A proof of Roth’s theorem

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

Theorem 1 (Roth’s theorem) Let be a compact abelian group, with Haar probability measure , and let be a measurable subset of with for some . Then we have

where denotes the bound for some depending only on .

This theorem is usually formulated in the case that is a finite abelian group (in which case the result is essentially due to Meshulam) or more specifically a cyclic group (in which case it is essentially due to Varnavides), but is also valid for the more general setting of compact abelian groups, as we shall shortly see. One can be more precise about the dependence of the implied constant on , but to keep the exposition simple we will work at the qualitative level here, without trying at all to get good quantitative bounds.

We can deduce Theorem 1 from the following more general Khintchine-type statement. Let denote the Pontryagin dual of a compact abelian group , that is to say the set of all continuous homomorphisms from to the (additive) unit circle . Thus is a discrete abelian group, and functions have a Fourier transform defined by

For any finite set and any radius , define the Bohr set

where denotes the distance of to the nearest integer. We refer to the cardinality of as the rank of the Bohr set. We record a simple volume bound on Bohr sets:

Lemma 2 (Volume packing bound) Let be a compact abelian group with Haar probability measure . For any Bohr set , we have

Proof: We can cover the torus by translates of the cube . Then the sets form an cover of . But all of these sets lie in a translate of , and the claim then follows from the translation invariance of .

Given any Bohr set , we define a normalised “Lipschitz” cutoff function by the formula

where is the constant such that

thus

The function should be viewed as an -normalised “tent function” cutoff to . Note from Lemma 2 that

We then have the following sharper version of Theorem 1:

Theorem 3 (Roth-Khintchine theorem) Let be a compact abelian group, with Haar probability measure , and let . Then for any measurable function , there exists a Bohr set with and such that

where denotes the convolution operation

A variant of this result (expressed in the language of ergodic theory) appears in this paper of Bergelson, Host, and Kra; a combinatorial version of the Bergelson-Host-Kra result that is closer to Theorem 3 subsequently appeared in this paper of Ben Green and myself, but this theorem arguably appears implicitly in a much older paper of Bourgain. To see why Theorem 3 implies Theorem 1, we apply the theorem with and equal to a small multiple of to conclude that there is a Bohr set with and such that

But from (2) we have the pointwise bound , and Theorem 1 follows.

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

Polymath8b, X: writing the paper, and chasing down loose ends

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

;

the previous thread may be found here.

Numerical progress on these bounds have slowed in recent months, although we have very recently lowered the unconditional bound on from 252 to 246 (see the wiki page for more detailed results).  While there may still be scope for further improvement (particularly with respect to bounds for with , which we have not focused on for a while, it looks like we have reached the point of diminishing returns, and it is time to turn to the task of writing up the results.

A draft version of the paper so far may be found here (with the directory of source files here).  Currently, the introduction and the sieve-theoretic portions of the paper are written up, although the sieve-theoretic arguments are surprisingly lengthy, and some simplification (or other reorganisation) may well be possible.  Other portions of the paper that have not yet been written up include the asymptotic analysis of for large k (leading in particular to results for m=2,3,4,5), and a description of the quadratic programming that is used to estimate for small and medium k.  Also we will eventually need an appendix to summarise the material from Polymath8a that we would use to generate various narrow admissible tuples.

One issue here is that our current unconditional bounds on for m=2,3,4,5 rely on a distributional estimate on the primes which we believed to be true in Polymath8a, but never actually worked out (among other things, there was some delicate algebraic geometry issues concerning the vanishing of certain cohomology groups that was never resolved).  This issue does not affect the m=1 calculations, which only use the Bombieri-Vinogradov theorem or else assume the generalised Elliott-Halberstam conjecture.  As such, we will have to rework the computations for these , given that the task of trying to attain the conjectured distributional estimate on the primes would be a significant amount of work that is rather disjoint from the rest of the Polymath8b writeup.  One could simply dust off the old maple code for this (e.g. one could tweak the code here, with the constraint  1080*varpi/13+ 330*delta/13<1  being replaced by 600*varpi/7+180*delta/7<1), but there is also a chance that our asymptotic bounds for (currently given in messy detail here) could be sharpened.  I plan to look at this issue fairly soon.

Also, there are a number of smaller observations (e.g. the parity problem barrier that prevents us from ever getting a better bound on than 6) that should also go into the paper at some point; the current outline of the paper as given in the draft is not necessarily comprehensive.

The Cayley-Salmon theorem via classical differential geometry

Let be an irreducible polynomial in three variables. As is not algebraically closed, the zero set can split into various components of dimension between and . For instance, if , the zero set is a line; more interestingly, if , then is the union of a line and a surface (or the product of an acnodal cubic curve with a line). We will assume that the -dimensional component is non-empty, thus defining a real surface in . In particular, this hypothesis implies that is not just irreducible over , but is in fact absolutely irreducible (i.e. irreducible over ), since otherwise one could use the complex factorisation of to contain inside the intersection of the complex zero locus of complex polynomial and its complex conjugate, with having no common factor, forcing to be at most one-dimensional. (For instance, in the case , one can take .) Among other things, this makes a Zariski-dense subset of , thus any polynomial identity which holds true at every point of , also holds true on all of . This allows us to easily use tools from algebraic geometry in this real setting, even though the reals are not quite algebraically closed.

The surface is said to be ruled if, for a Zariski open dense set of points , there exists a line through for some non-zero which is completely contained in , thus

for all . Also, a point is said to be a flecnode if there exists a line through for some non-zero which is tangent to to third order, in the sense that

as , or equivalently that

for . Clearly, if is a ruled surface, then a Zariski open dense set of points on are a flecnode. We then have the remarkable theorem of Cayley and Salmon asserting the converse:

Theorem 1 (Cayley-Salmon theorem) Let be an irreducible polynomial with non-empty. Suppose that a Zariski dense set of points in are flecnodes. Then is a ruled surface.

Among other things, this theorem was used in the celebrated result of Guth and Katz that almost solved the Erdos distance problem in two dimensions, as discussed in this previous blog post. Vanishing to third order is necessary: observe that in a surface of negative curvature, such as the saddle , every point on the surface is tangent to second order to a line (the line in the direction for which the second fundamental form vanishes).

The original proof of the Cayley-Salmon theorem, dating back to at least 1915, is not easily accessible and not written in modern language. A modern proof of this theorem (together with substantial generalisations, for instance to higher dimensions) is given by Landsberg; the proof uses the machinery of modern algebraic geometry. The purpose of this post is to record an alternate proof of the Cayley-Salmon theorem based on classical differential geometry (in particular, the notion of torsion of a curve) and basic ODE methods (in particular, Gronwall’s inequality and the Picard existence theorem). The idea is to “integrate” the lines indicated by the flecnode to produce smooth curves on the surface ; one then uses the vanishing (1) and some basic calculus to conclude that these curves have zero torsion and are thus planar curves. Some further manipulation using (1) (now just to second order instead of third) then shows that these curves are in fact straight lines, giving the ruling on the surface.

Update: Janos Kollar has informed me that the above theorem was essentially known to Monge in 1809; see his recent arXiv note for more details.

I thank Larry Guth and Micha Sharir for conversations leading to this post.

Metric entropy analogues of sum set theory

A core foundation of the subject now known as arithmetic combinatorics (and particularly the subfield of additive combinatorics) are the elementary sum set estimates (sometimes known as “Ruzsa calculus”) that relate the cardinality of various sum sets

and difference sets

as well as iterated sumsets such as , , and so forth. Here, are finite non-empty subsets of some additive group (classically one took or , but nowadays one usually considers more general additive groups). Some basic estimates in this vein are the following:

Lemma 1 (Ruzsa covering lemma) Let be finite non-empty subsets of . Then may be covered by at most translates of .

Proof: Consider a maximal set of disjoint translates of by elements . These translates have cardinality , are disjoint, and lie in , so there are at most of them. By maximality, for any , must intersect at least one of the selected , thus , and the claim follows.

Lemma 2 (Ruzsa triangle inequality) Let be finite non-empty subsets of . Then .

Proof: Consider the addition map from to . Every element of has a preimage of this map of cardinality at least , thanks to the obvious identity for each . Since has cardinality , the claim follows.

Such estimates (which are covered, incidentally, in Section 2 of my book with Van Vu) are particularly useful for controlling finite sets of small doubling, in the sense that for some bounded . (There are deeper theorems, most notably Freiman’s theorem, which give more control than what elementary Ruzsa calculus does, however the known bounds in the latter theorem are worse than polynomial in (although it is conjectured otherwise), whereas the elementary estimates are almost all polynomial in .)

However, there are some settings in which the standard sum set estimates are not quite applicable. One such setting is the continuous setting, where one is dealing with bounded open sets in an additive Lie group (e.g. or a torus ) rather than a finite setting. Here, one can largely replicate the discrete sum set estimates by working with a Haar measure in place of cardinality; this is the approach taken for instance in this paper of mine. However, there is another setting, which one might dub the “discretised” setting (as opposed to the “discrete” setting or “continuous” setting), in which the sets remain finite (or at least discretisable to be finite), but for which there is a certain amount of “roundoff error” coming from the discretisation. As a typical example (working now in a non-commutative multiplicative setting rather than an additive one), consider the orthogonal group of orthogonal matrices, and let be the matrices obtained by starting with all of the orthogonal matrice in and rounding each coefficient of each matrix in this set to the nearest multiple of , for some small . This forms a finite set (whose cardinality grows as like a certain negative power of ). In the limit , the set is not a set of small doubling in the discrete sense. However, is still close to in a metric sense, being contained in the -neighbourhood of . Another key example comes from graphs of maps from a subset of one additive group to another . If is “approximately additive” in the sense that for all , is close to in some metric, then might not have small doubling in the discrete sense (because could take a large number of values), but could be considered a set of small doubling in a discretised sense.

One would like to have a sum set (or product set) theory that can handle these cases, particularly in “high-dimensional” settings in which the standard methods of passing back and forth between continuous, discrete, or discretised settings behave poorly from a quantitative point of view due to the exponentially large doubling constant of balls. One way to do this is to impose a translation invariant metric on the underlying group (reverting back to additive notation), and replace the notion of cardinality by that of metric entropy. There are a number of almost equivalent ways to define this concept:

Definition 3 Let be a metric space, let be a subset of , and let be a radius.

• The packing number is the largest number of points one can pack inside such that the balls are disjoint.
• The internal covering number is the fewest number of points such that the balls cover .
• The external covering number is the fewest number of points such that the balls cover .
• The metric entropy is the largest number of points one can find in that are -separated, thus for all .

It is an easy exercise to verify the inequalities

for any , and that is non-increasing in and non-decreasing in for the three choices (but monotonicity in can fail for !). It turns out that the external covering number is slightly more convenient than the other notions of metric entropy, so we will abbreviate . The cardinality can be viewed as the limit of the entropies as .

If we have the bounded doubling property that is covered by translates of for each , and one has a Haar measure on which assigns a positive finite mass to each ball, then any of the above entropies is comparable to , as can be seen by simple volume packing arguments. Thus in the bounded doubling setting one can usually use the measure-theoretic sum set theory to derive entropy-theoretic sumset bounds (see e.g. this paper of mine for an example of this). However, it turns out that even in the absence of bounded doubling, one still has an entropy analogue of most of the elementary sum set theory, except that one has to accept some degradation in the radius parameter by some absolute constant. Such losses can be acceptable in applications in which the underlying sets are largely “transverse” to the balls , so that the -entropy of is largely independent of ; this is a situation which arises in particular in the case of graphs discussed above, if one works with “vertical” metrics whose balls extend primarily in the vertical direction. (I hope to present a specific application of this type here in the near future.)

Henceforth we work in an additive group equipped with a translation-invariant metric . (One can also generalise things slightly by allowing the metric to attain the values or , without changing much of the analysis below.) By the Heine-Borel theorem, any precompact set will have finite entropy for any . We now have analogues of the two basic Ruzsa lemmas above:

Lemma 4 (Ruzsa covering lemma) Let be precompact non-empty subsets of , and let . Then may be covered by at most translates of .

Proof: Let be a maximal set of points such that the sets are all disjoint. Then the sets are disjoint in and have entropy , and furthermore any ball of radius can intersect at most one of the . We conclude that , so . If , then must intersect one of the , so , and the claim follows.

Lemma 5 (Ruzsa triangle inequality) Let be precompact non-empty subsets of , and let . Then .

Proof: Consider the addition map from to . The domain may be covered by product balls . Every element of has a preimage of this map which projects to a translate of , and thus must meet at least of these product balls. However, if two elements of are separated by a distance of at least , then no product ball can intersect both preimages. We thus see that , and the claim follows.

Below the fold we will record some further metric entropy analogues of sum set estimates (basically redoing much of Chapter 2 of my book with Van Vu). Unfortunately there does not seem to be a direct way to abstractly deduce metric entropy results from their sum set analogues (basically due to the failure of a certain strong version of Freiman’s theorem, as discussed in this previous post); nevertheless, the proofs of the discrete arguments are elementary enough that they can be modified with a small amount of effort to handle the entropy case. (In fact, there should be a very general model-theoretic framework in which both the discrete and entropy arguments can be processed in a unified manner; see this paper of Hrushovski for one such framework.)

It is also likely that many of the arguments here extend to the non-commutative setting, but for simplicity we will not pursue such generalisations here.

Conserved quantities for the surface quasi-geostrophic equation

As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space and Eulerian space , we let be the space of volume-preserving, orientation-preserving maps from Lagrangian space to Eulerian space. Given a curve , we can define the Lagrangian velocity field as the time derivative of , and the Eulerian velocity field . The volume-preserving nature of ensures that is a divergence-free vector field:

If we formally define the functional

then one can show that the critical points of this functional (with appropriate boundary conditions) obey the Euler equations

for some pressure field . As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

the rigid motion symmetries of Eulerian space give conservation of the total momentum

and total angular momentum

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

for any closed loop in , or equivalently pointwise conservation of the Lagrangian vorticity , where is the -form associated with the vector field using the Euclidean metric on , with denoting pullback by .

It turns out that one can generalise the above calculations. Given any self-adjoint operator on divergence-free vector fields , we can define the functional

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

for some pressure field , where in coordinates is with the usual summation conventions. (When , , and this term can be absorbed into the pressure , and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian

If the operator commutes with rigid motions on , then we have conservation of total momentum

and total angular momentum

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

or pointwise conservation of the Lagrangian vorticity . These applications of Noether’s theorem proceed exactly as the previous post; we leave the details to the interested reader.

One particular special case of interest arises in two dimensions , when is the inverse derivative . The vorticity is a -form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write , then

Since is also divergence-free, we may therefore write

where the stream function is given by the formula

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As commutes with rigid motions, we also have (formally, at least) conservation of momentum

(which up to trivial transformations is also expressible in impulse form as , after integration by parts), and conservation of angular momentum

(which up to trivial transformations is ). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity , thus is transported by the flow (which is also evident from (3). In particular, all integrals of the form for a fixed function are conserved by the flow.

Noether’s theorem, and the conservation laws for the Euler equations

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection of one or more fields on a domain taking values in a space will solve the differential equation of interest if and only if is a critical point to the functional

involving the fields and their first derivatives , where the Lagrangian is a function on the vector bundle over consisting of triples with , , and a linear transformation; we also usually keep the boundary data of fixed in case has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on and , which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise as and as , in which case can be viewed locally as a function on .

Example 1 (Geodesic flow) Take and to be a Riemannian manifold, which we will write locally in coordinates as with metric for . A geodesic is then a critical point (keeping fixed) of the energy functional

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold is now infinite dimensional).

More generally, if is itself a Riemannian manifold, which we write locally in coordinates as with metric for , then a harmonic map is a critical point of the energy functional

or in coordinates (again ignoring coordinate patch issues)

If we replace the Riemannian manifold by a Lorentzian manifold, such as Minkowski space , then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ).

Example 2 (-particle interactions) Take and ; then a function can be interpreted as a collection of trajectories in space, which we give a physical interpretation as the trajectories of particles. If we assign each particle a positive mass , and also introduce a potential energy function , then it turns out that Newton’s laws of motion in this context (with the force on the particle being given by the conservative force ) are equivalent to the trajectories being a critical point of the action functional

Formally, if is a critical point of a functional , this means that

whenever is a (smooth) deformation with (and with respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

Write for the infinitesimal deformation of . By the chain rule, can be expressed in terms of . In coordinates, we have

where we parameterise by , and we use subscripts on to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of , and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves , where vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form for some class of functions that vanishes on the boundary, then there must be an associated differential identity that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let be a critical point of the functional defined in (1). Then for any deformation with , we have

where is the vector field that is expressible in coordinates as

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

The same computation, together with an integration by parts, shows that (2) may be rewritten as

Since is unconstrained on the interior of , the claim (6) follows (at a formal level, at least).

Many variational problems also enjoy one-parameter continuous symmetries: given any field (not necessarily a critical point), one can place that field in a one-parameter family with , such that

for all ; in particular,

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

whenever arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain is the Euclidean space , and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable , thus

then we obtain translation symmetries

for , where is the standard basis for . For a fixed , the left-hand side of (7) then becomes

where . Another common type of symmetry is a pointwise symmetry, in which

for all , in which case (7) clearly holds with .

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that is a critical point of the functional (1), and let be a one-parameter continuous symmetry with . Let be the vector field in (5), and let be the vector field in (7). Then we have the pointwise conservation law

In particular, for one-dimensional variational problems, in which , we have the conservation law for all (assuming of course that is connected and contains ).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if and the Lagrangian has no explicit time dependence, thus

then by using the time translation symmetry , we have

as discussed previously, whereas we have , and hence by (5)

and so Noether’s theorem gives conservation of the Hamiltonian

For instance, for geodesic flow, the Hamiltonian works out to be

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), vanishes, and so Noether’s theorem simplifies to ; in the one-dimensional case , we thus see from (5) that the quantity

is conserved in time. For instance, for the -particle system in Example 2, if we have the translation invariance

for all , then we have the pointwise translation symmetry

for all , and some , in which case , and the conserved quantity (11) becomes

as was arbitrary, this establishes conservation of the total momentum

Similarly, if we have the rotation invariance

for any and , then we have the pointwise rotation symmetry

for any skew-symmetric real matrix , in which case , and the conserved quantity (11) becomes

since is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

Conserved quantities for the Euler equations

The Euler equations for incompressible inviscid fluids may be written as

where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting.

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).

Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy

 

(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine, I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has conservation of helicity

 

and (formally, at least) one has conservation of momentum

and angular momentum

(although, as we shall discuss below, due to the slow decay of at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity

is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem, and can also be used to deduce the helicity conservation law mentioned above. In my opinion, none of these laws actually prohibits a self-replicating computer from existing within the laws of ideal fluid flow, but they do significantly complicate the task of actually designing such a computer, or of the basic “gates” that such a computer would consist of.

Below the fold I would like to record and derive all the conservation laws mentioned above, which to my knowledge essentially form the complete set of known conserved quantities for the Euler equations. The material here (although not the notation) is drawn from this text of Majda and Bertozzi.

Polymath8b, IX: Large quadratic programs

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

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The focus is now on bounding unconditionally (in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations). We can bound whenever one can find a symmetric square-integrable function supported on the simplex such that

Our strategy for establishing this has been to restrict to be a linear combination of symmetrised monomials (restricted of course to ), where the degree is small; actually, it seems convenient to work with the slightly different basis where the are restricted to be even. The criterion (1) then becomes a large quadratic program with explicit but complicated rational coefficients. This approach has lowered down to , which led to the bound .

Actually, we know that the more general criterion

will suffice, whenever and is supported now on and obeys the vanishing marginal condition whenever . The latter is in particular obeyed when is supported on . A modification of the preceding strategy has lowered slightly to , giving the bound which is currently our best record.

However, the quadratic programs here have become extremely large and slow to run, and more efficient algorithms (or possibly more computer power) may be required to advance further.

Polymath8b, VIII: Time to start writing up the results?

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

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The big news since the last thread is that we have managed to obtain the (sieve-theoretically) optimal bound of assuming the generalised Elliott-Halberstam conjecture (GEH), which pretty much closes off that part of the story. Unconditionally, our bound on is still . This bound was obtained using the “vanilla” Maynard sieve, in which the cutoff was supported in the original simplex , and only Bombieri-Vinogradov was used. In principle, we can enlarge the sieve support a little bit further now; for instance, we can enlarge to , but then have to shrink the J integrals to , provided that the marginals vanish for . However, we do not yet know how to numerically work with these expanded problems.

Given the substantial progress made so far, it looks like we are close to the point where we should declare victory and write up the results (though we should take one last look to see if there is any room to improve the bounds). There is actually a fair bit to write up:

• Improvements to the Maynard sieve (pushing beyond the simplex, the epsilon trick, and pushing beyond the cube);
• Asymptotic bounds for and hence ;
• Explicit bounds for (using the Polymath8a results)
• ;
• on GEH (and parity obstructions to any further improvement).

I will try to create a skeleton outline of such a paper in the Polymath8 Dropbox folder soon. It shouldn’t be nearly as big as the Polymath8a paper, but it will still be quite sizeable.

“New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective

There are multiple purposes to this blog post.

The first purpose is to announce the uploading of the paper “New equidistribution estimates of Zhang type, and bounded gaps between primes” by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes, to the arXiv, and to describe the main results of this paper below the fold.

The second purpose is to roll over the previous thread on all remaining Polymath8a-related matters (e.g. updates on the submission status of the paper) to a fresh thread. (Discussion of the ongoing Polymath8b project is however being kept on a separate thread, to try to reduce confusion.)

The final purpose of this post is to coordinate the writing of a retrospective article on the Polymath8 experience, which has been solicited for the Newsletter of the European Mathematical Society. I suppose that this could encompass both the Polymath8a and Polymath8b projects, even though the second one is still ongoing (but I think we will soon be entering the endgame there). I think there would be two main purposes of such a retrospective article. The first one would be to tell a story about the process of conducting mathematical research, rather than just describe the outcome of such research; this is an important aspect of the subject which is given almost no attention in most mathematical writing, and it would be good to be able to capture some sense of this process while memories are still relatively fresh. The other would be to draw some tentative conclusions with regards to what the strengths and weaknesses of a Polymath project are, and how appropriate such a format would be for other mathematical problems than bounded gaps between primes. In my opinion, the bounded gaps problem had some fairly unique features that made it particularly amenable to a Polymath project, such as (a) a high level of interest amongst the mathematical community in the problem; (b) a very focused objective (“improve !”), which naturally provided an obvious metric to measure progress; (c) the modular nature of the project, which allowed for people to focus on one aspect of the problem only, and still make contributions to the final goal; and (d) a very reasonable level of ambition (for instance, we did not attempt to prove the twin prime conjecture, which in my opinion would make a terrible Polymath project at our current level of mathematical technology). This is not an exhaustive list of helpful features of the problem; I would welcome other diagnoses of the project by other participants.

With these two objectives in mind, I propose a format for the retrospective article consisting of a brief introduction to the polymath concept in general and the polymath8 project in particular, followed by a collection of essentially independent contributions by different participants on their own experiences and thoughts. Finally we could have a conclusion section in which we make some general remarks on the polymath project (such as the remarks above). I’ve started a dropbox subfolder for this article (currently in a very skeletal outline form only), and will begin writing a section on my own experiences; other participants are of course encouraged to add their own sections (it is probably best to create separate files for these, and then input them into the main file retrospective.tex, to reduce edit conflicts. If there are participants who wish to contribute but do not currently have access to the Dropbox folder, please email me and I will try to have you added (or else you can supply your thoughts by email, or in the comments to this post; we may have a section for shorter miscellaneous comments from more casual participants, for people who don’t wish to write a lengthy essay on the subject).

As for deadlines, the EMS Newsletter would like a submitted article by mid-April in order to make the June issue, but in the worst case, it will just be held over until the issue after that.