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How many groups of order four are there? Technically, there are an enormous number, so much so, in fact, that the class of groups of order four is not even a set, but merely a proper class. This is because any four objects can be turned into a group by designating one of the four objects, say , to be the group identity, and imposing a suitable multiplication table (and inversion law) on the four elements in a manner that obeys the usual group axioms. Since all sets are themselves objects, the class of four-element groups is thus at least as large as the class of all sets, which by Russell’s paradox is known not to itself be a set (assuming the usual ZFC axioms of set theory).
A much better question is to ask how many groups of order four there are up to isomorphism, counting each isomorphism class of groups exactly once. Now, as one learns in undergraduate group theory classes, the answer is just “two”: the cyclic group and the Klein four-group .
More generally, given a class of objects and some equivalence relation on (which one should interpret as describing the property of two objects in being “isomorphic”), one can consider the number of objects in “up to isomorphism”, which is simply the cardinality of the collection of equivalence classes of . In the case where is finite, one can express this cardinality by the formula
thus one counts elements in , weighted by the reciprocal of the number of objects they are isomorphic to.
Example 1 Let be the five-element set of integers between and . Let us say that two elements of are isomorphic if they have the same magnitude: . Then the quotient space consists of just three equivalence classes: , , and . Thus there are three objects in up to isomorphism, and the identity (1) is then just
Thus, to count elements in up to equivalence, the elements are given a weight of because they are each isomorphic to two elements in , while the element is given a weight of because it is isomorphic to just one element in (namely, itself).
Given a finite probability set , there is also a natural probability distribution on , namely the uniform distribution, according to which a random variable is set equal to any given element of with probability :
Given a notion of isomorphism on , one can then define the random equivalence class that the random element belongs to. But if the isomorphism classes are unequal in size, we now encounter a biasing effect: even if was drawn uniformly from , the equivalence class need not be uniformly distributed in . For instance, in the above example, if was drawn uniformly from , then the equivalence class will not be uniformly distributed in the three-element space , because it has a probability of being equal to the class or to the class , and only a probability of being equal to the class .
However, it is possible to remove this bias by changing the weighting in (1), and thus changing the notion of what cardinality means. To do this, we generalise the previous situation. Instead of considering sets with an equivalence relation to capture the notion of isomorphism, we instead consider groupoids, which are sets in which there are allowed to be multiple isomorphisms between elements in (and in particular, there are allowed to be multiple automorphisms from an element to itself). More precisely:
Definition 2 A groupoid is a set (or proper class) , together with a (possibly empty) collection of “isomorphisms” between any pair of elements of , and a composition map from isomorphisms , to isomorphisms in for every , obeying the following group-like axioms:
- (Identity) For every , there is an identity isomorphism , such that for all and .
- (Associativity) If , , and for some , then .
- (Inverse) If for some , then there exists an inverse isomorphism such that and .
We say that two elements of a groupoid are isomorphic, and write , if there is at least one isomorphism from to .
Example 3 Any category gives a groupoid by taking to be the set (or class) of objects, and to be the collection of invertible morphisms from to . For instance, in the category of sets, would be the collection of bijections from to ; in the category of linear vector spaces over some given base field , would be the collection of invertible linear transformations from to ; and so forth.
Every set equipped with an equivalence relation can be turned into a groupoid by assigning precisely one isomorphism from to for any pair with , and no isomorphisms from to when , with the groupoid operations of identity, composition, and inverse defined in the only way possible consistent with the axioms. We will call this the simply connected groupoid associated with this equivalence relation. For instance, with as above, if we turn into a simply connected groupoid, there will be precisely one isomorphism from to , and also precisely one isomorphism from to , but no isomorphisms from to , , or .
However, one can also form multiply-connected groupoids in which there can be multiple isomorphisms from one element of to another. For instance, one can view as a space that is acted on by multiplication by the two-element group . This gives rise to two types of isomorphisms, an identity isomorphism from to for each , and a negation isomorphism from to for each ; in particular, there are two automorphisms of (i.e., isomorphisms from to itself), namely and , whereas the other four elements of only have a single automorphism (the identity isomorphism). One defines composition, identity, and inverse in this groupoid in the obvious fashion (using the group law of the two-element group ); for instance, we have .
For a finite multiply-connected groupoid, it turns out that the natural notion of “cardinality” (or as I prefer to call it, “cardinality up to isomorphism”) is given by the variant
of (1). That is to say, in the multiply connected case, the denominator is no longer the number of objects isomorphic to , but rather the number of isomorphisms from to other objects . Grouping together all summands coming from a single equivalence class in , we can also write this expression as
where is the automorphism group of , that is to say the group of isomorphisms from to itself. (Note that if belong to the same equivalence class , then the two groups and will be isomorphic and thus have the same cardinality, and so the above expression is well-defined.
For instance, if we take to be the simply connected groupoid on , then the number of elements of up to isomorphism is
exactly as before. If however we take the multiply connected groupoid on , in which has two automorphisms, the number of elements of up to isomorphism is now the smaller quantity
the equivalence class is now counted with weight rather than due to the two automorphisms on . Geometrically, one can think of this groupoid as being formed by taking the five-element set , and “folding it in half” around the fixed point , giving rise to two “full” quotient points and one “half” point . More generally, given a finite group acting on a finite set , and forming the associated multiply connected groupoid, the cardinality up to isomorphism of this groupoid will be , since each element of will have isomorphisms on it (whether they be to the same element , or to other elements of ).
The definition (2) can also make sense for some infinite groupoids; to my knowledge this was first explicitly done in this paper of Baez and Dolan. Consider for instance the category of finite sets, with isomorphisms given by bijections as in Example 3. Every finite set is isomorphic to for some natural number , so the equivalence classes of may be indexed by the natural numbers. The automorphism group of has order , so the cardinality of up to isomorphism is
(This fact is sometimes loosely stated as “the number of finite sets is “, but I view this statement as somewhat misleading if the qualifier “up to isomorphism” is not added.) Similarly, when one allows for multiple isomorphisms from a group to itself, the number of groups of order four up to isomorphism is now
because the cyclic group has two automorphisms, whereas the Klein four-group has six.
In the case that the cardinality of a groupoid up to isomorphism is finite and non-empty, one can now define the notion of a random isomorphism class in drawn “uniformly up to isomorphism”, by requiring the probability of attaining any given isomorphism class to be
thus the probability of being isomorphic to a given element will be inversely proportional to the number of automorphisms that has. For instance, if we take to be the set with the simply connected groupoid, will be drawn uniformly from the three available equivalence classes , with a probability of attaining each; but if instead one uses the multiply connected groupoid coming from the action of , and draws uniformly up to isomorphism, then and will now be selected with probability each, and will be selected with probability . Thus this distribution has accounted for the bias mentioned previously: if a finite group acts on a finite space , and is drawn uniformly from , then now still be drawn uniformly up to isomorphism from , if we use the multiply connected groupoid coming from the action, rather than the simply connected groupoid coming from just the -orbit structure on .
Using the groupoid of finite sets, we see that a finite set chosen uniformly up to isomorphism will have a cardinality that is distributed according to the Poisson distribution of parameter , that is to say it will be of cardinality with probability .
One important source of groupoids are the fundamental groupoids of a manifold (one can also consider more general topological spaces than manifolds, but for simplicity we will restrict this discussion to the manifold case), in which the underlying space is simply , and the isomorphisms from to are the equivalence classes of paths from to up to homotopy; in particular, the automorphism group of any given point is just the fundamental group of at that base point. The equivalence class of a point in is then the connected component of in . The cardinality up to isomorphism of the fundamental groupoid is then
where is the collection of connected components of , and is the order of the fundamental group of . Thus, simply connected components of count for a full unit of cardinality, whereas multiply connected components (which can be viewed as quotients of their simply connected cover by their fundamental group) will count for a fractional unit of cardinality, inversely to the order of their fundamental group.
This notion of cardinality up to isomorphism of a groupoid behaves well with respect to various basic notions. For instance, suppose one has an -fold covering map of one finite groupoid by another . This means that is a functor that is surjective, with all preimages of cardinality , with the property that given any pair in the base space and any in the preimage of , every isomorphism has a unique lift from the given initial point (and some in the preimage of ). Then one can check that the cardinality up to isomorphism of is times the cardinality up to isomorphism of , which fits well with the geometric picture of as the -fold cover of . (For instance, if one covers a manifold with finite fundamental group by its universal cover, this is a -fold cover, the base has cardinality up to isomorphism, and the universal cover has cardinality one up to isomorphism.) Related to this, if one draws an equivalence class of uniformly up to isomorphism, then will be an equivalence class of drawn uniformly up to isomorphism also.
Indeed, one can show that this notion of cardinality up to isomorphism for groupoids is uniquely determined by a small number of axioms such as these (similar to the axioms that determine Euler characteristic); see this blog post of Qiaochu Yuan for details.
The probability distributions on isomorphism classes described by the above recipe seem to arise naturally in many applications. For instance, if one draws a profinite abelian group up to isomorphism at random in this fashion (so that each isomorphism class of a profinite abelian group occurs with probability inversely proportional to the number of automorphisms of this group), then the resulting distribution is known as the Cohen-Lenstra distribution, and seems to emerge as the natural asymptotic distribution of many randomly generated profinite abelian groups in number theory and combinatorics, such as the class groups of random quadratic fields; see this previous blog post for more discussion. For a simple combinatorial example, the set of fixed points of a random permutation on elements will have a cardinality that converges in distribution to the Poisson distribution of rate (as discussed in this previous post), thus we see that the fixed points of a large random permutation asymptotically are distributed uniformly up to isomorphism. I’ve been told that this notion of cardinality up to isomorphism is also particularly compatible with stacks (which are a good framework to describe such objects as moduli spaces of algebraic varieties up to isomorphism), though I am not sufficiently acquainted with this theory to say much more than this.
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 “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest than I had hoped, but they did at least seem to indicate a potential strategy to make further progress on the problem, and also highlight some of the difficulties of the problem.
One can rephrase the lonely runner conjecture as the following covering problem. Given any integer “velocity” and radius , define the Bohr set to be the subset of the unit circle given by the formula
where denotes the distance of to the nearest integer. Thus, for positive, is simply the union of the intervals for , projected onto the unit circle ; in the language of the usual formulation of the lonely runner conjecture, represents those times in which a runner moving at speed returns to within of his or her starting position. For any non-zero integers , let be the smallest radius such that the Bohr sets cover the unit circle:
Then define to be the smallest value of , as ranges over tuples of distinct non-zero integers. The Dirichlet approximation theorem quickly gives that
for any . The lonely runner conjecture is equivalent to the assertion that this bound is in fact optimal:
Conjecture 1 (Lonely runner conjecture) For any , one has .
This conjecture is currently known for (see this paper of Barajas and Serra), but remains open for higher .
It is natural to try to attack the problem by establishing lower bounds on the quantity . We have the following “trivial” bound, that gets within a factor of two of the conjecture:
Proposition 2 (Trivial bound) For any , one has .
we see that the covering (1) is only possible if , giving the claim.
So, in some sense, all the difficulty is coming from the need to improve upon the trivial union bound (2) by a factor of two.
which was improved a little by Chen and Cusick in 1999 to
when was prime. In a recent paper of Perarnau and Serra, the bound
was obtained for arbitrary . These bounds only improve upon the trivial bound by a multiplicative factor of . Heuristically, one reason for this is as follows. The union bound (2) would of course be sharp if the Bohr sets were all disjoint. Strictly speaking, such disjointness is not possible, because all the Bohr sets have to contain the origin as an interior point. However, it is possible to come up with a large number of Bohr sets which are almost disjoint. For instance, suppose that we had velocities that were all prime numbers between and , and that was equal to (and in particular was between and . Then each set can be split into a “kernel” interval , together with the “petal” intervals . Roughly speaking, as the prime varies, the kernel interval stays more or less fixed, but the petal intervals range over disjoint sets, and from this it is not difficult to show that
so that the union bound is within a multiplicative factor of of the truth in this case.
This does not imply that is within a multiplicative factor of of , though, because there are not enough primes between and to assign to distinct velocities; indeed, by the prime number theorem, there are only about such velocities that could be assigned to a prime. So, while the union bound could be close to tight for up to Bohr sets, the above counterexamples don’t exclude improvements to the union bound for larger collections of Bohr sets. Following this train of thought, I was able to obtain a logarithmic improvement to previous lower bounds:
Theorem 3 For sufficiently large , one has for some absolute constant .
The factors of in the denominator are for technical reasons and might perhaps be removable by a more careful argument. However it seems difficult to adapt the methods to improve the in the numerator, basically because of the obstruction provided by the near-counterexample discussed above.
Roughly speaking, the idea of the proof of this theorem is as follows. If we have the covering (1) for very close to , then the multiplicity function will then be mostly equal to , but occasionally be larger than . On the other hand, one can compute that the norm of this multiplicity function is significantly larger than (in fact it is at least ). Because of this, the norm must be very large, which means that the triple intersections must be quite large for many triples . Using some basic Fourier analysis and additive combinatorics, one can deduce from this that the velocities must have a large structured component, in the sense that there exists an arithmetic progression of length that contains of these velocities. For simplicity let us take the arithmetic progression to be , thus of the velocities lie in . In particular, from the prime number theorem, most of these velocities will not be prime, and will in fact likely have a “medium-sized” prime factor (in the precise form of the argument, “medium-sized” is defined to be “between and “). Using these medium-sized prime factors, one can show that many of the will have quite a large overlap with many of the other , and this can be used after some elementary arguments to obtain a more noticeable improvement on the union bound (2) than was obtained previously.
In my previous blog post, I showed that in order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that all of the velocities are of size ; I reproduce this argument (slightly cleaned up for publication) in the current preprint. There is unfortunately a huge gap between and , so the above bound (3) does not immediately give any new bounds for . However, one could perhaps try to start attacking the lonely runner conjecture by increasing the range for which one has good results, and by decreasing the range that one can reduce to. For instance, in the current preprint I give an elementary argument (using a certain amount of case-checking) that shows that the lonely runner bound
holds if all the velocities are assumed to lie between and . This upper threshold of is only a tiny improvement over the trivial threshold of , but it seems to be an interesting sub-problem of the lonely runner conjecture to increase this threshold further. One key target would be to get up to , as there are actually a number of -tuples in this range for which (4) holds with equality. The Dirichlet approximation theorem of course gives the tuple , but there is also the double of this tuple, and furthermore there is an additional construction of Goddyn and Wong that gives some further examples such as , or more generally one can start with the standard tuple and accelerate one of the velocities to ; this turns out to work as long as shares a common factor with every integer between and . There are a few more examples of this type in the paper of Goddyn and Wong, but all of them can be placed in an arithmetic progression of length at most, so if one were very optimistic, one could perhaps envision a strategy in which the upper bound of mentioned earlier was reduced all the way to something like , and then a separate argument deployed to treat this remaining case, perhaps isolating the constructions of Goddyn and Wong (and possible variants thereof) as the only extreme cases.
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.
[This blog post was written jointly by Terry Tao and Will Sawin.]
In the previous blog post, one of us (Terry) implicitly introduced a notion of rank for tensors which is a little different from the usual notion of tensor rank, and which (following BCCGNSU) we will call “slice rank”. This notion of rank could then be used to encode the Croot-Lev-Pach-Ellenberg-Gijswijt argument that uses the polynomial method to control capsets.
Afterwards, several papers have applied the slice rank method to further problems – to control tri-colored sum-free sets in abelian groups (BCCGNSU, KSS) and from there to the triangle removal lemma in vector spaces over finite fields (FL), to control sunflowers (NS), and to bound progression-free sets in -groups (P).
In this post we investigate the notion of slice rank more systematically. In particular, we show how to give lower bounds for the slice rank. In many cases, we can show that the upper bounds on slice rank given in the aforementioned papers are sharp to within a subexponential factor. This still leaves open the possibility of getting a better bound for the original combinatorial problem using the slice rank of some other tensor, but for very long arithmetic progressions (at least eight terms), we show that the slice rank method cannot improve over the trivial bound using any tensor.
It will be convenient to work in a “basis independent” formalism, namely working in the category of abstract finite-dimensional vector spaces over a fixed field . (In the applications to the capset problem one takes to be the finite field of three elements, but most of the discussion here applies to arbitrary fields.) Given such vector spaces , we can form the tensor product , generated by the tensor products with for , subject to the constraint that the tensor product operation is multilinear. For each , we have the smaller tensor products , as well as the tensor product
defined in the obvious fashion. Elements of of the form for some and will be called rank one functions, and the slice rank (or rank for short) of an element of is defined to be the least nonnegative integer such that is a linear combination of rank one functions. If are finite-dimensional, then the rank is always well defined as a non-negative integer (in fact it cannot exceed . It is also clearly subadditive:
For , is when is zero, and otherwise. For , is the usual rank of the -tensor (which can for instance be identified with a linear map from to the dual space ). The usual notion of tensor rank for higher order tensors uses complete tensor products , as the rank one objects, rather than , giving a rank that is greater than or equal to the slice rank studied here.
From basic linear algebra we have the following equivalences:
- (i) One has .
- (ii) One has a representation of the form
where are finite sets of total cardinality at most , and for each and , and .
- (iii) One has
where for each , is a subspace of of total dimension at most , and we view as a subspace of in the obvious fashion.
- (iv) (Dual formulation) There exist subspaces of the dual space for , of total dimension at least , such that is orthogonal to , in the sense that one has the vanishing
for all , where is the obvious pairing.
Proof: The equivalence of (i) and (ii) is clear from definition. To get from (ii) to (iii) one simply takes to be the span of the , and conversely to get from (iii) to (ii) one takes the to be a basis of the and computes by using a basis for the tensor product consisting entirely of functions of the form for various . To pass from (iii) to (iv) one takes to be the annihilator of , and conversely to pass from (iv) to (iii).
One corollary of the formulation (iv), is that the set of tensors of slice rank at most is Zariski closed (if the field is algebraically closed), and so the slice rank itself is a lower semi-continuous function. This is in contrast to the usual tensor rank, which is not necessarily semicontinuous.
Corollary 2 Let be finite-dimensional vector spaces over an algebraically closed field . Let be a nonnegative integer. The set of elements of of slice rank at most is closed in the Zariski topology.
Proof: In view of Lemma 1(i and iv), this set is the union over tuples of integers with of the projection from of the set of tuples with orthogonal to , where is the Grassmanian parameterizing -dimensional subspaces of .
One can check directly that the set of tuples with orthogonal to is Zariski closed in using a set of equations of the form locally on . Hence because the Grassmanian is a complete variety, the projection of this set to is also Zariski closed. So the finite union over tuples of these projections is also Zariski closed.
We also have good behaviour with respect to linear transformations:
Furthermore, if the are all injective, then one has equality in (2).
Thus, for instance, the rank of a tensor is intrinsic in the sense that it is unaffected by any enlargements of the spaces .
Computing the rank of a tensor is difficult in general; however, the problem becomes a combinatorial one if one has a suitably sparse representation of that tensor in some basis, where we will measure sparsity by the property of being an antichain.
Now suppose that the coefficients are all non-zero, that each of the are equipped with a total ordering , and is the set of maximal elements of , thus there do not exist distinct , such that for all . Then one has
In particular, if is an antichain (i.e. every element is maximal), then equality holds in (4).
Proof: By Lemma 3 (or by enlarging the bases ), we may assume without loss of generality that each of the is spanned by the . By relabeling, we can also assume that each is of the form
with the usual ordering, and by Lemma 3 we may take each to be , with the standard basis.
Let denote the rank of . To show (4), it suffices to show the inequality
can (after collecting terms) be written as
holds for some covering . By Lemma 1(iv), there exist subspaces of whose dimension sums to
Let . Using Gaussian elimination, one can find a basis of whose representation in the standard dual basis of is in row-echelon form. That is to say, there exist natural numbers
such that for all , is a linear combination of the dual vectors , with the coefficient equal to one.
We now claim that is disjoint from . Suppose for contradiction that this were not the case, thus there exists for each such that
As is the set of maximal elements of , this implies that
for any tuple other than . On the other hand, we know that is a linear combination of , with the coefficient one. We conclude that the tensor product is equal to
plus a linear combination of other tensor products with not in . Taking inner products with (3), we conclude that , contradicting the fact that is orthogonal to . Thus we have disjoint from .
As an instance of this proposition, we recover the computation of diagonal rank from the previous blog post:
Example 5 Let be finite-dimensional vector spaces over a field for some . Let be a natural number, and for , let be a linearly independent set in . Let be non-zero coefficients in . Then
has rank . Indeed, one applies the proposition with all equal to , with the diagonal in ; this is an antichain if we give one of the the standard ordering, and another of the the opposite ordering (and ordering the remaining arbitrarily). In this case, the are all bijective, and so it is clear that the minimum in (4) is simply .
The combinatorial minimisation problem in the above proposition can be solved asymptotically when working with tensor powers, using the notion of the Shannon entropy of a discrete random variable .
Let be a tensor of the form (3) for some coefficients . For each natural number , let be the tensor power of copies of , viewed as an element of . Then
Now suppose that the coefficients are all non-zero and that each of the are equipped with a total ordering . Let be the set of maximal elements of in the product ordering, and let where range over random variables taking values in . Then
as , where is the projection map. Then the same thing will apply to and . Then applying Proposition 4, using the lexicographical ordering on and noting that, if are the maximal elements of , then are the maximal elements of , we obtain both (9) and (11).
Let be a small positive quantity that goes to zero sufficiently slowly with . Let denote the set of all tuples in that are within of being distributed according to the law of , in the sense that for all , one has
By the asymptotic equipartition property, the cardinality of can be computed to be
which by (13) implies that
noting that the factor can be absorbed into the error). This gives the lower bound in (12).
Now we prove the upper bound. We can cover by sets of the form for various choices of random variables taking values in . For each such random variable , we can find such that ; we then place all of in . It is then clear that the cover and that
for all , giving the required upper bound.
It is of interest to compute the quantity in (10). We have the following criterion for when a maximiser occurs:
Proposition 7 Let be finite sets, and be non-empty. Let be the quantity in (10). Let be a random variable taking values in , and let denote the essential range of , that is to say the set of tuples such that is non-zero. Then the following are equivalent:
- (i) attains the maximum in (10).
- (ii) There exist weights and a finite quantity , such that whenever , and such that
for all , with equality if . (In particular, must vanish if there exists a with .)
Proof: We first show that (i) implies (ii). The function is concave on . As a consequence, if we define to be the set of tuples such that there exists a random variable taking values in with , then is convex. On the other hand, by (10), is disjoint from the orthant . Thus, by the hyperplane separation theorem, we conclude that there exists a half-space
where are reals that are not all zero, and is another real, which contains on its boundary and in its interior, such that avoids the interior of the half-space. Since is also on the boundary of , we see that the are non-negative, and that whenever .
By construction, the quantity
is maximised when . At this point we could use the method of Lagrange multipliers to obtain the required constraints, but because we have some boundary conditions on the (namely, that the probability that they attain a given element of has to be non-negative) we will work things out by hand. Let be an element of , and an element of . For small enough, we can form a random variable taking values in , whose probability distribution is the same as that for except that the probability of attaining is increased by , and the probability of attaining is decreased by . If there is any for which and , then one can check that
for sufficiently small , contradicting the maximality of ; thus we have whenever . Taylor expansion then gives
for small , where
and similarly for . We conclude that for all and , thus there exists a quantity such that for all , and for all . By construction must be nonnegative. Sampling using the distribution of , one has
almost surely; taking expectations we conclude that
The inner sum is , which equals when is non-zero, giving (17).
for any (note the right-hand side may be infinite when and ). Let be any random variable taking values in , then on applying the above inequality with and , multiplying by , and summing over and gives
By construction, one has
so to prove that (which would give (i)), it suffices to show that
or equivalently that the quantity
is maximised when . Since
it suffices to show this claim for the quantity
One can view this quantity as
By (ii), this quantity is bounded by , with equality if is equal to (and is in particular ranging in ), giving the claim.
The second half of the proof of Proposition 7 only uses the marginal distributions and the equation(16), not the actual distribution of , so it can also be used to prove an upper bound on when the exact maximizing distribution is not known, given suitable probability distributions in each variable. The logarithm of the probability distribution here plays the role that the weight functions do in BCCGNSU.
Remark 8 Suppose one is in the situation of (i) and (ii) above; assume the nondegeneracy condition that is positive (or equivalently that is positive). We can assign a “degree” to each element by the formula
then every tuple in has total degree at most , and those tuples in have degree exactly . In particular, every tuple in has degree at most , and hence by (17), each such tuple has a -component of degree less than or equal to for some with . On the other hand, we can compute from (19) and the fact that for that . Thus, by asymptotic equipartition, and assuming , the number of “monomials” in of total degree at most is at most ; one can in fact use (19) and (18) to show that this is in fact an equality. This gives a direct way to cover by sets with , which is in the spirit of the Croot-Lev-Pach-Ellenberg-Gijswijt arguments from the previous post.
We can now show that the rank computation for the capset problem is sharp:
Proof: In , we have
Thus, if we let be the space of functions from to (with domain variable denoted respectively), and define the basis functions
of indexed by (with the usual ordering), respectively, and set to be the set
then is a linear combination of the with , and all coefficients non-zero. Then we have . We will show that the quantity of (10) agrees with the quantity of (20), and that the optimizing distribution is supported on , so that by Proposition 6 the rank of is .
To compute the quantity at (10), we use the criterion in Proposition 7. We take to be the random variable taking values in that attains each of the values with a probability of , and each of with a probability of ; then each of the attains the values of with probabilities respectively, so in particular is equal to the quantity in (20). If we now set and
This statement already follows from the result of Kleinberg-Sawin-Speyer, which gives a “tri-colored sum-free set” in of size , as the slice rank of this tensor is an upper bound for the size of a tri-colored sum-free set. If one were to go over the proofs more carefully to evaluate the subexponential factors, this argument would give a stronger lower bound than KSS, as it does not deal with the substantial loss that comes from Behrend’s construction. However, because it actually constructs a set, the KSS result rules out more possible approaches to give an exponential improvement of the upper bound for capsets. The lower bound on slice rank shows that the bound cannot be improved using only the slice rank of this particular tensor, whereas KSS shows that the bound cannot be improved using any method that does not take advantage of the “single-colored” nature of the problem.
We can also show that the slice rank upper bound in a result of Naslund-Sawin is similarly sharp:
Proposition 10 Let denote the space of functions from to . Then the function from , viewed as an element of , has slice rank
Proof: Let and be a basis for the space of functions on , itself indexed by . Choose similar bases for and , with and .
Set . Then is a linear combination of the with , and all coefficients non-zero. Order the usual way so that is an antichain. We will show that the quantity of (10) is , so that applying the last statement of Proposition 6, we conclude that the rank of is ,
Let be the random variable taking values in that attains each of the values with a probability of . Then each of the attains the value with probability and with probability , so
We used a slightly different method in each of the last two results. In the first one, we use the most natural bases for all three vector spaces, and distinguish from its set of maximal elements . In the second one we modify one basis element slightly, with instead of the more obvious choice , which allows us to work with instead of . Because is an antichain, we do not need to distinguish and . Both methods in fact work with either problem, and they are both about equally difficult, but we include both as either might turn out to be substantially more convenient in future work.
Proposition 11 Let be a natural number and let be a finite abelian group. Let be any field. Let denote the space of functions from to .
Let be any -valued function on that is nonzero only when the elements of form a -term arithmetic progression, and is nonzero on every -term constant progression.
Then the slice rank of is .
Proof: We apply Proposition 4, using the standard bases of . Let be the support of . Suppose that we have orderings on such that the constant progressions are maximal elements of and thus all constant progressions lie in . Then for any partition of , can contain at most constant progressions, and as all constant progressions must lie in one of the , we must have . By Proposition 4, this implies that the slice rank of is at least . Since is a tensor, the slice rank is at most , hence exactly .
So it is sufficient to find orderings on such that the constant progressions are maximal element of . We make several simplifying reductions: We may as well assume that consists of all the -term arithmetic progressions, because if the constant progressions are maximal among the set of all progressions then they are maximal among its subset . So we are looking for an ordering in which the constant progressions are maximal among all -term arithmetic progressions. We may as well assume that is cyclic, because if for each cyclic group we have an ordering where constant progressions are maximal, on an arbitrary finite abelian group the lexicographic product of these orderings is an ordering for which the constant progressions are maximal. We may assume , as if we have an -tuple of orderings where constant progressions are maximal, we may add arbitrary orderings and the constant progressions will remain maximal.
So it is sufficient to find orderings on the cyclic group such that the constant progressions are maximal elements of the set of -term progressions in in the -fold product ordering. To do that, let the first, second, third, and fifth orderings be the usual order on and let the fourth, sixth, seventh, and eighth orderings be the reverse of the usual order on .
Then let be a constant progression and for contradiction assume that is a progression greater than in this ordering. We may assume that , because otherwise we may reverse the order of the progression, which has the effect of reversing all eight orderings, and then apply the transformation , which again reverses the eight orderings, bringing us back to the original problem but with .
Take a representative of the residue class in the interval . We will abuse notation and call this . Observe that , and are all contained in the interval modulo . Take a representative of the residue class in the interval . Then is in the interval for some . The distance between any distinct pair of intervals of this type is greater than , but the distance between and is at most , so is in the interval . By the same reasoning, is in the interval . Therefore . But then the distance between and is at most , so by the same reasoning is in the interval . Because is between and , it also lies in the interval . Because is in the interval , and by assumption it is congruent mod to a number in the set greater than or equal to , it must be exactly . Then, remembering that and lie in , we have and , so , hence , thus , which contradicts the assumption that .
In fact, given a -term progressions mod and a constant, we can form a -term binary sequence with a for each step of the progression that is greater than the constant and a for each step that is less. Because a rotation map, viewed as a dynamical system, has zero topological entropy, the number of -term binary sequences that appear grows subexponentially in . Hence there must be, for large enough , at least one sequence that does not appear. In this proof we exploit a sequence that does not appear for .
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.
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 two related papers to the arXiv:
- The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, submitted to Forum of Mathematics, Pi; and
- The Erdos discrepancy problem, submitted to the new arXiv overlay journal, Discrete Analysis (see this recent announcement on Tim Gowers’ blog).
This pair of papers is an outgrowth of these two recent blog posts and the ensuing discussion. In the first paper, we establish the following logarithmically averaged version of the Chowla conjecture (in the case of two-point correlations (or “pair correlations”)):
Theorem 1 (Logarithmically averaged Chowla conjecture) Let be natural numbers, and let be integers such that . Let be a quantity depending on that goes to infinity as . Let denote the Liouville function. Then one has
which is a strictly stronger estimate than (2), and remains open.
The arguments also extend to other completely multiplicative functions than the Liouville function. In particular, one obtains a slightly averaged version of the non-asymptotic Elliott conjecture that was shown in the previous blog post to imply a positive solution to the Erdos discrepancy problem. The averaged version of the conjecture established in this paper is slightly weaker than the one assumed in the previous blog post, but it turns out that the arguments there can be modified without much difficulty to accept this averaged Elliott conjecture as input. In particular, we obtain an unconditional solution to the Erdos discrepancy problem as a consequence; this is detailed in the second paper listed above. In fact we can also handle the vector-valued version of the Erdos discrepancy problem, in which the sequence takes values in the unit sphere of an arbitrary Hilbert space, rather than in .
Estimates such as (2) or (3) are known to be subject to the “parity problem” (discussed numerous times previously on this blog), which roughly speaking means that they cannot be proven solely using “linear” estimates on functions such as the von Mangoldt function. However, it is known that the parity problem can be circumvented using “bilinear” estimates, and this is basically what is done here.
We now describe in informal terms the proof of Theorem 1, focusing on the model case (2) for simplicity. Suppose for contradiction that the left-hand side of (2) was large and (say) positive. Using the multiplicativity , we conclude that
is also large and positive for all primes that are not too large; note here how the logarithmic averaging allows us to leave the constraint unchanged. Summing in , we conclude that
is large for many choices of , where is a medium-sized parameter at our disposal to choose, and we take to be some set of primes that are somewhat smaller than . (A similar approach was taken in this recent paper of Matomaki, Radziwill, and myself to study sign patterns of the Möbius function.) To obtain the required contradiction, one thus wants to demonstrate significant cancellation in the expression (4). As in that paper, we view as a random variable, in which case (4) is essentially a bilinear sum of the random sequence along a random graph on , in which two vertices are connected if they differ by a prime in that divides . A key difficulty in controlling this sum is that for randomly chosen , the sequence and the graph need not be independent. To get around this obstacle we introduce a new argument which we call the “entropy decrement argument” (in analogy with the “density increment argument” and “energy increment argument” that appear in the literature surrounding Szemerédi’s theorem on arithmetic progressions, and also reminiscent of the “entropy compression argument” of Moser and Tardos, discussed in this previous post). This argument, which is a simple consequence of the Shannon entropy inequalities, can be viewed as a quantitative version of the standard subadditivity argument that establishes the existence of Kolmogorov-Sinai entropy in topological dynamical systems; it allows one to select a scale parameter (in some suitable range ) for which the sequence and the graph exhibit some weak independence properties (or more precisely, the mutual information between the two random variables is small).
Informally, the entropy decrement argument goes like this: if the sequence has significant mutual information with , then the entropy of the sequence for will grow a little slower than linearly, due to the fact that the graph has zero entropy (knowledge of more or less completely determines the shifts of the graph); this can be formalised using the classical Shannon inequalities for entropy (and specifically, the non-negativity of conditional mutual information). But the entropy cannot drop below zero, so by increasing as necessary, at some point one must reach a metastable region (cf. the finite convergence principle discussed in this previous blog post), within which very little mutual information can be shared between the sequence and the graph . Curiously, for the application it is not enough to have a purely quantitative version of this argument; one needs a quantitative bound (which gains a factor of a bit more than on the trivial bound for mutual information), and this is surprisingly delicate (it ultimately comes down to the fact that the series diverges, which is only barely true).
Once one locates a scale with the low mutual information property, one can use standard concentration of measure results such as the Hoeffding inequality to approximate (4) by the significantly simpler expression
The important thing here is that Hoeffding’s inequality gives exponentially strong bounds on the failure probability, which is needed to counteract the logarithms that are inevitably present whenever trying to use entropy inequalities. The expression (5) can then be controlled in turn by an application of the Hardy-Littlewood circle method and a non-trivial estimate
for averaged short sums of a modulated Liouville function established in another recent paper by Matomäki, Radziwill and myself.
When one uses this method to study more general sums such as
one ends up having to consider expressions such as
where is the coefficient . When attacking this sum with the circle method, one soon finds oneself in the situation of wanting to locate the large Fourier coefficients of the exponential sum
In many cases (such as in the application to the Erdös discrepancy problem), the coefficient is identically , and one can understand this sum satisfactorily using the classical results of Vinogradov: basically, is large when lies in a “major arc” and is small when it lies in a “minor arc”. For more general functions , the coefficients are more or less arbitrary; the large values of are no longer confined to the major arc case. Fortunately, even in this general situation one can use a restriction theorem for the primes established some time ago by Ben Green and myself to show that there are still only a bounded number of possible locations (up to the uncertainty mandated by the Heisenberg uncertainty principle) where is large, and we can still conclude by using (6). (Actually, as recently pointed out to me by Ben, one does not need the full strength of our result; one only needs the restriction theorem for the primes, which can be proven fairly directly using Plancherel’s theorem and some sieve theory.)
It is tempting to also use the method to attack higher order cases of the (logarithmically) averaged Chowla conjecture, for instance one could try to prove the estimate
The above arguments reduce matters to obtaining some non-trivial cancellation for sums of the form
A little bit of “higher order Fourier analysis” (as was done for very similar sums in the ergodic theory context by Frantzikinakis-Host-Kra and Wooley-Ziegler) lets one control this sort of sum if one can establish a bound of the form
where goes to infinity and is a very slowly growing function of . This looks very similar to (6), but the fact that the supremum is now inside the integral makes the problem much more difficult. However it looks worth attacking (7) further, as this estimate looks like it should have many nice applications (beyond just the case of the logarithmically averaged Chowla or Elliott conjectures, which is already interesting).
For higher than , the same line of analysis requires one to replace the linear phase by more complicated phases, such as quadratic phases or even -step nilsequences. Given that (7) is already beyond the reach of current literature, these even more complicated expressions are also unavailable at present, but one can imagine that they will eventually become tractable, in which case we would obtain an averaged form of the Chowla conjecture for all , which would have a number of consequences (such as a logarithmically averaged version of Sarnak’s conjecture, as per this blog post).
It would of course be very nice to remove the logarithmic averaging, and be able to establish bounds such as (3). I did attempt to do so, but I do not see a way to use the entropy decrement argument in a manner that does not require some sort of averaging of logarithmic type, as it requires one to pick a scale that one cannot specify in advance, which is not a problem for logarithmic averages (which are quite stable with respect to dilations) but is problematic for ordinary averages. But perhaps the problem can be circumvented by some clever modification of the argument. One possible approach would be to start exploiting multiplicativity at products of primes, and not just individual primes, to try to keep the scale fixed, but this makes the concentration of measure part of the argument much more complicated as one loses some independence properties (coming from the Chinese remainder theorem) which allowed one to conclude just from the Hoeffding inequality.
This week I have been at a Banff workshop “Combinatorics meets Ergodic theory“, focused on the combinatorics surrounding Szemerédi’s theorem and the Gowers uniformity norms on one hand, and the ergodic theory surrounding Furstenberg’s multiple recurrence theorem and the Host-Kra structure theory on the other. This was quite a fruitful workshop, and directly inspired the various posts this week on this blog. Incidentally, BIRS being as efficient as it is, videos for this week’s talks are already online.
Then there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that
There is a higher dimensional generalisation, which first appeared explicitly (in a more general form) in this preprint of Szegedy (which used a slightly different argument than the one of Ben, Tammy, and myself; see also this previous preprint of Szegedy with related results):
The case of this theorem was recently used by Wenbo Sun. One can replace the polynomial sequence with a linear sequence if desired by using a lifting trick (essentially due to Furstenberg, but which appears explicitly in Appendix C of my paper with Ben and Tammy).
In this post I would like to record a very neat and simple observation of Ben Green and Nikos Frantzikinakis, that uses the tool of Freiman isomorphisms to derive Theorem 2 as a corollary of the one-dimensional theorem. Namely, consider the linear map defined by
that is to say is the digit string base that has digits . This map is a linear map from to a subset of of density . Furthermore it has the following “Freiman isomorphism” property: if lie in with in the image set of for all , then there exist (unique) lifts such that
for all . Indeed, the injectivity of on uniquely determines the sum for each , and one can use base arithmetic to verify that the alternating sum of these sums on any -facet of the cube vanishes, which gives the claim. (In the language of additive combinatorics, the point is that is a Freiman isomorphism of order (say) on .)
Now let be the function defined by setting whenever , with vanishing outside of . If obeys (1), then from the above Freiman isomorphism property we have
Applying the one-dimensional inverse theorem (Theorem 1), with reduced by a factor of and replaced by , this implies the existence of a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that
which by the Freiman isomorphism property again implies that
But the map is clearly a polynomial map from to (the composition of two polynomial maps is polynomial, see e.g. Appendix B of my paper with Ben and Tammy), and the claim follows.
Remark 3 This trick appears to be largely restricted to the case of boundedly generated groups such as ; I do not see any easy way to deduce an inverse theorem for, say, from the -inverse theorem by this method.
Remark 4 By combining this argument with the one in the previous post, one can obtain a weak ergodic inverse theorem for -actions. Interestingly, the Freiman isomorphism argument appears to be difficult to implement directly in the ergodic category; in particular, there does not appear to be an obvious direct way to derive the Host-Kra inverse theorem for actions (a result first obtained in the PhD thesis of Griesmer) from the counterpart for actions.