particular shows that the derivative map annihilate {Z} almost everywhere, which is incompatible with being bilipschitz.

On the other hand, if one snowflakes the Heisenberg group by replacing the metric {d} with {d^{1-\varepsilon}} for some {0 < \varepsilon < 1}, then it follows from the general theory of Assouad on embedding snowflaked metrics of doubling spaces that {(H,d^{1-\varepsilon})} may be embedded in a bilipschitz fashion into {\ell^2}, or even to {{\bf R}^{D_\varepsilon}} for some {D_\varepsilon} depending on {\varepsilon}.

Of course, the distortion of this bilipschitz embedding must degenerate in the limit {\varepsilon \rightarrow 0}. From the work of Austin-Naor-Tessera and Naor-Neiman it follows that {(H,d^{1-\varepsilon})} may be embedded into {\ell^2} with a distortion of {O( \varepsilon^{-1/2} )}, but no better. The Naor-Neiman paper also embeds {(H,d^{1-\varepsilon})} into a finite-dimensional space {{\bf R}^D} with {D} independent of {\varepsilon}, but at the cost of worsening the distortion to {O(\varepsilon^{-1})}. They then posed the question of whether this worsening of the distortion is necessary.

The main result of this paper answers this question in the negative:

Theorem 1 There exists an absolute constant {D} such that {(H,d^{1-\varepsilon})} may be embedded into {{\bf R}^D} in a bilipschitz fashion with distortion {O(\varepsilon^{-1/2})} for any {0 < \varepsilon \leq 1/2}.

To motivate the proof of this theorem, let us first present a bilipschitz map {\Phi: {\bf R} \rightarrow \ell^2} from the snowflaked line {({\bf R},d_{\bf R}^{1-\varepsilon})} (with {d_{\bf R}} being the usual metric on {{\bf R}}) into complex Hilbert space {\ell^2({\bf C})}. The map is given explicitly as a Weierstrass type function

\displaystyle \Phi(x) := \sum_{k \in {\bf Z}} 2^{-\varepsilon k} (\phi_k(x) - \phi_k(0))

where for each {k}, {\phi_k: {\bf R} \rightarrow \ell^2} is the function

\displaystyle \phi_k(x) := 2^k e^{2\pi i x / 2^k} e_k.

and {(e_k)_{k \in {\bf Z}}} are an orthonormal basis for {\ell^2({\bf C})}. The subtracting of the constant {\phi_k(0)} is purely in order to make the sum convergent as {k \rightarrow \infty}. If {x,y \in {\bf R}} are such that {2^{k_0-2} \leq d_{\bf R}(x,y) \leq 2^{k_0-1}} for some integer {k_0}, one can easily check the bounds

\displaystyle |\phi_k(x) - \phi_k(y)| \lesssim d_{\bf R}(x,y)^{(1-\varepsilon)} \min( 2^{-(1-\varepsilon) (k_0-k)}, 2^{-\varepsilon (k-k_0)} )

with the lower bound

\displaystyle |\phi_{k_0}(x) - \phi_{k_0}(y)| \gtrsim d_{\bf R}(x,y)^{(1-\varepsilon)}

at which point one finds that

\displaystyle d_{\bf R}(x,y)^{1-\varepsilon} \lesssim |\Phi(x) - \Phi(y)| \lesssim \varepsilon^{-1/2} d_{\bf R}(x,y)^{1-\varepsilon}

as desired.

The key here was that each function {\phi_k} oscillated at a different spatial scale {2^k}, and the functions were all orthogonal to each other (so that the upper bound involved a factor of {\varepsilon^{-1/2}} rather than {\varepsilon^{-1}}). One can replicate this example for the Heisenberg group without much difficulty. Indeed, if we let {\Gamma := \{ [a,b,c]: a,b,c \in {\bf Z} \}} be the discrete Heisenberg group, then the nilmanifold {H/\Gamma} is a three-dimensional smooth compact manifold; thus, by the Whitney embedding theorem, it smoothly embeds into {{\bf R}^6}. This gives a smooth immersion {\phi: H \rightarrow {\bf R}^6} which is {\Gamma}-automorphic in the sense that {\phi(p\gamma) = \phi(p)} for all {p \in H} and {\gamma \in \Gamma}. If one then defines {\phi_k: H \rightarrow \ell^2 \otimes {\bf R}^6} to be the function

\displaystyle \phi_k(p) := 2^k \phi( \delta_{2^{-k}}(p) ) \otimes e_k

where {\delta_\lambda: H \rightarrow H} is the scaling map

\displaystyle \delta_\lambda([x,y,z]) := [\lambda x, \lambda y, \lambda^2 z],

then one can repeat the previous arguments to obtain the required bilipschitz bounds

\displaystyle d(p,q)^{1-\varepsilon} \lesssim |\Phi(p) - \Phi(q) \lesssim \varepsilon^{-1/2} d(p,q)^{1-\varepsilon}

for the function

\displaystyle \Phi(p) :=\sum_{k \in {\bf Z}} 2^{-\varepsilon k} (\phi_k(p) - \phi_k(0)).

To adapt this construction to bounded dimension, the main obstruction was the requirement that the {\phi_k} took values in orthogonal subspaces. But if one works things out carefully, it is enough to require the weaker orthogonality requirement

\displaystyle B( \phi_{k_0}, \sum_{k>k_0} 2^{-\varepsilon(k-k_0)} \phi_k ) = 0

for all {k_0 \in {\bf Z}}, where {B(\phi, \psi): H \rightarrow {\bf R}^2} is the bilinear form

\displaystyle B(\phi,\psi) := (X \phi \cdot X \psi, Y \phi \cdot Y \psi ).

One can then try to construct the {\phi_k: H \rightarrow {\bf R}^D} for bounded dimension {D} by an iterative argument. After some standard reductions, the problem becomes this (roughly speaking): given a smooth, slowly varying function {\psi: H \rightarrow {\bf R}^{D}} whose derivatives obey certain quantitative upper and lower bounds, construct a smooth oscillating function {\phi: H \rightarrow {\bf R}^{D}}, whose derivatives also obey certain quantitative upper and lower bounds, which obey the equation

\displaystyle B(\phi,\psi) = 0. \ \ \ \ \ (1)

 

We view this as an underdetermined system of differential equations for {\phi} (two equations in {D} unknowns; after some reductions, our {D} can be taken to be the explicit value {36}). The trivial solution {\phi=0} to this equation will be inadmissible for our purposes due to the lower bounds we will require on {\phi} (in order to obtain the quantitative immersion property mentioned previously, as well as for a stronger “freeness” property that is needed to close the iteration). Because this construction will need to be iterated, it will be essential that the regularity control on {\phi} is the same as that on {\psi}; one cannot afford to “lose derivatives” when passing from {\psi} to {\phi}.

This problem has some formal similarities with the isometric embedding problem (discussed for instance in this previous post), which can be viewed as the problem of solving an equation of the form {Q(\phi,\phi) = g}, where {(M,g)} is a Riemannian manifold and {Q} is the bilinear form

\displaystyle Q(\phi,\psi)_{ij} = \partial_i \phi \cdot \partial_j \psi.

The isometric embedding problem also has the key obstacle that naive attempts to solve the equation {Q(\phi,\phi)=g} iteratively can lead to an undesirable “loss of derivatives” that prevents one from iterating indefinitely. This obstacle was famously resolved by the Nash-Moser iteration scheme in which one alternates between perturbatively adjusting an approximate solution to improve the residual error term, and mollifying the resulting perturbation to counteract the loss of derivatives. The current equation (1) differs in some key respects from the isometric embedding equation {Q(\phi,\phi)=g}, in particular being linear in the unknown field {\phi} rather than quadratic; nevertheless the key obstacle is the same, namely that naive attempts to solve either equation lose derivatives. Our approach to solving (1) was inspired by the Nash-Moser scheme; in retrospect, I also found similarities with Uchiyama’s constructive proof of the Fefferman-Stein decomposition theorem, discussed in this previous post (and in this recent one).

To motivate this iteration, we first express {B(\phi,\psi)} using the product rule in a form that does not place derivatives directly on the unknown {\phi}:

\displaystyle B(\phi,\psi) = \left( W(\phi \cdot W \psi) - \phi \cdot WW \psi\right)_{W = X,Y} \ \ \ \ \ (2)

 

This reveals that one can construct solutions {\phi} to (1) by solving the system of equations

\displaystyle \phi \cdot W \psi = \phi \cdot WW \psi = 0 \ \ \ \ \ (3)

 

for {W \in \{X, Y \}}. Because this system is zeroth order in {\phi}, this can easily be done by linear algebra (even in the presence of a forcing term {B(\phi,\psi)=F}) if one imposes a “freeness” condition (analogous to the notion of a free embedding in the isometric embedding problem) that {X \psi(p), Y \psi(p), XX \psi(p), YY \psi(p)} are linearly independent at each point {p}, which (together with some other technical conditions of a similar nature) one then adds to the list of upper and lower bounds required on {\psi} (with a related bound then imposed on {\phi}, in order to close the iteration). However, as mentioned previously, there is a “loss of derivatives” problem with this construction: due to the presence of the differential operators {W} in (3), a solution {\phi} constructed by this method can only be expected to have two degrees less regularity than {\psi} at best, which makes this construction unsuitable for iteration.

To get around this obstacle (which also prominently appears when solving (linearisations of) the isometric embedding equation {Q(\phi,\phi)=g}), we instead first construct a smooth, low-frequency solution {\phi_{\leq N_0} \colon H \rightarrow {\bf R}^{D}} to a low-frequency equation

\displaystyle B( \phi_{\leq N_0}, P_{\leq N_0} \psi ) = 0 \ \ \ \ \ (4)

 

where {P_{\leq N_0} \psi} is a mollification of {\psi} (of Littlewood-Paley type) applied at a small spatial scale {1/N_0} for some {N_0}, and then gradually relax the frequency cutoff {P_{\leq N_0}} to deform this low frequency solution {\phi_{\leq N_0}} to a solution {\phi} of the actual equation (1).

We will construct the low-frequency solution {\phi_{\leq N_0}} rather explicitly, using the Whitney embedding theorem to construct an initial oscillating map {f} into a very low dimensional space {{\bf R}^6}, composing it with a Veronese type embedding into a slightly larger dimensional space {{\bf R}^{27}} to obtain a required “freeness” property, and then composing further with a slowly varying isometry {U(p) \colon {\bf R}^{27} \rightarrow {\bf R}^{36}} depending on {P_{\leq N_0}} and constructed by a quantitative topological lemma (relying ultimately on the vanishing of the first few homotopy groups of high-dimensional spheres), in order to obtain the required orthogonality (4). (This sort of “quantitative null-homotopy” was first proposed by Gromov, with some recent progress on optimal bounds by Chambers-Manin-Weinberger and by Chambers-Dotterer-Manin-Weinberger, but we will not need these more advanced results here, as one can rely on the classical qualitative vanishing {\pi^k(S^d)=0} for {k < d} together with a compactness argument to obtain (ineffective) quantitative bounds, which suffice for this application).

To perform the deformation of {\phi_{\leq N_0}} into {\phi}, we must solve what is essentially the linearised equation

\displaystyle B( \dot \phi, \psi ) + B( \phi, \dot \psi ) = 0 \ \ \ \ \ (5)

 

of (1) when {\phi}, {\psi} (viewed as low frequency functions) are both being deformed at some rates {\dot \phi, \dot \psi} (which should be viewed as high frequency functions). To avoid losing derivatives, the magnitude of the deformation {\dot \phi} in {\phi} should not be significantly greater than the magnitude of the deformation {\dot \psi} in {\psi}, when measured in the same function space norms.

As before, if one directly solves the difference equation (5) using a naive application of (2) with {B(\phi,\dot \psi)} treated as a forcing term, one will lose at least one derivative of regularity when passing from {\dot \psi} to {\dot \phi}. However, observe that (2) (and the symmetry {B(\phi, \dot \psi) = B(\dot \psi,\phi)}) can be used to obtain the identity

\displaystyle B( \dot \phi, \psi ) + B( \phi, \dot \psi ) = \left( W(\dot \phi \cdot W \psi + \dot \psi \cdot W \phi) - (\dot \phi \cdot WW \psi + \dot \psi \cdot WW \phi)\right)_{W = X,Y} \ \ \ \ \ (6)

 

and then one can solve (5) by solving the system of equations

\displaystyle \dot \phi \cdot W \psi = - \dot \psi \cdot W \phi

for {W \in \{X,XX,Y,YY\}}. The key point here is that this system is zeroth order in both {\dot \phi} and {\dot \psi}, so one can solve this system without losing any derivatives when passing from {\dot \psi} to {\dot \phi}; compare this situation with that of the superficially similar system

\displaystyle \dot \phi \cdot W \psi = - \phi \cdot W \dot \psi

that one would obtain from naively linearising (3) without exploiting the symmetry of {B}. There is still however one residual “loss of derivatives” problem arising from the presence of a differential operator {W} on the {\phi} term, which prevents one from directly evolving this iteration scheme in time without losing regularity in {\phi}. It is here that we borrow the final key idea of the Nash-Moser scheme, which is to replace {\phi} by a mollified version {P_{\leq N} \phi} of itself (where the projection {P_{\leq N}} depends on the time parameter). This creates an error term in (5), but it turns out that this error term is quite small and smooth (being a “high-high paraproduct” of {\nabla \phi} and {\nabla\psi}, it ends up being far more regular than either {\phi} or {\psi}, even with the presence of the derivatives) and can be iterated away provided that the initial frequency cutoff {N_0} is large and the function {\psi} has a fairly high (but finite) amount of regularity (we will eventually use the Hölder space {C^{20,\alpha}} on the Heisenberg group to measure this).

About six years ago on this blog, I started thinking about trying to make a web-based game based around high-school algebra, and ended up using Scratch to write a short but playable puzzle game in which one solves linear equations for an unknown {x} using a restricted set of moves. (At almost the same time, there were a number of more professionally made games released along similar lines, most notably Dragonbox.)

Since then, I have thought a couple times about whether there were other parts of mathematics which could be gamified in a similar fashion. Shortly after my first blog posts on this topic, I experimented with a similar gamification of Lewis Carroll’s classic list of logic puzzles, but the results were quite clunky, and I was never satisfied with the results.

Over the last few weeks I returned to this topic though, thinking in particular about how to gamify the rules of inference of propositional logic, in a manner that at least vaguely resembles how mathematicians actually go about making logical arguments (e.g., splitting into cases, arguing by contradiction, using previous result as lemmas to help with subsequent ones, and so forth). The rules of inference are a list of a dozen or so deductive rules concerning propositional sentences (things like “({A} AND {B}) OR (NOT {C})”, where {A,B,C} are some formulas). A typical such rule is Modus Ponens: if the sentence {A} is known to be true, and the implication “{A} IMPLIES {B}” is also known to be true, then one can deduce that {B} is also true. Furthermore, in this deductive calculus it is possible to temporarily introduce some unproven statements as an assumption, only to discharge them later. In particular, we have the deduction theorem: if, after making an assumption {A}, one is able to derive the statement {B}, then one can conclude that the implication “{A} IMPLIES {B}” is true without any further assumption.

It took a while for me to come up with a workable game-like graphical interface for all of this, but I finally managed to set one up, now using Javascript instead of Scratch (which would be hopelessly inadequate for this task); indeed, part of the motivation of this project was to finally learn how to program in Javascript, which turned out to be not as formidable as I had feared (certainly having experience with other C-like languages like C++, Java, or lua, as well as some prior knowledge of HTML, was very helpful). The main code for this project is available here. Using this code, I have created an interactive textbook in the style of a computer game, which I have titled “QED”. This text contains thirty-odd exercises arranged in twelve sections that function as game “levels”, in which one has to use a given set of rules of inference, together with a given set of hypotheses, to reach a desired conclusion. The set of available rules increases as one advances through the text; in particular, each new section gives one or more rules, and additionally each exercise one solves automatically becomes a new deduction rule one can exploit in later levels, much as lemmas and propositions are used in actual mathematics to prove more difficult theorems. The text automatically tries to match available deduction rules to the sentences one clicks on or drags, to try to minimise the amount of manual input one needs to actually make a deduction.

Most of one’s proof activity takes place in a “root environment” of statements that are known to be true (under the given hypothesis), but for more advanced exercises one has to also work in sub-environments in which additional assumptions are made. I found the graphical metaphor of nested boxes to be useful to depict this tree of sub-environments, and it seems to combine well with the drag-and-drop interface.

The text also logs one’s moves in a more traditional proof format, which shows how the mechanics of the game correspond to a traditional mathematical argument. My hope is that this will give students a way to understand the underlying concept of forming a proof in a manner that is more difficult to achieve using traditional, non-interactive textbooks.

I have tried to organise the exercises in a game-like progression in which one first works with easy levels that train the player on a small number of moves, and then introduce more advanced moves one at a time. As such, the order in which the rules of inference are introduced is a little idiosyncratic. The most powerful rule (the law of the excluded middle, which is what separates classical logic from intuitionistic logic) is saved for the final section of the text.

Anyway, I am now satisfied enough with the state of the code and the interactive text that I am willing to make both available (and open source; I selected a CC-BY licence for both), and would be happy to receive feedback on any aspect of the either. In principle one could extend the game mechanics to other mathematical topics than the propositional calculus – the rules of inference for first-order logic being an obvious next candidate – but it seems to make sense to focus just on propositional logic for now.

I have just uploaded to the arXiv my paper “Commutators close to the identity“, submitted to the Journal of Operator Theory. This paper resulted from some progress I made on the problem discussed in this previous post. Recall in that post the following result of Popa: if {D,X \in B(H)} are bounded operators on a Hilbert space {H} whose commutator {[D,X] := DX-XD} is close to the identity in the sense that

\displaystyle  \| [D,X] - I \|_{op} \leq \varepsilon \ \ \ \ \ (1)

for some {\varepsilon > 0}, then one has the lower bound

\displaystyle  \| X \|_{op} \|D \|_{op} \geq \frac{1}{2} \log \frac{1}{\varepsilon}. \ \ \ \ \ (2)

In the other direction, for any {0 < \varepsilon < 1}, there are examples of operators {D,X \in B(H)} obeying (1) such that

\displaystyle  \| X \|_{op} \|D \|_{op} \ll \varepsilon^{-2}. \ \ \ \ \ (3)

In this paper we improve the upper bound to come closer to the lower bound:

Theorem 1 For any {0 < \varepsilon < 1/2}, and any infinite-dimensional {H}, there exist operators {D,X \in B(H)} obeying (1) such that

\displaystyle  \| X \|_{op} \|D \|_{op} \ll \log^{16} \frac{1}{\varepsilon}. \ \ \ \ \ (4)

One can probably improve the exponent {16} somewhat by a modification of the methods, though it does not seem likely that one can lower it all the way to {1} without a substantially new idea. Nevertheless I believe it plausible that the lower bound (2) is close to optimal.

We now sketch the methods of proof. The construction giving (3) proceeded by first identifying {B(H)} with the algebra {M_2(B(H))} of {2 \times 2} matrices that have entries in {B(H)}. It is then possible to find two matrices {D, X \in M_2(B(H))} whose commutator takes the form

\displaystyle  [D,X] = \begin{pmatrix} I & u \\ 0 & I \end{pmatrix}

for some bounded operator {u \in B(H)} (for instance one can take {u} to be an isometry). If one then conjugates {D, X} by the diagonal operator {\mathrm{diag}(\varepsilon,1)}, one can eusure that (1) and (3) both hold.

It is natural to adapt this strategy to {n \times n} matrices {D,X \in M_n(B(H))} rather than {2 \times 2} matrices, where {n} is a parameter at one’s disposal. If one can find matrices {D,X \in M_n(B(H))} that are almost upper triangular (in that only the entries on or above the lower diagonal are non-zero), whose commutator {[D,X]} only differs from the identity in the top right corner, thus

\displaystyle  [D, X] = \begin{pmatrix} I & 0 & 0 & \dots & 0 & S \\ 0 & I & 0 & \dots & 0 & 0 \\ 0 & 0 & I & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & I & 0 \\ 0 & 0 & 0 & \dots & 0 & I \end{pmatrix}.

for some {S}, then by conjugating by a diagonal matrix such as {\mathrm{diag}( \mu^{n-1}, \mu^{n-2}, \dots, 1)} for some {\mu} and optimising in {\mu}, one can improve the bound {\varepsilon^{-2}} in (3) to {O_n( \varepsilon^{-\frac{2}{n-1}} )}; if the bounds in the implied constant in the {O_n(1)} are polynomial in {n}, one can then optimise in {n} to obtain a bound of the form (4) (perhaps with the exponent {16} replaced by a different constant).

The task is then to find almost upper triangular matrices {D, X} whose commutator takes the required form. The lower diagonals of {D,X} must then commute; it took me a while to realise then that one could (usually) conjugate one of the matrices, say {X} by a suitable diagonal matrix, so that the lower diagonal consisted entirely of the identity operator, which would make the other lower diagonal consist of a single operator, say {u}. After a lot of further lengthy experimentation, I eventually realised that one could conjugate {X} further by unipotent upper triangular matrices so that all remaining entries other than those on the far right column vanished. Thus, without too much loss of generality, one can assume that {X} takes the normal form

\displaystyle  X := \begin{pmatrix} 0 & 0 & 0 & \dots & 0 & b_1 \\ I & 0 & 0 & \dots & 0 & b_2 \\ 0 & I & 0 & \dots & 0 & b_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 0 & b_{n-1} \\ 0 & 0 & 0 & \dots & I & b_n \end{pmatrix}.

\displaystyle  D := \begin{pmatrix} v & I & 0 & \dots & 0 & b_1 u \\ u & v & 2 I & \dots & 0 & b_2 u \\ 0 & u & v & \dots & 0 & b_3 u \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & v & (n-1) I + b_{n-1} u \\ 0 & 0 & 0 & \dots & u & v + b_n u \end{pmatrix}

for some {u,v \in B(H)}, solving the system of equations

\displaystyle  [v, b_i] + [u, b_{i-1}] + i b_{i+1} + b_i [u, b_n] = 0 \ \ \ \ \ (5)

for {i=2,\dots,n-1}, and also

\displaystyle  [v, b_n] + [u, b_{n-1}] + b_n [u, b_n] = n \cdot 1_{B(H)}. \ \ \ \ \ (6)

It turns out to be possible to solve this system of equations by a contraction mapping argument if one takes {u,v} to be a “Hilbert’s hotel” pair of isometries as in the previous post, though the contraction is very slight, leading to polynomial losses in {n} in the implied constant.

There is a further question raised in Popa’s paper which I was unable to resolve. As a special case of one of the main theorems (Theorem 2.1) of that paper, the following result was shown: if {A \in B(H)} obeys the bounds

\displaystyle  \|A \| = O(1)

and

\displaystyle  \| A \| = O( \mathrm{dist}( A, {\bf C} + K(H) )^{2/3} ) \ \ \ \ \ (7)

(where {{\bf C} + K(H)} denotes the space of all operators of the form {\lambda I + T} with {\lambda \in {\bf C}} and {T} compact), then there exist operators {D,X \in B(H)} with {\|D\|, \|X\| = O(1)} such that {A = [D,X]}. (In fact, Popa’s result covers a more general situation in which one is working in a properly infinite {W^*} algebra with non-trivial centre.) We sketch a proof of this result as follows. Suppose that {\mathrm{dist}(A, {\bf C} + K(H)) = \varepsilon} and {\|A\| = O( \varepsilon^{2/3})} for some {0 < \varepsilon \ll 1}. A standard greedy algorithm argument (see this paper of Brown and Pearcy) allows one to find orthonormal vectors {e_n, f_n, g_n} for {n=1,2,\dots} such that for each {n}, one has {A e_n = \varepsilon_n f_n + v_n} for some {\varepsilon_n} comparable to {\varepsilon}, and some {v_n} orthogonal to all of the {e_n,f_n,g_n}. After some conjugation (and a suitable identification of {B(H)} with {M_2(B(H))}, one can thus place {A} in a normal form

\displaystyle  A = \begin{pmatrix} \varepsilon^{2/3} x & \varepsilon v^* \\ \varepsilon^{2/3} y & \varepsilon^{2/3} z \end{pmatrix}

where {v \in B(H)} is a isometry with infinite deficiency, and {x,y,z \in B(H)} have norm {O(1)}. Setting {\varepsilon' := \varepsilon^{1/3}}, it then suffices to solve the commutator equation

\displaystyle  [D,X] = \begin{pmatrix} x & \varepsilon' v^* \\ y & z \end{pmatrix}

with {\|D\|_{op} \|X\|_{op} \ll (\varepsilon')^{-2}}; note the similarity with (3).

By the usual Hilbert’s hotel construction, one can complement {v} with another isometry {u} obeying the “Hilbert’s hotel” identity

\displaystyle  uu^* + vv^* = I

and also {u^* u = v^* v = I}, {u^* v = v^* u = 0}. Proceeding as in the previous post, we can try the ansatz

\displaystyle  D = \begin{pmatrix} \frac{1}{2} u^* & 0 \\ a & \frac{1}{2} u^* - v^* \end{pmatrix}, X = \begin{pmatrix} b & \varepsilon' I \\ c & d \end{pmatrix}

for some operators {a,b,c,d \in B(H)}, leading to the system of equations

\displaystyle  [\frac{1}{2} u^*, b] + [\frac{1}{2} u^* - v^*, c] = x+z

\displaystyle  \varepsilon' a = [\frac{1}{2} u^*, b] - x

\displaystyle  \frac{1}{2} u^* c + c (\frac{1}{2} u^* - v^*) + ab-da = y.

Using the first equation to solve for {b,c}, the second to then solve for {a}, and the third to then solve for {c}, one can obtain matrices {D,X} with the required properties.

Thus far, my attempts to extend this construction to larger matrices with good bounds on {D,X} have been unsuccessful. A model problem would be to express

\displaystyle  \begin{pmatrix} I & 0 & \varepsilon v^* \\ 0 & I & 0 \\ 0 & 0 & I \end{pmatrix}

as a commutator {[D,X]} with {\|D\| \|X\|} significantly smaller than {O(\varepsilon^{-2})}. The construction in my paper achieves something like this, but with {v^*} replaced by a more complicated operator. One would also need variants of this result in which one is allowed to perturb the above operator by an arbitrary finite rank operator of bounded operator norm.

Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and I have uploaded to the arXiv our paper “Long gaps in sieved sets“, submitted to J. Europ. Math. Soc..

This paper originated from the MSRI program in analytic number theory last year, and was centred around variants of the question of finding large gaps between primes. As discussed for instance in this previous post, it is now known that within the set of primes {{\mathcal P} = \{2,3,5,\dots\}}, one can find infinitely many adjacent elements {a,b} whose gap {b-a} obeys a lower bound of the form

\displaystyle b-a \gg \log a \frac{\log_2 a \log_4 a}{\log_3 a}

where {\log_k} denotes the {k}-fold iterated logarithm. This compares with the trivial bound of {b-a \gg \log a} that one can obtain from the prime number theorem and the pigeonhole principle. Several years ago, Pomerance posed the question of whether analogous improvements to the trivial bound can be obtained for such sets as

\displaystyle {\mathcal P}_2 = \{ n \in {\bf N}: n^2+1 \hbox{ prime} \}.

Here there is the obvious initial issue that this set is not even known to be infinite (this is the fourth Landau problem), but let us assume for the sake of discussion that this set is indeed infinite, so that we have an infinite number of gaps to speak of. Standard sieve theory techniques give upper bounds for the density of {{\mathcal P}_2} that is comparable (up to an absolute constant) to the prime number theorem bounds for {{\mathcal P}}, so again we can obtain a trivial bound of {b-a \gg \log a} for the gaps of {{\mathcal P}_2}. In this paper we improve this to

\displaystyle b-a \gg \log a \log^c_2 a

for an absolute constant {c>0}; this is not as strong as the corresponding bound for {{\mathcal P}}, but still improves over the trivial bound. In fact we can handle more general “sifted sets” than just {{\mathcal P}_2}. Recall from the sieve of Eratosthenes that the elements of {{\mathcal P}} in, say, the interval {[x/2, x]} can be obtained by removing from {[x/2, x]} one residue class modulo {p} for each prime up to {\sqrt{x}}, namely the class {0} mod {p}. In a similar vein, the elements of {{\mathcal P}_2} in {[x/2,x]} can be obtained by removing for each prime {p} up to {x} zero, one, or two residue classes modulo {p}, depending on whether {-1} is a quadratic residue modulo {p}. On the average, one residue class will be removed (this is a very basic case of the Chebotarev density theorem), so this sieving system is “one-dimensional on the average”. Roughly speaking, our arguments apply to any other set of numbers arising from a sieving system that is one-dimensional on average. (One can consider other dimensions also, but unfortunately our methods seem to give results that are worse than a trivial bound when the dimension is less than or greater than one.)

The standard “Erdős-Rankin” method for constructing long gaps between primes proceeds by trying to line up some residue classes modulo small primes {p} so that they collectively occupy a long interval. A key tool in doing so are the smooth number estimates of de Bruijn and others, which among other things assert that if one removes from an interval such as {[1,x]} all the residue classes {0} mod {p} for {p} between {x^{1/u}} and {x} for some fixed {u>1}, then the set of survivors has exceptionally small density (roughly of the order of {u^{-u}}, with the precise density given by the Dickman function), in marked contrast to the situation in which one randomly removes one residue class for each such prime {p}, in which the density is more like {1/u}. One generally exploits this phenomenon to sieve out almost all the elements of a long interval using some of the primes available, and then using the remaining primes to cover up the remaining elements that have not already been sifted out. In the more recent work on this problem, advanced combinatorial tools such as hypergraph covering lemmas are used for the latter task.

In the case of {{\mathcal P}_2}, there does not appear to be any analogue of smooth numbers, in the sense that there is no obvious way to arrange the residue classes so that they have significantly fewer survivors than a random arrangement. Instead we adopt the following semi-random strategy to cover an interval {[1,y]} by residue classes. Firstly, we randomly remove residue classes for primes {p} up to some intermediate threshold {z} (smaller than {y} by a logarithmic factor), leaving behind a preliminary sifted set {S_{[2,z]}}. Then, for each prime {p} between {z} and another intermediate threshold {x/2}, we remove a residue class mod {p} that maximises (or nearly maximises) its intersection with {S_{[2,z]}}. This ends up reducing the number of survivors to be significantly below what one would achieve if one selects residue classes randomly, particularly if one also uses the hypergraph covering lemma from our previous paper. Finally, we cover each the remaining survivors by a residue class from a remaining available prime.

Brad Rodgers and I have uploaded to the arXiv our paper “The De Bruijn-Newman constant is non-negative“. This paper affirms a conjecture of Newman regarding to the extent to which the Riemann hypothesis, if true, is only “barely so”. To describe the conjecture, let us begin with the Riemann xi function

\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s)

where {\Gamma(s) := \int_0^\infty e^{-t} t^{s-1}\ dt} is the Gamma function and {\zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s}} is the Riemann zeta function. Initially, this function is only defined for {\mathrm{Re} s > 1}, but, as was already known to Riemann, we can manipulate it into a form that extends to the entire complex plane as follows. Firstly, in view of the standard identity {s \Gamma(s) = \Gamma(s+1)}, we can write

\displaystyle \frac{s(s-1)}{2} \Gamma(\frac{s}{2}) = 2 \Gamma(\frac{s+4}{2}) - 3 \Gamma( \frac{s+2}{2} )

and hence

\displaystyle \xi(s) = \sum_{n=1}^\infty 2 \pi^{-s/2} n^{-s} \int_0^\infty e^{-t} t^{\frac{s+4}{2}-1}\ dt - 3 \pi^{-s/2} n^{-s} \int_0^\infty e^{-t} t^{\frac{s+2}{2}-1}\ dt.

By a rescaling, one may write

\displaystyle \int_0^\infty e^{-t} t^{\frac{s+4}{2}-1}\ dt = (\pi n^2)^{\frac{s+4}{2}} \int_0^\infty e^{-\pi n^2 t} t^{\frac{s+4}{2}-1}\ dt

and similarly

\displaystyle \int_0^\infty e^{-t} t^{\frac{s+2}{2}-1}\ dt = (\pi n^2)^{\frac{s+2}{2}} \int_0^\infty e^{-\pi n^2 t} t^{\frac{s+2}{2}-1}\ dt

and thus (after applying Fubini’s theorem)

\displaystyle \xi(s) = \int_0^\infty \sum_{n=1}^\infty 2 \pi^2 n^4 e^{-\pi n^2 t} t^{\frac{s+4}{2}-1} - 3 \pi n^2 e^{-\pi n^2 t} t^{\frac{s+2}{2}-1}\ dt.

We’ll make the change of variables {t = e^{4u}} to obtain

\displaystyle \xi(s) = 4 \int_{\bf R} \sum_{n=1}^\infty (2 \pi^2 n^4 e^{8u} - 3 \pi n^2 e^{4u}) \exp( 2su - \pi n^2 e^{4u} )\ du.

If we introduce the mild renormalisation

\displaystyle H_0(z) := \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} )

of {\xi}, we then conclude (at least for {\mathrm{Im} z > 1}) that

\displaystyle H_0(z) = \frac{1}{2} \int_{\bf R} \Phi(u)\exp(izu)\ du \ \ \ \ \ (1)

 

where {\Phi: {\bf R} \rightarrow {\bf C}} is the function

\displaystyle \Phi(u) := \sum_{n=1}^\infty (2 \pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp( - \pi n^2 e^{4u} ), \ \ \ \ \ (2)

 

which one can verify to be rapidly decreasing both as {u \rightarrow +\infty} and as {u \rightarrow -\infty}, with the decrease as {u \rightarrow +\infty} faster than any exponential. In particular {H_0} extends holomorphically to the upper half plane.

If we normalize the Fourier transform {{\mathcal F} f(\xi)} of a (Schwartz) function {f(x)} as {{\mathcal F} f(\xi) := \int_{\bf R} f(x) e^{-2\pi i \xi x}\ dx}, it is well known that the Gaussian {x \mapsto e^{-\pi x^2}} is its own Fourier transform. The creation operator {2\pi x - \frac{d}{dx}} interacts with the Fourier transform by the identity

\displaystyle {\mathcal F} (( 2\pi x - \frac{d}{dx} ) f) (\xi) = -i (2 \pi \xi - \frac{d}{d\xi} ) {\mathcal F} f(\xi).

Since {(-i)^4 = 1}, this implies that the function

\displaystyle x \mapsto (2\pi x - \frac{d}{dx})^4 e^{-\pi x^2} = 128 \pi^2 (2 \pi^2 x^4 - 3 \pi x^2) e^{-\pi x^2} + 48 \pi^2 e^{-\pi x^2}

is its own Fourier transform. (One can view the polynomial {128 \pi^2 (2\pi^2 x^4 - 3 \pi x^2) + 48 \pi^2} as a renormalised version of the fourth Hermite polynomial.) Taking a suitable linear combination of this with {x \mapsto e^{-\pi x^2}}, we conclude that

\displaystyle x \mapsto (2 \pi^2 x^4 - 3 \pi x^2) e^{-\pi x^2}

is also its own Fourier transform. Rescaling {x} by {e^{2u}} and then multiplying by {e^u}, we conclude that the Fourier transform of

\displaystyle x \mapsto (2 \pi^2 x^4 e^{9u} - 3 \pi x^2 e^{5u}) \exp( - \pi x^2 e^{4u} )

is

\displaystyle x \mapsto (2 \pi^2 x^4 e^{-9u} - 3 \pi x^2 e^{-5u}) \exp( - \pi x^2 e^{-4u} ),

and hence by the Poisson summation formula (using symmetry and vanishing at {n=0} to unfold the {n} summation in (2) to the integers rather than the natural numbers) we obtain the functional equation

\displaystyle \Phi(-u) = \Phi(u),

which implies that {\Phi} and {H_0} are even functions (in particular, {H_0} now extends to an entire function). From this symmetry we can also rewrite (1) as

\displaystyle H_0(z) = \int_0^\infty \Phi(u) \cos(zu)\ du,

which now gives a convergent expression for the entire function {H_0(z)} for all complex {z}. As {\Phi} is even and real-valued on {{\bf R}}, {H_0(z)} is even and also obeys the functional equation {H_0(\overline{z}) = \overline{H_0(z)}}, which is equivalent to the usual functional equation for the Riemann zeta function. The Riemann hypothesis is equivalent to the claim that all the zeroes of {H_0} are real.

De Bruijn introduced the family {H_t: {\bf C} \rightarrow {\bf C}} of deformations of {H_0: {\bf C} \rightarrow {\bf C}}, defined for all {t \in {\bf R}} and {z \in {\bf C}} by the formula

\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du.

From a PDE perspective, one can view {H_t} as the evolution of {H_0} under the backwards heat equation {\partial_t H_t(z) = - \partial_{zz} H_t(z)}. As with {H_0}, the {H_t} are all even entire functions that obey the functional equation {H_t(\overline{z}) = \overline{H_t(z)}}, and one can ask an analogue of the Riemann hypothesis for each such {H_t}, namely whether all the zeroes of {H_t} are real. De Bruijn showed that these hypotheses were monotone in {t}: if {H_t} had all real zeroes for some {t}, then {H_{t'}} would also have all zeroes real for any {t' \geq t}. Newman later sharpened this claim by showing the existence of a finite number {\Lambda \leq 1/2}, now known as the de Bruijn-Newman constant, with the property that {H_t} had all zeroes real if and only if {t \geq \Lambda}. Thus, the Riemann hypothesis is equivalent to the inequality {\Lambda \leq 0}. Newman then conjectured the complementary bound {\Lambda \geq 0}; in his words, this conjecture asserted that if the Riemann hypothesis is true, then it is only “barely so”, in that the reality of all the zeroes is destroyed by applying heat flow for even an arbitrarily small amount of time. Over time, a significant amount of evidence was established in favour of this conjecture; most recently, in 2011, Saouter, Gourdon, and Demichel showed that {\Lambda \geq -1.15 \times 10^{-11}}.

In this paper we finish off the proof of Newman’s conjecture, that is we show that {\Lambda \geq 0}. The proof is by contradiction, assuming that {\Lambda < 0} (which among other things, implies the truth of the Riemann hypothesis), and using the properties of backwards heat evolution to reach a contradiction.

Very roughly, the argument proceeds as follows. As observed by Csordas, Smith, and Varga (and also discussed in this previous blog post, the backwards heat evolution of the {H_t} introduces a nice ODE dynamics on the zeroes {x_j(t)} of {H_t}, namely that they solve the ODE

\displaystyle \frac{d}{dt} x_j(t) = -2 \sum_{j \neq k} \frac{1}{x_k(t) - x_j(t)} \ \ \ \ \ (3)

 

for all {j} (one has to interpret the sum in a principal value sense as it is not absolutely convergent, but let us ignore this technicality for the current discussion). Intuitively, this ODE is asserting that the zeroes {x_j(t)} repel each other, somewhat like positively charged particles (but note that the dynamics is first-order, as opposed to the second-order laws of Newtonian mechanics). Formally, a steady state (or equilibrium) of this dynamics is reached when the {x_k(t)} are arranged in an arithmetic progression. (Note for instance that for any positive {u}, the functions {z \mapsto e^{tu^2} \cos(uz)} obey the same backwards heat equation as {H_t}, and their zeroes are on a fixed arithmetic progression {\{ \frac{2\pi (k+\tfrac{1}{2})}{u}: k \in {\bf Z} \}}.) The strategy is to then show that the dynamics from time {-\Lambda} to time {0} creates a convergence to local equilibrium, in which the zeroes {x_k(t)} locally resemble an arithmetic progression at time {t=0}. This will be in contradiction with known results on pair correlation of zeroes (or on related statistics, such as the fluctuations on gaps between zeroes), such as the results of Montgomery (actually for technical reasons it is slightly more convenient for us to use related results of Conrey, Ghosh, Goldston, Gonek, and Heath-Brown). Another way of thinking about this is that even very slight deviations from local equilibrium (such as a small number of gaps that are slightly smaller than the average spacing) will almost immediately lead to zeroes colliding with each other and leaving the real line as one evolves backwards in time (i.e., under the forward heat flow). This is a refinement of the strategy used in previous lower bounds on {\Lambda}, in which “Lehmer pairs” (pairs of zeroes of the zeta function that were unusually close to each other) were used to limit the extent to which the evolution continued backwards in time while keeping all zeroes real.

How does one obtain this convergence to local equilibrium? We proceed by broad analogy with the “local relaxation flow” method of Erdos, Schlein, and Yau in random matrix theory, in which one combines some initial control on zeroes (which, in the case of the Erdos-Schlein-Yau method, is referred to with terms such as “local semicircular law”) with convexity properties of a relevant Hamiltonian that can be used to force the zeroes towards equilibrium.

We first discuss the initial control on zeroes. For {H_0}, we have the classical Riemann-von Mangoldt formula, which asserts that the number of zeroes in the interval {[0,T]} is {\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + O(\log T)} as {T \rightarrow \infty}. (We have a factor of {4\pi} here instead of the more familiar {2\pi} due to the way {H_0} is normalised.) This implies for instance that for a fixed {\alpha}, the number of zeroes in the interval {[T, T+\alpha]} is {\frac{\alpha}{4\pi} \log T + O(\log T)}. Actually, because we get to assume the Riemann hypothesis, we can sharpen this to {\frac{\alpha}{4\pi} \log T + o(\log T)}, a result of Littlewood (see this previous blog post for a proof). Ideally, we would like to obtain similar control for the other {H_t}, {\Lambda \leq t < 0}, as well. Unfortunately we were only able to obtain the weaker claims that the number of zeroes of {H_t} in {[0,T]} is {\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + O(\log^2 T)}, and that the number of zeroes in {[T, T+\alpha \log T]} is {\frac{\alpha}{4 \pi} \log^2 T + o(\log^2 T)}, that is to say we only get good control on the distribution of zeroes at scales {\gg \log T} rather than at scales {\gg 1}. Ultimately this is because we were only able to get control (and in particular, lower bounds) on {|H_t(x-iy)|} with high precision when {y \gg \log x} (whereas {|H_0(x-iy)|} has good estimates as soon as {y} is larger than (say) {2}). This control is obtained by the expressing {H_t(x-iy)} in terms of some contour integrals and using the method of steepest descent (actually it is slightly simpler to rely instead on the Stirling approximation for the Gamma function, which can be proven in turn by steepest descent methods). Fortunately, it turns out that this weaker control is still (barely) enough for the rest of our argument to go through.

Once one has the initial control on zeroes, we now need to force convergence to local equilibrium by exploiting convexity of a Hamiltonian. Here, the relevant Hamiltonian is

\displaystyle H(t) := \sum_{j,k: j \neq k} \log \frac{1}{|x_j(t) - x_k(t)|},

ignoring for now the rather important technical issue that this sum is not actually absolutely convergent. (Because of this, we will need to truncate and renormalise the Hamiltonian in a number of ways which we will not detail here.) The ODE (3) is formally the gradient flow for this Hamiltonian. Furthermore, this Hamiltonian is a convex function of the {x_j} (because {t \mapsto \log \frac{1}{t}} is a convex function on {(0,+\infty)}). We therefore expect the Hamiltonian to be a decreasing function of time, and that the derivative should be an increasing function of time. As time passes, the derivative of the Hamiltonian would then be expected to converge to zero, which should imply convergence to local equilibrium.

Formally, the derivative of the above Hamiltonian is

\displaystyle \partial_t H(t) = -4 E(t), \ \ \ \ \ (4)

 

where {E(t)} is the “energy”

\displaystyle E(t) := \sum_{j,k: j \neq k} \frac{1}{|x_j(t) - x_k(t)|^2}.

Again, there is the important technical issue that this quantity is infinite; but it turns out that if we renormalise the Hamiltonian appropriately, then the energy will also become suitably renormalised, and in particular will vanish when the {x_j} are arranged in an arithmetic progression, and be positive otherwise. One can also formally calculate the derivative of {E(t)} to be a somewhat complicated but manifestly non-negative quantity (a sum of squares); see this previous blog post for analogous computations in the case of heat flow on polynomials. After flowing from time {\Lambda} to time {0}, and using some crude initial bounds on {H(t)} and {E(t)} in this region (coming from the Riemann-von Mangoldt type formulae mentioned above and some further manipulations), we can eventually show that the (renormalisation of the) energy {E(0)} at time zero is small, which forces the {x_j} to locally resemble an arithmetic progression, which gives the required convergence to local equilibrium.

There are a number of technicalities involved in making the above sketch of argument rigorous (for instance, justifying interchanges of derivatives and infinite sums turns out to be a little bit delicate). I will highlight here one particular technical point. One of the ways in which we make expressions such as the energy {E(t)} finite is to truncate the indices {j,k} to an interval {I} to create a truncated energy {E_I(t)}. In typical situations, we would then expect {E_I(t)} to be decreasing, which will greatly help in bounding {E_I(0)} (in particular it would allow one to control {E_I(0)} by time-averaged quantities such as {\int_{\Lambda/2}^0 E_I(t)\ dt}, which can in turn be controlled using variants of (4)). However, there are boundary effects at both ends of {I} that could in principle add a large amount of energy into {E_I}, which is bad news as it could conceivably make {E_I(0)} undesirably large even if integrated energies such as {\int_{\Lambda/2}^0 E_I(t)\ dt} remain adequately controlled. As it turns out, such boundary effects are negligible as long as there is a large gap between adjacent zeroes at boundary of {I} – it is only narrow gaps that can rapidly transmit energy across the boundary of {I}. Now, narrow gaps can certainly exist (indeed, the GUE hypothesis predicts these happen a positive fraction of the time); but the pigeonhole principle (together with the Riemann-von Mangoldt formula) can allow us to pick the endpoints of the interval {I} so that no narrow gaps appear at the boundary of {I} for any given time {t}. However, there was a technical problem: this argument did not allow one to find a single interval {I} that avoided gaps for all times {\Lambda/2 \leq t \leq 0} simultaneously – the pigeonhole principle could produce a different interval {I} for each time {t}! Since the number of times was uncountable, this was a serious issue. (In physical terms, the problem was that there might be very fast “longitudinal waves” in the dynamics that, at each time, cause some gaps between zeroes to be highly compressed, but the specific gap that was narrow changed very rapidly with time. Such waves could, in principle, import a huge amount of energy into {E_I} by time {0}.) To resolve this, we borrowed a PDE trick of Bourgain’s, in which the pigeonhole principle was coupled with local conservation laws. More specifically, we use the phenomenon that very narrow gaps {g_i = x_{i+1}-x_i} take a nontrivial amount of time to expand back to a reasonable size (this can be seen by comparing the evolution of this gap with solutions of the scalar ODE {\partial_t g = \frac{4}{g^2}}, which represents the fastest at which a gap such as {g_i} can expand). Thus, if a gap {g_i} is reasonably large at some time {t_0}, it will also stay reasonably large at slightly earlier times {t \in [t_0-\delta, t_0]} for some moderately small {\delta>0}. This lets one locate an interval {I} that has manageable boundary effects during the times in {[t_0-\delta, t_0]}, so in particular {E_I} is basically non-increasing in this time interval. Unfortunately, this interval is a little bit too short to cover all of {[\Lambda/2,0]}; however it turns out that one can iterate the above construction and find a nested sequence of intervals {I_k}, with each {E_{I_k}} non-increasing in a different time interval {[t_k - \delta, t_k]}, and with all of the time intervals covering {[\Lambda/2,0]}. This turns out to be enough (together with the obvious fact that {E_I} is monotone in {I}) to still control {E_I(0)} for some reasonably sized interval {I}, as required for the rest of the arguments.

ADDED LATER: the following analogy (involving functions with just two zeroes, rather than an infinite number of zeroes) may help clarify the relation between this result and the Riemann hypothesis (and in particular why this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis). Suppose one had a quadratic polynomial {P} of the form {P(z) = z^2 + \Lambda}, where {\Lambda} was an unknown real constant. Suppose that one was for some reason interested in the analogue of the “Riemann hypothesis” for {P}, namely that all the zeroes of {P} are real. A priori, there are three scenarios:

  • (Riemann hypothesis false) {\Lambda > 0}, and {P} has zeroes {\pm i |\Lambda|^{1/2}} off the real axis.
  • (Riemann hypothesis true, but barely so) {\Lambda = 0}, and both zeroes of {P} are on the real axis; however, any slight perturbation of {\Lambda} in the positive direction would move zeroes off the real axis.
  • (Riemann hypothesis true, with room to spare) {\Lambda < 0}, and both zeroes of {P} are on the real axis. Furthermore, any slight perturbation of {P} will also have both zeroes on the real axis.

The analogue of our result in this case is that {\Lambda \geq 0}, thus ruling out the third of the three scenarios here. In this simple example in which only two zeroes are involved, one can think of the inequality {\Lambda \geq 0} as asserting that if the zeroes of {P} are real, then they must be repeated. In our result (in which there are an infinity of zeroes, that become increasingly dense near infinity), and in view of the convergence to local equilibrium properties of (3), the analogous assertion is that if the zeroes of {H_0} are real, then they do not behave locally as if they were in arithmetic progression.

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges“. This is a sequel of sorts to our previous paper on divisor correlations, though the proof techniques in this paper are rather different. As with the previous paper, our interest is in correlations such as

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) \ \ \ \ \ (1)

for medium-sized {h} and large {X}, where {k \geq l \geq 1} are natural numbers and {d_k(n) = \sum_{n = m_1 \dots m_k} 1} is the {k^{th}} divisor function (actually our methods can also treat a generalisation in which {k} is non-integer, but for simplicity let us stick with the integer case for this discussion). Our methods also allow for one of the divisor function factors to be replaced with a von Mangoldt function, but (in contrast to the previous paper) we cannot treat the case when both factors are von Mangoldt.

As discussed in this previous post, one heuristically expects an asymptotic of the form

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) = P_{k,l,h}( \log X ) X + O( X^{1/2+\varepsilon})

for any fixed {\varepsilon>0}, where {P_{k,l,h}} is a certain explicit (but rather complicated) polynomial of degree {k+l-1}. Such asymptotics are known when {l \leq 2}, but remain open for {k \geq l \geq 3}. In the previous paper, we were able to obtain a weaker bound of the form

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) = P_{k,l,h}( \log X ) X + O_A( X \log^{-A} X)

for {1-O_A(\log^{-A} X)} of the shifts {-H \leq h \leq H}, whenever the shift range {H} lies between {X^{8/33+\varepsilon}} and {X^{1-\varepsilon}}. But the methods become increasingly hard to use as {H} gets smaller. In this paper, we use a rather different method to obtain the even weaker bound

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) = (1+o(1)) P_{k,l,h}( \log X ) X

for {1-o(1)} of the shifts {-H \leq h \leq H}, where {H} can now be as short as {H = \log^{10^4 k \log k} X}. The constant {10^4} can be improved, but there are serious obstacles to using our method to go below {\log^{k \log k} X} (as the exceptionally large values of {d_k} then begin to dominate). This can be viewed as an analogue to our previous paper on correlations of bounded multiplicative functions on average, in which the functions {d_k,d_l} are now unbounded, and indeed our proof strategy is based in large part on that paper (but with many significant new technical complications).

We now discuss some of the ingredients of the proof. Unsurprisingly, the first step is the circle method, expressing (1) in terms of exponential sums such as

\displaystyle  S(\alpha) := \sum_{n \leq X} d_k(n) e(\alpha).

Actually, it is convenient to first prune {d_k} slightly by zeroing out this function on “atypical” numbers {n} that have an unusually small or large number of factors in a certain sense, but let us ignore this technicality for this discussion. The contribution of {S(\alpha)} for “major arc” {\alpha} can be treated by standard techniques (and is the source of the main term {P_{k,l,h}(\log X) X}; the main difficulty comes from treating the contribution of “minor arc” {\alpha}.

In our previous paper on bounded multiplicative functions, we used Plancherel’s theorem to estimate the global {L^2} norm {\int_{{\bf R}/{\bf Z}} |S(\alpha)|^2\ d\alpha}, and then also used the Katai-Bourgain-Sarnak-Ziegler orthogonality criterion to control local {L^2} norms {\int_I |S(\alpha)|^2\ d\alpha}, where {I} was a minor arc interval of length about {1/H}, and these two estimates together were sufficient to get a good bound on correlations by an application of Hölder’s inequality. For {d_k}, it is more convenient to use Dirichlet series methods (and Ramaré-type factorisations of such Dirichlet series) to control local {L^2} norms on minor arcs, in the spirit of the proof of the Matomaki-Radziwill theorem; a key point is to develop “log-free” mean value theorems for Dirichlet series associated to functions such as {d_k}, so as not to wipe out the (rather small) savings one will get over the trivial bound from this method. On the other hand, the global {L^2} bound will definitely be unusable, because the {\ell^2} sum {\sum_{n \leq X} d_k(n)^2} has too many unwanted factors of {\log X}. Fortunately, we can substitute this global {L^2} bound with a “large values” bound that controls expressions such as

\displaystyle  \sum_{i=1}^J \int_{I_i} |S(\alpha)|^2\ d\alpha

for a moderate number of disjoint intervals {I_1,\dots,I_J}, with a bound that is slightly better (for {J} a medium-sized power of {\log X}) than what one would have obtained by bounding each integral {\int_{I_i} |S(\alpha)|^2\ d\alpha} separately. (One needs to save more than {J^{1/2}} for the argument to work; we end up saving a factor of about {J^{3/4}}.) This large values estimate is probably the most novel contribution of the paper. After taking the Fourier transform, matters basically reduce to getting a good estimate for

\displaystyle  \sum_{i=1}^J (\int_X^{2X} |\sum_{x \leq n \leq x+H} d_k(n) e(\alpha_i n)|^2\ dx)^{1/2},

where {\alpha_i} is the midpoint of {I_i}; thus we need some upper bound on the large local Fourier coefficients of {d_k}. These coefficients are difficult to calculate directly, but, in the spirit of a paper of Ben Green and myself, we can try to replace {d_k} by a more tractable and “pseudorandom” majorant {\tilde d_k} for which the local Fourier coefficients are computable (on average). After a standard duality argument, one ends up having to control expressions such as

\displaystyle  |\sum_{x \leq n \leq x+H} \tilde d_k(n) e((\alpha_i -\alpha_{i'}) n)|

after various averaging in the {x, i,i'} parameters. These local Fourier coefficients of {\tilde d_k} turn out to be small on average unless {\alpha_i -\alpha_{i'}} is “major arc”. One then is left with a mostly combinatorial problem of trying to bound how often this major arc scenario occurs. This is very close to a computation in the previously mentioned paper of Ben and myself; there is a technical wrinkle in that the {\alpha_i} are not as well separated as they were in my paper with Ben, but it turns out that one can modify the arguments in that paper to still obtain a satisfactory estimate in this case (after first grouping nearby frequencies {\alpha_i} together, and modifying the duality argument accordingly).

I have just uploaded to the arXiv the paper “An inverse theorem for an inequality of Kneser“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kneser discussed previously in this blog, namely that

\displaystyle \mu(A+B) \geq \min(\mu(A)+\mu(B), 1) \ \ \ \ \ (1)

whenever {A,B} are compact non-empty subsets of a compact connected additive group {G} with probability Haar measure {\mu}.  (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime

\displaystyle \mu(A), \mu(B), 1- \mu(A)-\mu(B) > 0. \ \ \ \ \ (2)

The connectedness of {G} is essential, otherwise one could form counterexamples involving proper subgroups of {G} of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as

\displaystyle \mu( (A_1 \cup A_2) + B) + \mu( (A_1 \cap A_2) + B)

\displaystyle \leq \mu(A_1+B) + \mu(A_2+B) \ \ \ \ \ (3)

which in turn easily follows from the identity {(A_1 \cup A_2) + B = (A_1+B) \cup (A_2+B)} and the inclusion {(A_1 \cap A_2) + B \subset (A_1 +B) \cap (A_2+B)}, combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking {G} to be the unit circle {G = {\bf R}/{\bf Z}} and {A,B} to be arcs in that circle (obeying (2)). A bit more generally, if {G} is an arbitrary connected compact abelian group and {\xi: G \rightarrow {\bf R}/{\bf Z}} is a non-trivial character (i.e., a continuous homomorphism), then {\xi} must be surjective (as {{\bf R}/{\bf Z}} has no non-trivial connected subgroups), and one can take {A = \xi^{-1}(I)} and {B = \xi^{-1}(J)} for some arcs {I,J} in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then {A,B} must be close (in measure) to an example of the above form {A = \xi^{-1}(I), B = \xi^{-1}(J)}. Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset {A+B} is replaced by the partial sumset {A +_\varepsilon B} consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call {(A,B)} a critical pair if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in {A,B}, and invariant with respect to translation of either {A} or {B}. Furthermore, from the submodularity inequality (3), one can show that if {(A_1,B)} and {(A_2,B)} are critical pairs (with {\mu(A_1 \cap A_2)} and {1 - \mu(A_1 \cup A_2) - \mu(B)} positive), then {(A_1 \cap A_2,B)} and {(A_1 \cup A_2, B)} are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when {A,B} come from arcs of a circle.) Similarly, from associativity {(A+B)+C = A+(B+C)}, one can show that if {(A,B)} and {(A+B,C)} are critical pairs, then so are {(B,C)} and {(A,B+C)}.

One can combine these closure properties to obtain further ones. For instance, suppose {A,B} is such that {\mu(A+B)  0}. Then (cheating a little bit), one can show that {(A+B,C)} is also a critical pair, basically because {A+B} is the union of the {A+b}, {b \in B}, the {(A+b,C)} are all critical pairs, and the {A+b} all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate {A+B} by the union of finitely many of the {A+b}, then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair {(A,B)} and end up with a small set {C} such that {(A,C)} and {(kC,C)} are also critical pairs for all {1 \leq k \leq 10^4} (say), where {kC} is the {k}-fold sumset of {C}. (Intuitively, if {A,B} are thought of as secretly coming from the pullback of arcs {I,J} by some character {\xi}, then {C} should be the pullback of a much shorter arc by the same character.) In particular, {C} exhibits linear growth, in that {\mu(kC) = k\mu(C)} for all {1 \leq k \leq 10^4}. One can now use standard technology from inverse sumset theory to show first that {C} has a very large Fourier coefficient (and thus is biased with respect to some character {\xi}), and secondly that {C} is in fact almost of the form {C = \xi^{-1}(K)} for some arc {K}, from which it is not difficult to conclude similar statements for {A} and {B} and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)

 

[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman.  Thanks to John Griesmer for pointing out the error.]

Joni Teräväinen and I have just uploaded to the arXiv our paper “Odd order cases of the logarithmically averaged Chowla conjecture“, submitted to J. Numb. Thy. Bordeaux. This paper gives an alternate route to one of the main results of our previous paper, and more specifically reproves the asymptotic

\displaystyle \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log x) \ \ \ \ \ (1)

 

for all odd {k} and all integers {h_1,\dots,h_k} (that is to say, all the odd order cases of the logarithmically averaged Chowla conjecture). Our previous argument relies heavily on some deep ergodic theory results of Bergelson-Host-Kra, Leibman, and Le (and was applicable to more general multiplicative functions than the Liouville function {\lambda}); here we give a shorter proof that avoids ergodic theory (but instead requires the Gowers uniformity of the (W-tricked) von Mangoldt function, established in several papers of Ben Green, Tamar Ziegler, and myself). The proof follows the lines sketched in the previous blog post. In principle, due to the avoidance of ergodic theory, the arguments here have a greater chance to be made quantitative; however, at present the known bounds on the Gowers uniformity of the von Mangoldt function are qualitative, except at the {U^2} level, which is unfortunate since the first non-trivial odd case {k=3} requires quantitative control on the {U^3} level. (But it may be possible to make the Gowers uniformity bounds for {U^3} quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)

Apoorva Khare and I have updated our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“, announced at this post from last month. The quantitative results are now sharpened using a new monotonicity property of ratios {s_{\lambda}(u)/s_{\mu}(u)} of Schur polynomials, namely that such ratios are monotone non-decreasing in each coordinate of {u} if {u} is in the positive orthant, and the partition {\lambda} is larger than that of {\mu}. (This monotonicity was also independently observed by Rachid Ait-Haddou, using the theory of blossoms.) In the revised version of the paper we give two proofs of this monotonicity. The first relies on a deep positivity result of Lam, Postnikov, and Pylyavskyy, which uses a representation-theoretic positivity result of Haiman to show that the polynomial combination

\displaystyle s_{(\lambda \wedge \nu) / (\mu \wedge \rho)} s_{(\lambda \vee \nu) / (\mu \vee \rho)} - s_{\lambda/\mu} s_{\nu/\rho} \ \ \ \ \ (1)

of skew-Schur polynomials is Schur-positive for any partitions {\lambda,\mu,\nu,\rho} (using the convention that the skew-Schur polynomial {s_{\lambda/\mu}} vanishes if {\mu} is not contained in {\lambda}, and where {\lambda \wedge \nu} and {\lambda \vee \nu} denotes the pointwise min and max of {\lambda} and {\nu} respectively). It is fairly easy to derive the monotonicity of {s_\lambda(u)/s_\mu(u)} from this, by using the expansion

\displaystyle s_\lambda(u_1,\dots, u_n) = \sum_k u_1^k s_{\lambda/(k)}(u_2,\dots,u_n)

of Schur polynomials into skew-Schur polynomials (as was done in this previous post).

The second proof of monotonicity avoids representation theory by a more elementary argument establishing the weaker claim that the above expression (1) is non-negative on the positive orthant. In fact we prove a more general determinantal log-supermodularity claim which may be of independent interest:

Theorem 1 Let {A} be any {n \times n} totally positive matrix (thus, every minor has a non-negative determinant). Then for any {k}-tuples {I_1,I_2,J_1,J_2} of increasing elements of {\{1,\dots,n\}}, one has

\displaystyle \det( A_{I_1 \wedge I_2, J_1 \wedge J_2} ) \det( A_{I_1 \vee I_2, J_1 \vee J_2} ) - \det(A_{I_1,J_1}) \det(A_{I_2,J_2}) \geq 0

where {A_{I,J}} denotes the {k \times k} minor formed from the rows in {I} and columns in {J}.

For instance, if {A} is the matrix

\displaystyle A = \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{pmatrix}

for some real numbers {a,\dots,p}, one has

\displaystyle a h - de\geq 0

(corresponding to the case {k=1}, {I_1 = (1), I_2 = (2), J_1 = (4), J_2 = (1)}), or

\displaystyle \det \begin{pmatrix} a & c \\ i & k \end{pmatrix} \det \begin{pmatrix} f & h \\ n & p \end{pmatrix} - \det \begin{pmatrix} e & h \\ i & l \end{pmatrix} \det \begin{pmatrix} b & c \\ n & o \end{pmatrix} \geq 0

(corresponding to the case {k=2}, {I_1 = (2,3)}, {I_2 = (1,4)}, {J_1 = (1,4)}, {J_2 = (2,3)}). It turns out that this claim can be proven relatively easy by an induction argument, relying on the Dodgson and Karlin identities from this previous post; the difficulties are largely notational in nature. Combining this result with the Jacobi-Trudi identity for skew-Schur polynomials (discussed in this previous post) gives the non-negativity of (1); it can also be used to directly establish the monotonicity of ratios {s_\lambda(u)/s_\mu(u)} by applying the theorem to a generalised Vandermonde matrix.

(Log-supermodularity also arises as the natural hypothesis for the FKG inequality, though I do not know of any interesting application of the FKG inequality in this current setting.)

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