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Daniel Kane and I have just uploaded to the arXiv our paper “A bound on partitioning clusters“, submitted to the Electronic Journal of Combinatorics. In this short and elementary paper, we consider a question that arose from biomathematical applications: given a finite family of sets (or “clusters”), how many ways can there be of partitioning a set in this family as the disjoint union of two other sets in this family? That is to say, what is the best upper bound one can place on the quantity
in terms of the cardinality of ? A trivial upper bound would be , since this is the number of possible pairs , and clearly determine . In our paper, we establish the improved bound
Actually, the latter claim is quite easy to show: one takes to be all the subsets of of cardinality either or , for a multiple of , and the claim follows readily from Stirling’s formula. So it is perhaps the former claim that is more interesting (since many combinatorial proof techniques, such as those based on inequalities such as the Cauchy-Schwarz inequality, tend to produce exponents that are rational or at least algebraic). We follow the common, though unintuitive, trick of generalising a problem to make it simpler. Firstly, one generalises the bound to the “trilinear” bound
for arbitrary finite collections of sets. One can place all the sets in inside a single finite set such as , and then by replacing every set in by its complement in , one can phrase the inequality in the equivalent form
for arbitrary collections of subsets of . We generalise further by turning sets into functions, replacing the estimate with the slightly stronger convolution estimate
for arbitrary functions on the Hamming cube , where the convolution is on the integer lattice rather than on the finite field vector space . The advantage of working in this general setting is that it becomes very easy to apply induction on the dimension ; indeed, to prove this estimate for arbitrary it suffices to do so for . This reduces matters to establishing the elementary inequality
for all , which can be done by a combination of undergraduate multivariable calculus and a little bit of numerical computation. (The left-hand side turns out to have local maxima at , with the latter being the cause of the numerology (1).)
The same sort of argument also gives an energy bound
for any subset of the Hamming cube, where
is the additive energy of . The example shows that the exponent cannot be improved.
I’ve just uploaded to the arXiv my paper “An integration approach to the Toeplitz square peg problem“, submitted to Forum of Mathematics, Sigma. This paper resulted from my attempts recently to solve the Toeplitz square peg problem (also known as the inscribed square problem):
See this recent survey of Matschke in the Notices of the AMS for the latest results on this problem.
The route I took to the results in this paper was somewhat convoluted. I was motivated to look at this problem after lecturing recently on the Jordan curve theorem in my class. The problem is superficially similar to the Jordan curve theorem in that the result is known (and rather easy to prove) if is sufficiently regular (e.g. if it is a polygonal path), but seems to be significantly more difficult when the curve is merely assumed to be continuous. Roughly speaking, all the known positive results on the problem have proceeded using (in some form or another) tools from homology: note for instance that one can view the conjecture as asking whether the four-dimensional subset of the eight-dimensional space necessarily intersects the four-dimensional space consisting of the quadruples traversing a square in (say) anti-clockwise order; this space is a four-dimensional linear subspace of , with a two-dimensional subspace of “degenerate” squares removed. If one ignores this degenerate subspace, one can use intersection theory to conclude (under reasonable “transversality” hypotheses) that intersects an odd number of times (up to the cyclic symmetries of the square), which is basically how Conjecture 1 is proven in the regular case. Unfortunately, if one then takes a limit and considers what happens when is just a continuous curve, the odd number of squares created by these homological arguments could conceivably all degenerate to points, thus blocking one from proving the conjecture in the general case.
Inspired by my previous work on finite time blowup for various PDEs, I first tried looking for a counterexample in the category of (locally) self-similar curves that are smooth (or piecewise linear) away from a single origin where it can oscillate infinitely often; this is basically the smoothest type of curve that was not already covered by previous results. By a rescaling and compactness argument, it is not difficult to see that such a counterexample would exist if there was a counterexample to the following periodic version of the conjecture:
Conjecture 2 (Periodic square peg problem) Let be two disjoint simple closed piecewise linear curves in the cylinder which have a winding number of one, that is to say they are homologous to the loop from to . Then the union of and contains the four vertices of a square.
In contrast to Conjecture 1, which is known for polygonal paths, Conjecture 2 is still open even under the hypothesis of polygonal paths; the homological arguments alluded to previously now show that the number of inscribed squares in the periodic setting is even rather than odd, which is not enough to conclude the conjecture. (This flipping of parity from odd to even due to an infinite amount of oscillation is reminiscent of the “Eilenberg-Mazur swindle“, discussed in this previous post.)
I therefore tried to construct counterexamples to Conjecture 2. I began perturbatively, looking at curves that were small perturbations of constant functions. After some initial Taylor expansion, I was blocked from forming such a counterexample because an inspection of the leading Taylor coefficients required one to construct a continuous periodic function of mean zero that never vanished, which of course was impossible by the intermediate value theorem. I kept expanding to higher and higher order to try to evade this obstruction (this, incidentally, was when I discovered this cute application of Lagrange reversion) but no matter how high an accuracy I went (I think I ended up expanding to sixth order in a perturbative parameter before figuring out what was going on!), this obstruction kept resurfacing again and again. I eventually figured out that this obstruction was being caused by a “conserved integral of motion” for both Conjecture 2 and Conjecture 1, which can in fact be used to largely rule out perturbative constructions. This yielded a new positive result for both conjectures:
We sketch the proof of Theorem 3(i) as follows (the proof of Theorem 3(ii) is very similar). Let be the curve , thus traverses one of the two graphs that comprise . For each time , there is a unique square with first vertex (and the other three vertices, traversed in anticlockwise order, denoted ) such that also lies in the graph of and also lies in the graph of (actually for technical reasons we have to extend by constants to all of in order for this claim to be true). To see this, we simply rotate the graph of clockwise by around , where (by the Lipschitz hypotheses) it must hit the graph of in a unique point, which is , and which then determines the other two vertices of the square. The curve has the same starting and ending point as the graph of or ; using the Lipschitz hypothesis one can show this graph is simple. If the curve ever hits the graph of other than at the endpoints, we have created an inscribed square, so we may assume for contradiction that avoids the graph of , and hence by the Jordan curve theorem the two curves enclose some non-empty bounded open region .
Now for the conserved integral of motion. If we integrate the -form on each of the four curves , we obtain the identity
This identity can be established by the following calculation: one can parameterise
for some Lipschitz functions ; thus for instance . Inserting these parameterisations and doing some canceling, one can write the above integral as
which vanishes because (which represent the sidelengths of the squares determined by vanish at the endpoints .
Using this conserved integral of motion, one can show that
which by Stokes’ theorem then implies that the bounded open region mentioned previously has zero area, which is absurd.
This argument hinged on the curve being simple, so that the Jordan curve theorem could apply. Once one left the perturbative regime of curves of small Lipschitz constant, it became possible for to be self-crossing, but nevertheless there still seemed to be some sort of integral obstruction. I eventually isolated the problem in the form of a strengthened version of Conjecture 2:
(note the -form is still well defined on the cylinder ; note also that the curves are allowed to cross each other.) Then there exists a (possibly degenerate) square with vertices (traversed in anticlockwise order) lying on respectively.
It is not difficult to see that Conjecture 4 implies Conjecture 2. Actually I believe that the converse implication is at least morally true, in that any counterexample to Conjecture 4 can be eventually transformed to a counterexample to Conjecture 2 and Conjecture 1. The conserved integral of motion argument can establish Conjecture 4 in many cases, for instance if are graphs of functions of Lipschitz constant less than one.
Conjecture 4 has a model special case, when one of the is assumed to just be a horizontal loop. In this case, the problem collapses to that of producing an intersection between two three-dimensional subsets of a six-dimensional space, rather than to four-dimensional subsets of an eight-dimensional space. More precisely, some elementary transformations reveal that this special case of Conjecture 4 can be formulated in the following fashion in which the geometric notion of a square is replaced by the additive notion of a triple of real numbers summing to zero:
Then there exist and with such that for .
This conjecture is easy to establish if one of the curves, say , is the graph of some piecewise linear function , since in that case the curve and the curve enclose the same area in the sense that , and hence must intersect by the Jordan curve theorem (otherwise they would enclose a non-zero amount of area between them), giving the claim. But when none of the are graphs, the situation becomes combinatorially more complicated.
Using some elementary homological arguments (e.g. breaking up closed -cycles into closed paths) and working with a generic horizontal slice of the curves, I was able to show that Conjecture 5 was equivalent to a one-dimensional problem that was largely combinatorial in nature, revolving around the sign patterns of various triple sums with drawn from various finite sets of reals.
- (i) For any , the sums are non-zero.
- (ii) (Non-crossing) For any and with the same parity, the pairs and are non-crossing in the sense that
- (iii) (Non-crossing sums) For any , , of the same parity, one has
Then one has
Roughly speaking, Conjecture 6 and Conjecture 5 are connected by constructing curves to connect to for by various paths, which either lie to the right of the axis (when is odd) or to the left of the axis (when is even). The axiom (ii) is asserting that the numbers are ordered according to the permutation of a meander (formed by gluing together two non-crossing perfect matchings).
Using various ad hoc arguments involving “winding numbers”, it is possible to prove this conjecture in many cases (e.g. if one of the is at most ), to the extent that I have now become confident that this conjecture is true (and have now come full circle from trying to disprove Conjecture 1 to now believing that this conjecture holds also). But it seems that there is some non-trivial combinatorial argument to be made if one is to prove this conjecture; purely homological arguments seem to partially resolve the problem, but are not sufficient by themselves.
While I was not able to resolve the square peg problem, I think these results do provide a roadmap to attacking it, first by focusing on the combinatorial conjecture in Conjecture 6 (or its equivalent form in Conjecture 5), then after that is resolved moving on to Conjecture 4, and then finally to Conjecture 1.
Fifteen years ago, I wrote a paper entitled Global regularity of wave maps. II. Small energy in two dimensions, in which I established global regularity of wave maps from two spatial dimensions to the unit sphere, assuming that the initial data had small energy. Recently, Hao Jia (personal communication) discovered a small gap in the argument that requires a slightly non-trivial fix. The issue does not really affect the subsequent literature, because the main result has since been reproven and extended by methods that avoid the gap (see in particular this subsequent paper of Tataru), but I have decided to describe the gap and its fix on this blog.
I will assume familiarity with the notation of my paper. In Section 10, some complicated spaces are constructed for each frequency scale , and then a further space is constructed for a given frequency envelope by the formula
where is the Littlewood-Paley projection of to frequency magnitudes . Then, given a spacetime slab , we define the restrictions
where the infimum is taken over all extensions of to the Minkowski spacetime ; similarly one defines
The gap in the paper is as follows: it was implicitly assumed that one could restrict (1) to the slab to obtain the equality
(This equality is implicitly used to establish the bound (36) in the paper.) Unfortunately, (1) only gives the lower bound, not the upper bound, and it is the upper bound which is needed here. The problem is that the extensions of that are optimal for computing are not necessarily the Littlewood-Paley projections of the extensions of that are optimal for computing .
To remedy the problem, one has to prove an upper bound of the form
the extension will then obey (5) (here we use Lemma 9 from my paper), but unfortunately is not guaranteed to obey (4) (the norm does control the norm, but a key point about frequency envelopes for the small energy regularity problem is that the coefficients , while bounded, are not necessarily summable).
This can be fixed as follows. For each we introduce a time cutoff supported on that equals on and obeys the usual derivative estimates in between (the time derivative of size for each ). Later we will prove the truncation estimate
Assuming this estimate, then if we set , then using Lemma 9 in my paper and (6), (7) (and the local stability of frequency envelopes) we have the required property (5). (There is a technical issue arising from the fact that is not necessarily Schwartz due to slow decay at temporal infinity, but by considering partial sums in the summation and taking limits we can check that is the strong limit of Schwartz functions, which suffices here; we omit the details for sake of exposition.) So the only issue is to establish (4), that is to say that
for all .
For this is immediate from (2). Now suppose that for some integer (the case when is treated similarly). Then we can split
The contribution of the term is acceptable by (6) and estimate (82) from my paper. The term sums to which is acceptable by (2). So it remains to control the norm of . By the triangle inequality and the fundamental theorem of calculus, we can bound
By hypothesis, . Using the first term in (79) of my paper and Bernstein’s inequality followed by (6) we have
and then we are done by summing the geometric series in .
It remains to prove the truncation estimate (7). This estimate is similar in spirit to the algebra estimates already in my paper, but unfortunately does not seem to follow immediately from these estimates as written, and so one has to repeat the somewhat lengthy decompositions and case checkings used to prove these estimates. We do this below the fold.
I’ve just posted to the arXiv my paper “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation“. This paper is loosely in the spirit of other recent papers of mine in which I explore how close one can get to supercritical PDE of physical interest (such as the Euler and Navier-Stokes equations), while still being able to rigorously demonstrate finite time blowup for at least some choices of initial data. Here, the PDE we are trying to get close to is the incompressible inviscid Euler equations
in three spatial dimensions, where is the velocity vector field and is the pressure field. In vorticity form, and viewing the vorticity as a -form (rather than a vector), we can rewrite this system using the language of differential geometry as
where is the Lie derivative along , is the codifferential (the adjoint of the differential , or equivalently the negative of the divergence operator) that sends -vector fields to -vector fields, is the Hodge Laplacian, and is the identification of -vector fields with -forms induced by the Euclidean metric . The equation can be viewed as the Biot-Savart law recovering velocity from vorticity, expressed in the language of differential geometry.
One can then generalise this system by replacing the operator by a more general operator from -forms to -vector fields, giving rise to what I call the generalised Euler equations
For example, the surface quasi-geostrophic (SQG) equations can be written in this form, as discussed in this previous post. One can view (up to Hodge duality) as a vector potential for the velocity , so it is natural to refer to as a vector potential operator.
The generalised Euler equations carry much of the same geometric structure as the true Euler equations. For instance, the transport equation is equivalent to the Kelvin circulation theorem, which in three dimensions also implies the transport of vortex streamlines and the conservation of helicity. If is self-adjoint and positive definite, then the famous Euler-Poincaré interpretation of the true Euler equations as geodesic flow on an infinite dimensional Riemannian manifold of volume preserving diffeomorphisms (as discussed in this previous post) extends to the generalised Euler equations (with the operator determining the new Riemannian metric to place on this manifold). In particular, the generalised Euler equations have a Lagrangian formulation, and so by Noether’s theorem we expect any continuous symmetry of the Lagrangian to lead to conserved quantities. Indeed, we have a conserved Hamiltonian , and any spatial symmetry of leads to a conserved impulse (e.g. translation invariance leads to a conserved momentum, and rotation invariance leads to a conserved angular momentum). If behaves like a pseudodifferential operator of order (as is the case with the true vector potential operator ), then it turns out that one can use energy methods to recover the same sort of classical local existence theory as for the true Euler equations (up to and including the famous Beale-Kato-Majda criterion for blowup).
The true Euler equations are suspected of admitting smooth localised solutions which blow up in finite time; there is now substantial numerical evidence for this blowup, but it has not been proven rigorously. The main purpose of this paper is to show that such finite time blowup can at least be established for certain generalised Euler equations that are somewhat close to the true Euler equations. This is similar in spirit to my previous paper on finite time blowup on averaged Navier-Stokes equations, with the main new feature here being that the modified equation continues to have a Lagrangian structure and a vorticity formulation, which was not the case with the averaged Navier-Stokes equation. On the other hand, the arguments here are not able to handle the presence of viscosity (basically because they rely crucially on the Kelvin circulation theorem, which is not available in the viscous case).
In fact, three different blowup constructions are presented (for three different choices of vector potential operator ). The first is a variant of one discussed previously on this blog, in which a “neck pinch” singularity for a vortex tube is created by using a non-self-adjoint vector potential operator, in which the velocity at the neck of the vortex tube is determined by the circulation of the vorticity somewhat further away from that neck, which when combined with conservation of circulation is enough to guarantee finite time blowup. This is a relatively easy construction of finite time blowup, and has the advantage of being rather stable (any initial data flowing through a narrow tube with a large positive circulation will blow up in finite time). On the other hand, it is not so surprising in the non-self-adjoint case that finite blowup can occur, as there is no conserved energy.
The second blowup construction is based on a connection between the two-dimensional SQG equation and the three-dimensional generalised Euler equations, discussed in this previous post. Namely, any solution to the former can be lifted to a “two and a half-dimensional” solution to the latter, in which the velocity and vorticity are translation-invariant in the vertical direction (but the velocity is still allowed to contain vertical components, so the flow is not completely horizontal). The same embedding also works to lift solutions to generalised SQG equations in two dimensions to solutions to generalised Euler equations in three dimensions. Conveniently, even if the vector potential operator for the generalised SQG equation fails to be self-adjoint, one can ensure that the three-dimensional vector potential operator is self-adjoint. Using this trick, together with a two-dimensional version of the first blowup construction, one can then construct a generalised Euler equation in three dimensions with a vector potential that is both self-adjoint and positive definite, and still admits solutions that blow up in finite time, though now the blowup is now a vortex sheet creasing at on a line, rather than a vortex tube pinching at a point.
This eliminates the main defect of the first blowup construction, but introduces two others. Firstly, the blowup is less stable, as it relies crucially on the initial data being translation-invariant in the vertical direction. Secondly, the solution is not spatially localised in the vertical direction (though it can be viewed as a compactly supported solution on the manifold , rather than ). The third and final blowup construction of the paper addresses the final defect, by replacing vertical translation symmetry with axial rotation symmetry around the vertical axis (basically, replacing Cartesian coordinates with cylindrical coordinates). It turns out that there is a more complicated way to embed two-dimensional generalised SQG equations into three-dimensional generalised Euler equations in which the solutions to the latter are now axially symmetric (but are allowed to “swirl” in the sense that the velocity field can have a non-zero angular component), while still keeping the vector potential operator self-adjoint and positive definite; the blowup is now that of a vortex ring creasing on a circle.
As with the previous papers in this series, these blowup constructions do not directly imply finite time blowup for the true Euler equations, but they do at least provide a barrier to establishing global regularity for these latter equations, in that one is forced to use some property of the true Euler equations that are not shared by these generalisations. They also suggest some possible blowup mechanisms for the true Euler equations (although unfortunately these mechanisms do not seem compatible with the addition of viscosity, so they do not seem to suggest a viable Navier-Stokes blowup mechanism).
I’ve just uploaded to the arXiv my paper “Equivalence of the logarithmically averaged Chowla and Sarnak conjectures“, submitted to the Festschrift “Number Theory – Diophantine problems, uniform distribution and applications” in honour of Robert F. Tichy. This paper is a spinoff of my previous paper establishing a logarithmically averaged version of the Chowla (and Elliott) conjectures in the two-point case. In that paper, the estimate
as was demonstrated, where was any positive integer and denoted the Liouville function. The proof proceeded using a method I call the “entropy decrement argument”, which ultimately reduced matters to establishing a bound of the form
It is natural to see to what extent the arguments can be adapted to attack the higher-point cases of the logarithmically averaged Chowla conjecture (ignoring for this post the more general Elliott conjecture for other bounded multiplicative functions than the Liouville function). That is to say, one would like to prove that
as for any fixed distinct integers . As it turns out (and as is detailed in the current paper), the entropy decrement argument extends to this setting (after using some known facts about linear equations in primes), and allows one to reduce the above estimate to an estimate of the form
for a slowly growing function of and some fixed (in fact we can take for ), where is the (normalised) local Gowers uniformity norm. (In the case , , this becomes the Fourier-uniformity conjecture discussed in this previous post.) If one then applied the (now proven) inverse conjecture for the Gowers norms, this estimate is in turn equivalent to the more complicated looking assertion
where the supremum is over all possible choices of nilsequences of controlled step and complexity (see the paper for definitions of these terms).
The main novelty in the paper (elaborating upon a previous comment I had made on this blog) is to observe that this latter estimate in turn follows from the logarithmically averaged form of Sarnak’s conjecture (discussed in this previous post), namely that
whenever is a zero entropy (i.e. deterministic) sequence. Morally speaking, this follows from the well-known fact that nilsequences have zero entropy, but the presence of the supremum in (1) means that we need a little bit more; roughly speaking, we need the class of nilsequences of a given step and complexity to have “uniformly zero entropy” in some sense.
On the other hand, it was already known (see previous post) that the Chowla conjecture implied the Sarnak conjecture, and similarly for the logarithmically averaged form of the two conjectures. Putting all these implications together, we obtain the pleasant fact that the logarithmically averaged Sarnak and Chowla conjectures are equivalent, which is the main result of the current paper. There have been a large number of special cases of the Sarnak conjecture worked out (when the deterministic sequence involved came from a special dynamical system), so these results can now also be viewed as partial progress towards the Chowla conjecture also (at least with logarithmic averaging). However, my feeling is that the full resolution of these conjectures will not come from these sorts of special cases; instead, conjectures like the Fourier-uniformity conjecture in this previous post look more promising to attack.
It would also be nice to get rid of the pesky logarithmic averaging, but this seems to be an inherent requirement of the entropy decrement argument method, so one would probably have to find a way to avoid that argument if one were to remove the log averaging.
Tamar Ziegler and I have just uploaded to the arXiv two related papers: “Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteoristic factors” and “polynomial patterns in primes“, with the former developing a “quantitative Bessel inequality” for local Gowers norms that is crucial in the latter.
We use the term “concatenation theorem” to denote results in which structural control of a function in two or more “directions” can be “concatenated” into structural control in a joint direction. A trivial example of such a concatenation theorem is the following: if a function is constant in the first variable (thus is constant for each ), and also constant in the second variable (thus is constant for each ), then it is constant in the joint variable . A slightly less trivial example: if a function is affine-linear in the first variable (thus, for each , there exist such that for all ) and affine-linear in the second variable (thus, for each , there exist such that for all ) then is a quadratic polynomial in ; in fact it must take the form
The same phenomenon extends to higher degree polynomials. Given a function from one additive group to another, we say that is of degree less than along a subgroup of if all the -fold iterated differences of along directions in vanish, that is to say
for all and , where is the difference operator
(We adopt the convention that the only of degree less than is the zero function.)
We then have the following simple proposition:
Proposition 1 (Concatenation of polynomiality) Let be of degree less than along one subgroup of , and of degree less than along another subgroup of , for some . Then is of degree less than along the subgroup of .
Note the previous example was basically the case when , , , , and .
Proof: The claim is trivial for or (in which is constant along or respectively), so suppose inductively and the claim has already been proven for smaller values of .
We take a derivative in a direction along to obtain
where is the shift of by . Then we take a further shift by a direction to obtain
leading to the cocycle equation
Since has degree less than along and degree less than along , has degree less than along and less than along , so is degree less than along by induction hypothesis. Similarly is also of degree less than along . Combining this with the cocycle equation we see that is of degree less than along for any , and hence is of degree less than along , as required.
While this proposition is simple, it already illustrates some basic principles regarding how one would go about proving a concatenation theorem:
- (i) One should perform induction on the degrees involved, and take advantage of the recursive nature of degree (in this case, the fact that a function is of less than degree along some subgroup of directions iff all of its first derivatives along are of degree less than ).
- (ii) Structure is preserved by operations such as addition, shifting, and taking derivatives. In particular, if a function is of degree less than along some subgroup , then any derivative of is also of degree less than along , even if does not belong to .
Here is another simple example of a concatenation theorem. Suppose an at most countable additive group acts by measure-preserving shifts on some probability space ; we call the pair (or more precisely ) a -system. We say that a function is a generalised eigenfunction of degree less than along some subgroup of and some if one has
almost everywhere for all , and some functions of degree less than along , with the convention that a function has degree less than if and only if it is equal to . Thus for instance, a function is an generalised eigenfunction of degree less than along if it is constant on almost every -ergodic component of , and is a generalised function of degree less than along if it is an eigenfunction of the shift action on almost every -ergodic component of . A basic example of a higher order eigenfunction is the function on the skew shift with action given by the generator for some irrational . One can check that for every integer , where is a generalised eigenfunction of degree less than along , so is of degree less than along .
We then have
Proposition 2 (Concatenation of higher order eigenfunctions) Let be a -system, and let be a generalised eigenfunction of degree less than along one subgroup of , and a generalised eigenfunction of degree less than along another subgroup of , for some . Then is a generalised eigenfunction of degree less than along the subgroup of .
The argument is almost identical to that of the previous proposition and is left as an exercise to the reader. The key point is the point (ii) identified earlier: the space of generalised eigenfunctions of degree less than along is preserved by multiplication and shifts, as well as the operation of “taking derivatives” even along directions that do not lie in . (To prove this latter claim, one should restrict to the region where is non-zero, and then divide by to locate .)
A typical example of this proposition in action is as follows: consider the -system given by the -torus with generating shifts
for some irrational , which can be checked to give a action
The function can then be checked to be a generalised eigenfunction of degree less than along , and also less than along , and less than along . One can view this example as the dynamical systems translation of the example (1) (see this previous post for some more discussion of this sort of correspondence).
The main results of our concatenation paper are analogues of these propositions concerning a more complicated notion of “polynomial-like” structure that are of importance in additive combinatorics and in ergodic theory. On the ergodic theory side, the notion of structure is captured by the Host-Kra characteristic factors of a -system along a subgroup . These factors can be defined in a number of ways. One is by duality, using the Gowers-Host-Kra uniformity seminorms (defined for instance here) . Namely, is the factor of defined up to equivalence by the requirement that
An equivalent definition is in terms of the dual functions of along , which can be defined recursively by setting and
where denotes the ergodic average along a Følner sequence in (in fact one can also define these concepts in non-amenable abelian settings as per this previous post). The factor can then be alternately defined as the factor generated by the dual functions for .
In the case when and is -ergodic, a deep theorem of Host and Kra shows that the factor is equivalent to the inverse limit of nilsystems of step less than . A similar statement holds with replaced by any finitely generated group by Griesmer, while the case of an infinite vector space over a finite field was treated in this paper of Bergelson, Ziegler, and myself. The situation is more subtle when is not -ergodic, or when is -ergodic but is a proper subgroup of acting non-ergodically, when one has to start considering measurable families of directional nilsystems; see for instance this paper of Austin for some of the subtleties involved (for instance, higher order group cohomology begins to become relevant!).
One of our main theorems is then
Proposition 3 (Concatenation of characteristic factors) Let be a -system, and let be measurable with respect to the factor and with respect to the factor for some and some subgroups of . Then is also measurable with respect to the factor .
We give two proofs of this proposition in the paper; an ergodic-theoretic proof using the Host-Kra theory of “cocycles of type (along a subgroup )”, which can be used to inductively describe the factors , and a combinatorial proof based on a combinatorial analogue of this proposition which is harder to state (but which roughly speaking asserts that a function which is nearly orthogonal to all bounded functions of small norm, and also to all bounded functions of small norm, is also nearly orthogonal to alll bounded functions of small norm). The combinatorial proof parallels the proof of Proposition 2. A key point is that dual functions obey a property analogous to being a generalised eigenfunction, namely that
where and is a “structured function of order ” along . (In the language of this previous paper of mine, this is an assertion that dual functions are uniformly almost periodic of order .) Again, the point (ii) above is crucial, and in particular it is key that any structure that has is inherited by the associated functions and . This sort of inheritance is quite easy to accomplish in the ergodic setting, as there is a ready-made language of factors to encapsulate the concept of structure, and the shift-invariance and -algebra properties of factors make it easy to show that just about any “natural” operation one performs on a function measurable with respect to a given factor, returns a function that is still measurable in that factor. In the finitary combinatorial setting, though, encoding the fact (ii) becomes a remarkably complicated notational nightmare, requiring a huge amount of “epsilon management” and “second-order epsilon management” (in which one manages not only scalar epsilons, but also function-valued epsilons that depend on other parameters). In order to avoid all this we were forced to utilise a nonstandard analysis framework for the combinatorial theorems, which made the arguments greatly resemble the ergodic arguments in many respects (though the two settings are still not equivalent, see this previous blog post for some comparisons between the two settings). Unfortunately the arguments are still rather complicated.
For combinatorial applications, dual formulations of the concatenation theorem are more useful. A direct dualisation of the theorem yields the following decomposition theorem: a bounded function which is small in norm can be split into a component that is small in norm, and a component that is small in norm. (One may wish to understand this type of result by first proving the following baby version: any function that has mean zero on every coset of , can be decomposed as the sum of a function that has mean zero on every coset, and a function that has mean zero on every coset. This is dual to the assertion that a function that is constant on every coset and constant on every coset, is constant on every coset.) Combining this with some standard “almost orthogonality” arguments (i.e. Cauchy-Schwarz) give the following Bessel-type inequality: if one has a lot of subgroups and a bounded function is small in norm for most , then it is also small in norm for most . (Here is a baby version one may wish to warm up on: if a function has small mean on for some large prime , then it has small mean on most of the cosets of most of the one-dimensional subgroups of .)
There is also a generalisation of the above Bessel inequality (as well as several of the other results mentioned above) in which the subgroups are replaced by more general coset progressions (of bounded rank), so that one has a Bessel inequailty controlling “local” Gowers uniformity norms such as by “global” Gowers uniformity norms such as . This turns out to be particularly useful when attempting to compute polynomial averages such as
(actually one ends up with more complicated expressions than this, but let’s use this example for sake of discussion). This can be viewed as an average of various Gowers uniformity norms of along arithmetic progressions of the form for various . Using the above Bessel inequality, this can be controlled in turn by an average of various Gowers uniformity norms along rank two generalised arithmetic progressions of the form for various . But for generic , this rank two progression is close in a certain technical sense to the “global” interval (this is ultimately due to the basic fact that two randomly chosen large integers are likely to be coprime, or at least have a small gcd). As a consequence, one can use the concatenation theorems from our first paper to control expressions such as (2) in terms of global Gowers uniformity norms. This is important in number theoretic applications, when one is interested in computing sums such as
where and are the Möbius and von Mangoldt functions respectively. This is because we are able to control global Gowers uniformity norms of such functions (thanks to results such as the proof of the inverse conjecture for the Gowers norms, the orthogonality of the Möbius function with nilsequences, and asymptotics for linear equations in primes), but much less control is currently available for local Gowers uniformity norms, even with the assistance of the generalised Riemann hypothesis (see this previous blog post for some further discussion).
By combining these tools and strategies with the “transference principle” approach from our previous paper (as improved using the recent “densification” technique of Conlon, Fox, and Zhao, discussed in this previous post), we are able in particular to establish the following result:
Theorem 4 (Polynomial patterns in the primes) Let be polynomials of degree at most , whose degree coefficients are all distinct, for some . Suppose that is admissible in the sense that for every prime , there are such that are all coprime to . Then there exist infinitely many pairs of natural numbers such that are prime.
Furthermore, we obtain an asymptotic for the number of such pairs in the range , (actually for minor technical reasons we reduce the range of to be very slightly less than ). In fact one could in principle obtain asymptotics for smaller values of , and relax the requirement that the degree coefficients be distinct with the requirement that no two of the differ by a constant, provided one had good enough local uniformity results for the Möbius or von Mangoldt functions. For instance, we can obtain an asymptotic for triplets of the form unconditionally for , and conditionally on GRH for all , using known results on primes in short intervals on average.
The case of this theorem was obtained in a previous paper of myself and Ben Green (using the aforementioned conjectures on the Gowers uniformity norm and the orthogonality of the Möbius function with nilsequences, both of which are now proven). For higher , an older result of Tamar and myself was able to tackle the case when (though our results there only give lower bounds on the number of pairs , and no asymptotics). Both of these results generalise my older theorem with Ben Green on the primes containing arbitrarily long arithmetic progressions. The theorem also extends to multidimensional polynomials, in which case there are some additional previous results; see the paper for more details. We also get a technical refinement of our previous result on narrow polynomial progressions in (dense subsets of) the primes by making the progressions just a little bit narrower in the case of the density of the set one is using is small.
. This latter Bessel type inequality is particularly useful in combinatorial and number-theoretic applications, as it allows one to convert “global” Gowers uniformity norm (basically, bounds on norms such as ) to “local” Gowers uniformity norm control.
Van Vu and I just posted to the arXiv our paper “sum-free sets in groups” (submitted to Discrete Analysis), as well as a companion survey article (submitted to J. Comb.). Given a subset of an additive group , define the quantity to be the cardinality of the largest subset of which is sum-free in in the sense that all the sums with distinct elements of lie outside of . For instance, if is itself a group, then , since no two elements of can sum to something outside of . More generally, if is the union of groups, then is at most , thanks to the pigeonhole principle.
If is the integers, then there are no non-trivial subgroups, and one can thus expect to start growing with . For instance, one has the following easy result:
Proof: We use an argument of Ruzsa, which is based in turn on an older argument of Choi. Let be the largest element of , and then recursively, once has been selected, let be the largest element of not equal to any of the , such that for all , terminating this construction when no such can be located. This gives a sequence of elements in which are sum-free in , and with the property that for any , either is equal to one of the , or else for some with . Iterating this, we see that any is of the form for some and . The number of such expressions is at most , thus which implies . Since , the claim follows.
In particular, we have for subsets of the integers. It has been possible to improve upon this easy bound, but only with remarkable effort. The best lower bound currently is
Using the standard tool of Freiman homomorphisms, the above results for the integers extend to other torsion-free abelian groups . In our paper we study the opposite case where is finite (but still abelian). In this paper of Erdös (in which the quantity was first introduced), the following question was posed: if is sufficiently large depending on , does this imply the existence of two elements with ? As it turns out, we were able to find some simple counterexamples to this statement. For instance, if is any finite additive group, then the set has but with no summing to zero; this type of example in fact works with replaced by any larger Mersenne prime, and we also have a counterexample in for arbitrarily large. However, in the positive direction, we can show that the answer to Erdös’s question is positive if is assumed to have no small prime factors. That is to say,
Theorem 2 For every there exists such that if is a finite abelian group whose order is not divisible by any prime less than or equal to , and is a subset of with order at least and , then there exist with .
There are two main tools used to prove this result. One is an “arithmetic removal lemma” proven by Král, Serra, and Vena. Note that the condition means that for any distinct , at least one of the , , must also lie in . Roughly speaking, the arithmetic removal lemma allows one to “almost” remove the requirement that be distinct, which basically now means that for almost all . This near-dilation symmetry, when combined with the hypothesis that has no small prime factors, gives a lot of “dispersion” in the Fourier coefficients of which can now be exploited to prove the theorem.
The second tool is the following structure theorem, which is the main result of our paper, and goes a fair ways towards classifying sets for which is small:
Theorem 3 Let be a finite subset of an arbitrary additive group , with . Then one can find finite subgroups with such that and . Furthermore, if , then the exceptional set is empty.
Roughly speaking, this theorem shows that the example of the union of subgroups mentioned earlier is more or less the “only” example of sets with , modulo the addition of some small exceptional sets and some refinement of the subgroups to dense subsets.
This theorem has the flavour of other inverse theorems in additive combinatorics, such as Freiman’s theorem, and indeed one can use Freiman’s theorem (and related tools, such as the Balog-Szemeredi theorem) to easily get a weaker version of this theorem. Indeed, if there are no sum-free subsets of of order , then a fraction of all pairs in must have their sum also in (otherwise one could take random elements of and they would be sum-free in with positive probability). From this and the Balog-Szemeredi theorem and Freiman’s theorem (in arbitrary abelian groups, as established by Green and Ruzsa), we see that must be “commensurate” with a “coset progression” of bounded rank. One can then eliminate the torsion-free component of this coset progression by a number of methods (e.g. by using variants of the argument in Proposition 1), with the upshot being that one can locate a finite group that has large intersection with .
At this point it is tempting to simply remove from and iterate. But one runs into a technical difficulty that removing a set such as from can alter the quantity in unpredictable ways, so one has to still keep around when analysing the residual set . A second difficulty is that the latter set could be considerably smaller than or , but still large in absolute terms, so in particular any error term whose size is only bounded by for a small could be massive compared with the residual set , and so such error terms would be unacceptable. One can get around these difficulties if one first performs some preliminary “normalisation” of the group , so that the residual set does not intersect any coset of too strongly. The arguments become even more complicated when one starts removing more than one group from and analyses the residual set ; indeed the “epsilon management” involved became so fearsomely intricate that we were forced to use a nonstandard analysis formulation of the problem in order to keep the complexity of the argument at a reasonable level (cf. my previous blog post on this topic). One drawback of doing so is that we have no effective bounds for the implied constants in our main theorem; it would be of interest to obtain a more direct proof of our main theorem that would lead to effective bounds.
I’ve just uploaded to the arXiv my paper Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity, submitted to Comm. PDE. This paper is in the same spirit as (though not directly related to) my previous paper on finite time blowup of supercritical NLW systems, and was inspired by a question posed to me some time ago by Jeffrey Rauch. Here, instead of looking at supercritical equations, we look at an extremely subcritical equation, namely a system of the form
where is the unknown field, and is the nonlinearity, which we assume to have all derivatives bounded. A typical example of such an equation is the higher-dimensional sine-Gordon equation
for a scalar field . Here is the d’Alembertian operator. We restrict attention here to classical (i.e. smooth) solutions to (1).
We do not assume any Hamiltonian structure, so we do not require to be a gradient of a potential . But even without such Hamiltonian structure, the equation (1) is very well behaved, with many a priori bounds available. For instance, if the initial position and initial velocity are smooth and compactly supported, then from finite speed of propagation has uniformly bounded compact support for all in a bounded interval. As the nonlinearity is bounded, this immediately places in in any bounded time interval, which by the energy inequality gives an a priori bound on in this time interval. Next, from the chain rule we have
which (from the assumption that is bounded) shows that is in , which by the energy inequality again now gives an a priori bound on .
One might expect that one could keep iterating this and obtain a priori bounds on in arbitrarily smooth norms. In low dimensions such as , this is a fairly easy task, since the above estimates and Sobolev embedding already place one in , and the nonlinear map is easily verified to preserve the space for any natural number , from which one obtains a priori bounds in any Sobolev space; from this and standard energy methods, one can then establish global regularity for this equation (that is to say, any smooth choice of initial data generates a global smooth solution). However, one starts running into trouble in higher dimensions, in which no bound is available. The main problem is that even a really nice nonlinearity such as is unbounded in higher Sobolev norms. The estimates
ensure that the map is bounded in low regularity spaces like or , but one already runs into trouble with the second derivative
where there is a troublesome lower order term of size which becomes difficult to control in higher dimensions, preventing the map to be bounded in . Ultimately, the issue here is that when is not controlled in , the function can oscillate at a much higher frequency than ; for instance, if is the one-dimensional wave for some and , then oscillates at frequency , but the function more or less oscillates at the larger frequency .
In medium dimensions, it is possible to use dispersive estimates for the wave equation (such as the famous Strichartz estimates) to overcome these problems. This line of inquiry was pursued (albeit for slightly different classes of nonlinearity than those considered here) by Heinz-von Wahl, Pecher (in a series of papers), Brenner, and Brenner-von Wahl; to cut a long story short, one of the conclusions of these papers was that one had global regularity for equations such as (1) in dimensions . (I reprove this result using modern Strichartz estimate and Littlewood-Paley techniques in an appendix to my paper. The references given also allow for some growth in the nonlinearity , but we will not detail the precise hypotheses used in these papers here.)
In my paper, I complement these positive results with an almost matching negative result:
Theorem 1 If and , then there exists a nonlinearity with all derivatives bounded, and a solution to (1) that is smooth at time zero, but develops a singularity in finite time.
The construction crucially relies on the ability to choose the nonlinearity , and also needs some injectivity properties on the solution (after making a symmetry reduction using an assumption of spherical symmetry to view as a function of variables rather than ) which restricts our counterexample to the case. Thus the model case of the higher-dimensional sine-Gordon equation is not covered by our arguments. Nevertheless (as with previous finite-time blowup results discussed on this blog), one can view this result as a barrier to trying to prove regularity for equations such as in eleven and higher dimensions, as any such argument must somehow use a property of that equation that is not applicable to the more general system (1).
Let us first give some back-of-the-envelope calculations suggesting why there could be finite time blowup in eleven and higher dimensions. For sake of this discussion let us restrict attention to the sine-Gordon equation . The blowup ansatz we will use is as follows: for each frequency in a sequence of large quantities going to infinity, there will be a spacetime “cube” on which the solution oscillates with “amplitude” and “frequency” , where is an exponent to be chosen later; this ansatz is of course compatible with the uncertainty principle. Since as , this will create a singularity at the spacetime origin . To make this ansatz plausible, we wish to make the oscillation of on driven primarily by the forcing term at . Thus, by Duhamel’s formula, we expect a relation roughly of the form
on , where is the usual free wave propagator, and is the indicator function of .
On , oscillates with amplitude and frequency , we expect the derivative to be of size about , and so from the principle of stationary phase we expect to oscillate at frequency about . Since the wave propagator preserves frequencies, and is supposed to be of frequency on we are thus led to the requirement
where is surface measure on the unit sphere , and is the volume of that sphere. In our setting, is comparable to , and so we have the informal approximation
Since is bounded, is bounded as well. This gives a (non-rigorous) upper bound
which when combined with our ansatz that has ampitude about on , gives the constraint
which on applying (2) gives the further constraint
which can be rearranged as
It is now clear that the optimal choice of is
and this blowup ansatz is only self-consistent when
or equivalently if .
To turn this ansatz into an actual blowup example, we will construct as the sum of various functions that solve the wave equation with forcing term in , and which concentrate in with the amplitude and frequency indicated by the above heuristic analysis. The remaining task is to show that can be written in the form for some with all derivatives bounded. For this one needs some injectivity properties of (after imposing spherical symmetry to impose a dimensional reduction on the domain of from dimensions to ). This requires one to construct some solutions to the free wave equation that have some unusual restrictions on the range (for instance, we will need a solution taking values in the plane that avoid one quadrant of that plane). In order to do this we take advantage of the very explicit nature of the fundamental solution to the wave equation in odd dimensions (such as ), particularly under the assumption of spherical symmetry. Specifically, one can show that in odd dimension , any spherically symmetric function of the form
for an arbitrary smooth function , will solve the free wave equation; this is ultimately due to iterating the “ladder operator” identity
This precise and relatively simple formula for allows one to create “bespoke” solutions that obey various unusual properties, without too much difficulty.
It is not clear to me what to conjecture for . The blowup ansatz given above is a little inefficient, in that the frequency component of the solution is only generated from a portion of the component, namely the portion close to a certain light cone. In particular, the solution does not saturate the Strichartz estimates that are used to establish the positive results for , which helps explain the slight gap between the positive and negative results. It may be that a more complicated ansatz could work to give a negative result in ten dimensions; conversely, it is also possible that one could use more advanced estimates than the Strichartz estimate (that somehow capture the “thinness” of the fundamental solution, and not just its dispersive properties) to stretch the positive results to ten dimensions. Which side the case falls in all come down to some rather delicate numerology.