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Chantal David, Andrew Granville, Emmanuel Kowalski, Phillipe Michel, Kannan Soundararajan, and I are running a program at MSRI in the Spring of 2017 (more precisely, from Jan 17, 2017 to May 26, 2017) in the area of analytic number theory, with the intention to bringing together many of the leading experts in all aspects of the subject and to present recent work on the many active areas of the subject (the discussion on previous blog posts here have mostly focused on advances in the study of the distribution of the prime numbers, but there have been many other notable recent developments too, such as refinements of the circle method, a deeper understanding of the asymptotics of bounded multiplicative functions and of the “pretentious” approach to analytic number theory, more “analysis-friendly” formulations of the theorems of Deligne and others involving trace functions over fields, and new subconvexity theorems for automorphic forms, to name a few). Like any other semester MSRI program, there will be a number of workshops, seminars, and similar activities taking place while the members are in residence. I’m personally looking forward to the program, which should be occurring in the midst of a particularly productive time for the subject. Needless to say, I (and the rest of the organising committee) plan to be present for most of the program.
Applications for Postdoctoral Fellowships, Research Memberships, and Research Professorships for this program (and for other MSRI programs in this time period, namely the companion program in Harmonic Analysis and the Fall program in Geometric Group Theory, as well as the complementary program in all other areas of mathematics) have just opened up today. Applications are open to everyone (until they close on Dec 1), but require supporting documentation, such as a CV, statement of purpose, and letters of recommendation from other mathematicians; see the application page for more details.
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.
Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.
Then there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that
This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:
for all non-zero (all spaces are real-valued in this post). Then is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree nilsystems, that is to say systems of the form for some degree filtered nilmanifold and a group element that acts ergodically on .
It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of -actions, the connection is less clear.
One can split Theorem 2 into two components:
for all non-zero , where . Then is a factor of an inverse limit of ergodic degree nilsystems.
Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)
The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:
As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.
A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:
Then there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that
For the definitions of “filtered nilmanifold”, “degree”, “complexity”, and “polynomial sequence”, see the paper of Ben, Tammy, and myself. (I should caution the reader that this blog post will presume a fair amount of familiarity with this subfield of additive combinatorics.) This result has a number of applications, for instance to establishing asymptotics for linear equations in the primes, but this will not be the focus of discussion here.
The purpose of this post is to record the observation that this “discrete” inverse theorem, together with an equidistribution theorem for nilsequences that Ben and I worked out in a separate paper, implies a continuous version:
Theorem 2 (Continuous inverse theorem for Gowers norms) Let be an integer, and let . Suppose that is a measurable function supported on such that
Then there exists a filtered nilmanifold of degree and complexity , a (smooth) polynomial sequence , and a Lipschitz function of Lipschitz constant such that
The interval can be easily replaced with any other fixed interval by a change of variables. A key point here is that the bounds are completely uniform in the choice of . Note though that the coefficients of can be arbitrarily large (and this is necessary, as can be seen just by considering functions of the form for some arbitrarily large frequency ).
It is likely that one could prove Theorem 2 by carefully going through the proof of Theorem 1 and replacing all instances of with (and making appropriate modifications to the argument to accommodate this). However, the proof of Theorem 1 is quite lengthy. Here, we shall proceed by the usual limiting process of viewing the continuous interval as a limit of the discrete interval as . However there will be some problems taking the limit due to a failure of compactness, and specifically with regards to the coefficients of the polynomial sequence produced by Theorem 1, after normalising these coefficients by . Fortunately, a factorisation theorem from a paper of Ben Green and myself resolves this problem by splitting into a “smooth” part which does enjoy good compactness properties, as well as “totally equidistributed” and “periodic” parts which can be eliminated using the measurability (and thus, approximate smoothness), of .
Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:
for some depending only on .
The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases as they are trivial and somewhat degenerate.
There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.
In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel lists of facts (stated somewhat informally). Firstly, some facts about the prime factorisation of large integers:
- Every positive integer has a prime factorisation
into (not necessarily distinct) primes , which is unique up to rearrangement. Taking logarithms, we obtain a partition
- (Prime number theorem) A randomly selected integer of size will be prime with probability when is large.
- If is a randomly selected large integer of size , and is a randomly selected prime factor of (with each index being chosen with probability ), then is approximately uniformly distributed between and . (See Proposition 9 of this previous blog post.)
- The set of real numbers arising from the prime factorisation of a large random number converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit . (See the previously mentioned blog post for a definition of the Poisson-Dirichlet process, and a proof of this claim.)
Now for the facts about the cycle decomposition of large permutations:
- Every permutation has a cycle decomposition
into disjoint cycles , which is unique up to rearrangement, and where we count each fixed point of as a cycle of length . If is the length of the cycle , we obtain a partition
- (Prime number theorem for permutations) A randomly selected permutation of will be an -cycle with probability exactly . (This was noted in this previous blog post.)
- If is a random permutation in , and is a randomly selected cycle of (with each being selected with probability ), then is exactly uniformly distributed on . (See Proposition 8 of this blog post.)
- The set of real numbers arising from the cycle decomposition of a random permutation converges (in a suitable sense) to the Poisson-Dirichlet process in the limit . (Again, see this previous blog post for details.)
See this previous blog post (or the aforementioned article of Granville, or the Notices article of Arratia, Barbour, and Tavaré) for further exploration of the analogy between prime factorisation of integers and cycle decomposition of permutations.
There is however something unsatisfying about the analogy, in that it is not clear why there should be such a kinship between integer prime factorisation and permutation cycle decomposition. It turns out that the situation is clarified if one uses another fundamental analogy in number theory, namely the analogy between integers and polynomials over a finite field , discussed for instance in this previous post; this is the simplest case of the more general function field analogy between number fields and function fields. Just as we restrict attention to positive integers when talking about prime factorisation, it will be reasonable to restrict attention to monic polynomials . We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:
- Every monic polynomial has a factorisation
into irreducible monic polynomials , which is unique up to rearrangement. Taking degrees, we obtain a partition
- (Prime number theorem for polynomials) A randomly selected monic polynomial of degree will be irreducible with probability when is fixed and is large.
- If is a random monic polynomial of degree , and is a random irreducible factor of (with each selected with probability ), then is approximately uniformly distributed in when is fixed and is large.
- The set of real numbers arising from the factorisation of a randomly selected polynomial of degree converges (in a suitable sense) to the Poisson-Dirichlet process when is fixed and is large.
The above list of facts addressed the large limit of the polynomial ring , where the order of the field is held fixed, but the degrees of the polynomials go to infinity. This is the limit that is most closely analogous to the integers . However, there is another interesting asymptotic limit of polynomial rings to consider, namely the large limit where it is now the degree that is held fixed, but the order of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the characteristic of the field goes to infinity (again keeping the degree fixed), although all of the results proven below for the large limit turn out to be true as well in the large limit.
The large (or large ) limit is technically a different limit than the large limit, but in practice the asymptotic statistics of the two limits often agree quite closely. For instance, here is the prime number theorem in the large limit:
Proof: There are monic polynomials of degree . If is irreducible, then the zeroes of are distinct and lie in the finite field , but do not lie in any proper subfield of that field. Conversely, every element of that does not lie in a proper subfield is the root of a unique monic polynomial in of degree (the minimal polynomial of ). Since the union of all the proper subfields of has size , the total number of irreducible polynomials of degree is thus , and the claim follows.
Remark 2 The above argument and inclusion-exclusion in fact gives the well known exact formula for the number of irreducible monic polynomials of degree .
Now we can give a precise connection between the cycle distribution of a random permutation, and (the large limit of) the irreducible factorisation of a polynomial, giving a (somewhat indirect, but still connected) link between permutation cycle decomposition and integer factorisation:
Theorem 3 The partition of a random monic polynomial of degree converges in distribution to the partition of a random permutation of length , in the limit where is fixed and the characteristic goes to infinity.
We can quickly prove this theorem as follows. We first need a basic fact:
Lemma 4 (Most polynomials square-free in large limit) A random monic polynomial of degree will be square-free with probability when is fixed and (or ) goes to infinity. In a similar spirit, two randomly selected monic polynomials of degree will be coprime with probability if are fixed and or goes to infinity.
Proof: For any polynomial of degree , the probability that is divisible by is at most . Summing over all polynomials of degree , and using the union bound, we see that the probability that is not squarefree is at most , giving the first claim. For the second, observe from the first claim (and the fact that has only a bounded number of factors) that is squarefree with probability , giving the claim.
Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation consisting of cycles of length for , where are nonnegative integers with , is precisely
since there are ways to write a given tuple of cycles in cycle notation in nondecreasing order of length, and ways to select the labels for the cycle notation. On the other hand, by Theorem 1 (and using Lemma 4 to isolate the small number of cases involving repeated factors) the number of monic polynomials of degree that are the product of irreducible polynomials of degree is
which simplifies to
and the claim follows.
This was a fairly short calculation, but it still doesn’t quite explain why there is such a link between the cycle decomposition of permutations and the factorisation of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in with the multiplication structure of polynomials; however, these structures are too dissimilar to set up a convincing analogy. For instance, the multiplication law on polynomials is abelian and non-invertible, whilst the multiplication law on is (extremely) non-abelian but invertible. Also, the multiplication of a degree and a degree polynomial is a degree polynomial, whereas the group multiplication law on permutations does not take a permutation in and a permutation in and return a permutation in .
I recently found (after some discussions with Ben Green) what I feel to be a satisfying conceptual (as opposed to computational) explanation of this link, which I will place below the fold.
I’ve just uploaded to the arXiv my paper “Inverse theorems for sets and measures of polynomial growth“. This paper was motivated by two related questions. The first question was to obtain a qualitatively precise description of the sets of polynomial growth that arise in Gromov’s theorem, in much the same way that Freiman’s theorem (and its generalisations) provide a qualitatively precise description of sets of small doubling. The other question was to obtain a non-abelian analogue of inverse Littlewood-Offord theory.
Let me discuss the former question first. Gromov’s theorem tells us that if a finite subset of a group exhibits polynomial growth in the sense that grows polynomially in , then the group generated by is virtually nilpotent (the converse direction also true, and is relatively easy to establish). This theorem has been strengthened a number of times over the years. For instance, a few years ago, I proved with Shalom that the condition that grew polynomially in could be replaced by for a single , as long as was sufficiently large depending on (in fact we gave a fairly explicit quantitative bound on how large needed to be). A little more recently, with Breuillard and Green, the condition was weakened to , that is to say it sufficed to have polynomial relative growth at a finite scale. In fact, the latter paper gave more information on in this case, roughly speaking it showed (at least in the case when was a symmetric neighbourhood of the identity) that was “commensurate” with a very structured object known as a coset nilprogression. This can then be used to establish further control on . For instance, it was recently shown by Breuillard and Tointon (again in the symmetric case) that if for a single that was sufficiently large depending on , then all the for have a doubling constant bounded by a bound depending only on , thus for all .
In this paper we are able to refine this analysis a bit further; under the same hypotheses, we can show an estimate of the form
for all and some piecewise linear, continuous, non-decreasing function with , where the error is bounded by a constant depending only on , and where has at most pieces, each of which has a slope that is a natural number of size . To put it another way, the function for behaves (up to multiplicative constants) like a piecewise polynomial function, where the degree of the function and number of pieces is bounded by a constant depending on .
One could ask whether the function has any convexity or concavity properties. It turns out that it can exhibit either convex or concave behaviour (or a combination of both). For instance, if is contained in a large finite group, then will eventually plateau to a constant, exhibiting concave behaviour. On the other hand, in nilpotent groups one can see convex behaviour; for instance, in the Heisenberg group , if one sets to be a set of matrices of the form for some large (abusing the notation somewhat), then grows cubically for but then grows quartically for .
To prove this proposition, it turns out (after using a somewhat difficult inverse theorem proven previously by Breuillard, Green, and myself) that one has to analyse the volume growth of nilprogressions . In the “infinitely proper” case where there are no unexpected relations between the generators of the nilprogression, one can lift everything to a simply connected Lie group (where one can take logarithms and exploit the Baker-Campbell-Hausdorff formula heavily), eventually describing with fair accuracy by a certain convex polytope with vertices depending polynomially on , which implies that depends polynomially on up to constants. If one is not in the “infinitely proper” case, then at some point the nilprogression develops a “collision”, but then one can use this collision to show (after some work) that the dimension of the “Lie model” of has dropped by at least one from the dimension of (the notion of a Lie model being developed in the previously mentioned paper of Breuillard, Greenm, and myself), so that this sort of collision can only occur a bounded number of times, with essentially polynomial volume growth behaviour between these collisions.
The arguments also give a precise description of the location of a set for which grows polynomially in . In the symmetric case, what ends up happening is that becomes commensurate to a “coset nilprogression” of bounded rank and nilpotency class, whilst is “virtually” contained in a scaled down version of that nilprogression. What “virtually” means is a little complicated; roughly speaking, it means that there is a set of bounded cardinality such that for all . Conversely, if is virtually contained in , then is commensurate to (and more generally, is commensurate to for any natural number ), giving quite a (qualitatively) precise description of in terms of coset nilprogressions.
The main tool used to prove these results is the structure theorem for approximate groups established by Breuillard, Green, and myself, which roughly speaking asserts that approximate groups are always commensurate with coset nilprogressions. A key additional trick is a pigeonholing argument of Sanders, which in this context is the assertion that if is comparable to , then there is an between and such that is very close in size to (up to a relative error of ). It is this fact, together with the comparability of to a coset nilprogression , that allows us (after some combinatorial argument) to virtually place inside .
Similar arguments apply when discussing iterated convolutions of (symmetric) probability measures on a (discrete) group , rather than combinatorial powers of a finite set. Here, the analogue of volume is given by the negative power of the norm of (thought of as a non-negative function on of total mass 1). One can also work with other norms here than , but this norm has some minor technical conveniences (and other measures of the “spread” of end up being more or less equivalent for our purposes). There is an analogous structure theorem that asserts that if spreads at most polynomially in , then is “commensurate” with the uniform probability distribution on a coset progression , and itself is largely concentrated near . The factor of here is the familiar scaling factor in random walks that arises for instance in the central limit theorem. The proof of (the precise version of) this statement proceeds similarly to the combinatorial case, using pigeonholing to locate a scale where has almost the same norm as .
A special case of this theory occurs when is the uniform probability measure on elements of and their inverses. The probability measure is then the distribution of a random product , where each is equal to one of or its inverse , selected at random with drawn uniformly from with replacement. This is very close to the Littlewood-Offord situation of random products where each is equal to or selected independently at random (thus is now fixed to equal rather than being randomly drawn from . In the case when is abelian, it turns out that a little bit of Fourier analysis shows that these two random walks have “comparable” distributions in a certain sense. As a consequence, the results in this paper can be used to recover an essentially optimal abelian inverse Littlewood-Offord theorem of Nguyen and Vu. In the nonabelian case, the only Littlewood-Offord theorem I am aware of is a recent result of Tiep and Vu for matrix groups, but in this case I do not know how to relate the above two random walks to each other, and so we can only obtain an analogue of the Tiep-Vu results for the symmetrised random walk instead of the ordered random walk .
Just a short post here to note that the cover story of this month’s Notices of the AMS, by John Friedlander, is about the recent work on bounded gaps between primes by Zhang, Maynard, our own Polymath project, and others.
I may as well take this opportunity to upload some slides of my own talks on this subject: here are my slides on small and large gaps between the primes that I gave at the “Latinos in the Mathematical Sciences” back in April, and here are my slides on the Polymath project for the Schock Prize symposium last October. (I also gave an abridged version of the latter talk at an AAAS Symposium in February, as well as the Breakthrough Symposium from last November.)
Suppose that are two subgroups of some ambient group , with the index of in being finite. Then is the union of left cosets of , thus for some set of cardinality . The elements of are not entirely arbitrary with regards to . For instance, if is a normal subgroup of , then for each , the conjugation map preserves . In particular, if we write for the conjugate of by , then
Even if is not normal in , it turns out that the conjugation map approximately preserves , if is bounded. To quantify this, let us call two subgroups -commensurate for some if one has
Proposition 1 Let be groups, with finite index . Then for every , the groups and are -commensurate, in fact
Proof: One can partition into left translates of , as well as left translates of . Combining the partitions, we see that can be partitioned into at most non-empty sets of the form . Each of these sets is easily seen to be a left translate of the subgroup , thus . Since
and , we obtain the claim.
One can replace the inclusion by commensurability, at the cost of some worsening of the constants:
Corollary 2 Let be -commensurate subgroups of . Then for every , the groups and are -commensurate.
Proof: Applying the previous proposition with replaced by , we see that for every , and are -commensurate. Since and have index at most in and respectively, the claim follows.
It turns out that a similar phenomenon holds for the more general concept of an approximate group, and gives a “classification” of all the approximate groups containing a given approximate group as a “bounded index approximate subgroup”. Recall that a -approximate group in a group for some is a symmetric subset of containing the identity, such that the product set can be covered by at most left translates of (and thus also right translates, by the symmetry of ). For simplicity we will restrict attention to finite approximate groups so that we can use their cardinality as a measure of size. We call finite two approximate groups -commensurate if one has
note that this is consistent with the previous notion of commensurability for genuine groups.
Theorem 3 Let be a group, and let be real numbers. Let be a finite -approximate group, and let be a symmetric subset of that contains .
- (i) If is a -approximate group with , then one has for some set of cardinality at most . Furthermore, for each , the approximate groups and are -commensurate.
- (ii) Conversely, if for some set of cardinality at most , and and are -commensurate for all , then , and is a -approximate group.
Informally, the assertion that is an approximate group containing as a “bounded index approximate subgroup” is equivalent to being covered by a bounded number of shifts of , where approximately normalises in the sense that and have large intersection. Thus, to classify all such , the problem essentially reduces to that of classifying those that approximately normalise .
To prove the theorem, we recall some standard lemmas from arithmetic combinatorics, which are the foundation stones of the “Ruzsa calculus” that we will use to establish our results:
Lemma 4 (Ruzsa covering lemma) If and are finite non-empty subsets of , then one has for some set with cardinality .
Proof: We take to be a subset of with the property that the translates are disjoint in , and such that is maximal with respect to set inclusion. The required properties of are then easily verified.
Lemma 5 (Ruzsa triangle inequality) If are finite non-empty subsets of , then
Proof: If is an element of with and , then from the identity we see that can be written as the product of an element of and an element of in at least distinct ways. The claim follows.
Now we can prove (i). By the Ruzsa covering lemma, can be covered by at most
left-translates of , and hence by at most left-translates of , thus for some . Since only intersects if , we may assume that
and hence for any
By the Ruzsa covering lemma again, this implies that can be covered by at most left-translates of , and hence by at most left-translates of . By the pigeonhole principle, there thus exists a group element with
the claim follows.
Now we prove (ii). Clearly
Now we control the size of . We have
From the Ruzsa triangle inequality and symmetry we have
By the Ruzsa covering lemma, this implies that is covered by at most left-translates of , hence by at most left-translates of . Since , the claim follows.
We now establish some auxiliary propositions about commensurability of approximate groups. The first claim is that commensurability is approximately transitive:
Proposition 6 Let be a -approximate group, be a -approximate group, and be a -approximate group. If and are -commensurate, and and are -commensurate, then and are -commensurate.
Proof: From two applications of the Ruzsa triangle inequality we have
By the Ruzsa covering lemma, we may thus cover by at most left-translates of , and hence by left-translates of . By the pigeonhole principle, there thus exists a group element such that
and so by arguing as in the proof of part (i) of the theorem we have
and the claim follows.
The next proposition asserts that the union and (modified) intersection of two commensurate approximate groups is again an approximate group:
Proposition 7 Let be a -approximate group, be a -approximate group, and suppose that and are -commensurate. Then is a -approximate subgroup, and is a -approximate subgroup.
Using this proposition, one may obtain a variant of the previous theorem where the containment is replaced by commensurability; we leave the details to the interested reader.
Proof: We begin with . Clearly is symmetric and contains the identity. We have . The set is already covered by left translates of , and hence of ; similarly is covered by left translates of . As for , we observe from the Ruzsa triangle inequality that
and hence by the Ruzsa covering lemma, is covered by at most left translates of , and hence by left translates of , and hence of . Similarly is covered by at most left translates of . The claim follows.
Now we consider . Again, this is clearly symmetric and contains the identity. Repeating the previous arguments, we see that is covered by at most left-translates of , and hence there exists a group element with
Now observe that
and so by the Ruzsa covering lemma, can be covered by at most left-translates of . But this latter set is (as observed previously) contained in , and the claim follows.
Here’s a cute identity I discovered by accident recently. Observe that
and so one can conjecture that one has
when is even, and
when is odd. This is obvious in the even case since is a polynomial of degree , but I struggled for a while with the odd case before finding a slick three-line proof. (I was first trying to prove the weaker statement that was non-negative, but for some strange reason I was only able to establish this by working out the derivative exactly, rather than by using more analytic methods, such as convexity arguments.) I thought other readers might like the challenge (and also I’d like to see some other proofs), so rather than post my own proof immediately, I’ll see if anyone would like to supply their own proofs or thoughts in the comments. Also I am curious to know if this identity is connected to any other existing piece of mathematics.