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A capset in the vector space over the finite field of three elements is a subset of that does not contain any lines , where and . A basic problem in additive combinatorics (discussed in one of the very first posts on this blog) is to obtain good upper and lower bounds for the maximal size of a capset in .
Trivially, one has . Using Fourier methods (and the density increment argument of Roth), the bound of was obtained by Meshulam, and improved only as late as 2012 to for some absolute constant by Bateman and Katz. But in a very recent breakthrough, Ellenberg (and independently Gijswijt) obtained the exponentially superior bound , using a version of the polynomial method recently introduced by Croot, Lev, and Pach. (In the converse direction, a construction of Edel gives capsets as large as .) Given the success of the polynomial method in superficially similar problems such as the finite field Kakeya problem (discussed in this previous post), it was natural to wonder that this method could be applicable to the cap set problem (see for instance this MathOverflow comment of mine on this from 2010), but it took a surprisingly long time before Croot, Lev, and Pach were able to identify the precise variant of the polynomial method that would actually work here.
The proof of the capset bound is very short (Ellenberg’s and Gijswijt’s preprints are both 3 pages long, and Croot-Lev-Pach is 6 pages), but I thought I would present a slight reformulation of the argument which treats the three points on a line in symmetrically (as opposed to treating the third point differently from the first two, as is done in the Ellenberg and Gijswijt papers; Croot-Lev-Pach also treat the middle point of a three-term arithmetic progression differently from the two endpoints, although this is a very natural thing to do in their context of ). The basic starting point is this: if is a capset, then one has the identity
for all , where is the Kronecker delta function, which we view as taking values in . Indeed, (1) reflects the fact that the equation has solutions precisely when are either all equal, or form a line, and the latter is ruled out precisely when is a capset.
To exploit (1), we will show that the left-hand side of (1) is “low rank” in some sense, while the right-hand side is “high rank”. Recall that a function taking values in a field is of rank one if it is non-zero and of the form for some , and that the rank of a general function is the least number of rank one functions needed to express as a linear combination. More generally, if , we define the rank of a function to be the least number of “rank one” functions of the form
for some and some functions , , that are needed to generate as a linear combination. For instance, when , the rank one functions take the form , , , and linear combinations of such rank one functions will give a function of rank at most .
It is a standard fact in linear algebra that the rank of a diagonal matrix is equal to the number of non-zero entries. This phenomenon extends to higher dimensions:
Proof: We induct on . As mentioned above, the case follows from standard linear algebra, so suppose now that and the claim has already been proven for .
It is clear that the function (2) has rank at most equal to the number of non-zero (since the summands on the right-hand side are rank one functions), so it suffices to establish the lower bound. By deleting from those elements with (which cannot increase the rank), we may assume without loss of generality that all the are non-zero. Now suppose for contradiction that (2) has rank at most , then we obtain a representation
Consider the space of functions that are orthogonal to all the , in the sense that
for all . This space is a vector space whose dimension is at least . A basis of this space generates a coordinate matrix of full rank, which implies that there is at least one non-singular minor. This implies that there exists a function in this space which is nowhere vanishing on some subset of of cardinality at least .
If we multiply (3) by and sum in , we conclude that
The right-hand side has rank at most , since the summands are rank one functions. On the other hand, from induction hypothesis the left-hand side has rank at least , giving the required contradiction.
On the other hand, we have the following (symmetrised version of a) beautifully simple observation of Croot, Lev, and Pach:
Proof: Using the identity for , we have
The right-hand side is clearly a polynomial of degree in , which is then a linear combination of monomials
In particular, from the pigeonhole principle, at least one of is at most .
Consider the contribution of the monomials for which . We can regroup this contribution as
where ranges over those with , is the monomial
and is some explicitly computable function whose exact form will not be of relevance to our argument. The number of such is equal to , so this contribution has rank at most . The remaining contributions arising from the cases and similarly have rank at most (grouping the monomials so that each monomial is only counted once), so the claim follows.
Upon restricting from to , the rank of is still at most . The two lemmas then combine to give the Ellenberg-Gijswijt bound
All that remains is to compute the asymptotic behaviour of . This can be done using the general tool of Cramer’s theorem, but can also be derived from Stirling’s formula (discussed in this previous post). Indeed, if , , for some summing to , Stirling’s formula gives
where is the entropy function
We then have
where is the maximum entropy subject to the constraints
A routine Lagrange multiplier computation shows that the maximum occurs when
and is approximately , giving rise to the claimed bound of .
Remark 3 As noted in the Ellenberg and Gijswijt papers, the above argument extends readily to other fields than to control the maximal size of subset of that has no non-trivial solutions to the equation , where are non-zero constants that sum to zero. Of course one replaces the function in Lemma 2 by in this case.
Remark 4 This symmetrised formulation suggests that one possible way to improve slightly on the numerical quantity by finding a more efficient way to decompose into rank one functions, however I was not able to do so (though such improvements are reminiscent of the Strassen type algorithms for fast matrix multiplication).
Remark 5 It is tempting to see if this method can get non-trivial upper bounds for sets with no length progressions, in (say) . One can run the above arguments, replacing the function
this leads to the bound where
Unfortunately, is asymptotic to and so this bound is in fact slightly worse than the trivial bound ! However, there is a slim chance that there is a more efficient way to decompose into rank one functions that would give a non-trivial bound on . I experimented with a few possible such decompositions but unfortunately without success.
Remark 6 Return now to the capset problem. Since Lemma 1 is valid for any field , one could perhaps hope to get better bounds by viewing the Kronecker delta function as taking values in another field than , such as the complex numbers . However, as soon as one works in a field of characteristic other than , one can adjoin a cube root of unity, and one now has the Fourier decomposition
Moving to the Fourier basis, we conclude from Lemma 1 that the function on now has rank exactly , and so one cannot improve upon the trivial bound of by this method using fields of characteristic other than three as the range field. So it seems one has to stick with (or the algebraic completion thereof).
Thanks to Jordan Ellenberg and Ben Green for helpful discussions.
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).
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.
When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology. However, there are some nice things one can do in an electronic medium, such as this blog. Here, I would like to experiment with the ability to animate images, which I think can convey some mathematical concepts in ways that cannot be easily replicated by traditional static text and images. Given that many readers may find these animations annoying, I am placing the rest of the post below the fold.
Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about these modular forms. This of course includes the most famous examples of L-functions and modular forms respectively, namely the Riemann zeta function and the discriminant modular form , but there are countless other examples of both. The connections between these classes of objects lie at the heart of the Langlands programme.
As of today, the “L-functions and modular forms database” is now out of beta, and open to the public; at present the database is mostly geared towards specialists in computational number theory, but will hopefully develop into a more broadly useful resource as time develops. An article by John Cremona summarising the purpose of the database can be found here.
In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number of useful topologies on them, including the operator norm topology, strong operator topology, and the weak operator topology. For function spaces, one can use topologies associated to various modes of convergence, such as uniform convergence, pointwise convergence, locally uniform convergence, or convergence in the sense of distributions. (A small minority of such modes are not topologisable, though, the most common of which is pointwise almost everywhere convergence; see Exercise 8 of this previous post).
Some of these topologies are much stronger than others (in that they contain many more open sets, or equivalently that they have many fewer convergent sequences and nets). However, even the weakest topologies used in analysis (e.g. convergence in distributions) tend to be Hausdorff, since this at least ensures the uniqueness of limits of sequences and nets, which is a fundamentally useful feature for analysis. On the other hand, some Hausdorff topologies used are “better” than others in that many more analysis tools are available for those topologies. In particular, topologies that come from Banach space norms are particularly valued, as such topologies (and their attendant norm and metric structures) grant access to many convenient additional results such as the Baire category theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.
Of course, most topologies placed on a vector space will not come from Banach space norms. For instance, if one takes the space of continuous functions on that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm ), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.
I recently realised (while teaching a graduate class in real analysis) that the closed graph theorem provides a quick explanation for why Banach space topologies are so rare:
Proposition 1 Let be a Hausdorff topological vector space. Then, up to equivalence of norms, there is at most one norm one can place on so that is a Banach space whose topology is at least as strong as . In particular, there is at most one topology stronger than that comes from a Banach space norm.
Proof: Suppose one had two norms on such that and were both Banach spaces with topologies stronger than . Now consider the graph of the identity function from the Banach space to the Banach space . This graph is closed; indeed, if is a sequence in this graph that converged in the product topology to , then converges to in norm and hence in , and similarly converges to in norm and hence in . But limits are unique in the Hausdorff topology , so . Applying the closed graph theorem (see also previous discussions on this theorem), we see that the identity map is continuous from to ; similarly for the inverse. Thus the norms are equivalent as claimed.
By using various generalisations of the closed graph theorem, one can generalise the above proposition to Fréchet spaces, or even to F-spaces. The proposition can fail if one drops the requirement that the norms be stronger than a specified Hausdorff topology; indeed, if is infinite dimensional, one can use a Hamel basis of to construct a linear bijection on that is unbounded with respect to a given Banach space norm , and which can then be used to give an inequivalent Banach space structure on .
One can interpret Proposition 1 as follows: once one equips a vector space with some “weak” (but still Hausdorff) topology, there is a canonical choice of “strong” topology one can place on that space that is stronger than the “weak” topology but arises from a Banach space structure (or at least a Fréchet or F-space structure), provided that at least one such structure exists. In the case of function spaces, one can usually use the topology of convergence in distribution as the “weak” Hausdorff topology for this purpose, since this topology is weaker than almost all of the other topologies used in analysis. This helps justify the common practice of describing a Banach or Fréchet function space just by giving the set of functions that belong to that space (e.g. is the space of Schwartz functions on ) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on given by the usual Schwartz space seminorms).
Of course, there are still some topological vector spaces which have no “strong topology” arising from a Banach space at all. Consider for instance the space of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm stronger than this topology that makes this space a Banach space. For, if there were, then letting be the standard basis of , the series would have to converge in , and hence pointwise, to an element of , but the only available pointwise limit for this series lies outside of . But I do not know if there is an easily checkable criterion to test whether a given vector space (equipped with a Hausdorff “weak” toplogy) can be equipped with a stronger Banach space (or Fréchet space or -space) topology.
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.
There is a very nice recent paper by Lemke Oliver and Soundararajan (complete with a popular science article about it by the consistently excellent Erica Klarreich for Quanta) about a surprising (but now satisfactorily explained) bias in the distribution of pairs of consecutive primes when reduced to a small modulus .
This phenomenon is superficially similar to the more well known Chebyshev bias concerning the reduction of a single prime to a small modulus , but is in fact a rather different (and much stronger) bias than the Chebyshev bias, and seems to arise from a completely different source. The Chebyshev bias asserts, roughly speaking, that a randomly selected prime of a large magnitude will typically (though not always) be slightly more likely to be a quadratic non-residue modulo than a quadratic residue, but the bias is small (the difference in probabilities is only about for typical choices of ), and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function modulo with the zeroes of the -functions with period . This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function is quite unbiased modulo . The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo . (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias. (See this article of Rubinstein and Sarnak for a more technical discussion of the Chebyshev bias, and this survey of Granville and Martin for an accessible introduction. The story of the Chebyshev bias is also related to Skewes’ number, once considered the largest explicit constant to naturally appear in a mathematical argument.)
The paper of Lemke Oliver and Soundararajan considers instead the distribution of the pairs for small and for large consecutive primes , say drawn at random from the primes comparable to some large . For sake of discussion let us just take . Then all primes larger than are either or ; Chebyshev’s bias gives a very slight preference to the latter (of order , as discussed above), but apart from this, we expect the primes to be more or less equally distributed in both classes. For instance, assuming GRH, the probability that lands in would be , and similarly for .
In view of this, one would expect that up to errors of or so, the pair should be equally distributed amongst the four options , , , , thus for instance the probability that this pair is would naively be expected to be , and similarly for the other three tuples. These assertions are not yet proven (although some non-trivial upper and lower bounds for such probabilities can be obtained from recent work of Maynard).
However, Lemke Oliver and Soundararajan argue (backed by both plausible heuristic arguments (based ultimately on the Hardy-Littlewood prime tuples conjecture), as well as substantial numerical evidence) that there is a significant bias away from the tuples and – informally, adjacent primes don’t like being in the same residue class! For instance, they predict that the probability of attaining is in fact
with similar predictions for the other three pairs (in fact they give a somewhat more precise prediction than this). The magnitude of this bias, being comparable to , is significantly stronger than the Chebyshev bias of .
One consequence of this prediction is that the prime gaps are slightly less likely to be divisible by than naive random models of the primes would predict. Indeed, if the four options , , , all occurred with equal probability , then should equal with probability , and and with probability each (as would be the case when taking the difference of two random numbers drawn from those integers not divisible by ); but the Lemke Oliver-Soundararajan bias predicts that the probability of being divisible by three should be slightly lower, being approximately .
Below the fold we will give a somewhat informal justification of (a simplified version of) this phenomenon, based on the Lemke Oliver-Soundararajan calculation using the prime tuples conjecture.
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.