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Tanja Eisner and I have just uploaded to the arXiv our paper “Large values of the Gowers-Host-Kra seminorms“, submitted to Journal d’Analyse Mathematique. This paper is concerned with the properties of three closely related families of (semi)norms, indexed by a positive integer :
- The Gowers uniformity norms of a (bounded, measurable, compactly supported) function taking values on a locally compact abelian group , equipped with a Haar measure ;
- The Gowers uniformity norms of a function on a discrete interval ; and
- The Gowers-Host-Kra seminorms of a function on an ergodic measure-preserving system .
These norms have been discussed in depth in previous blog posts, so I will just quickly review the definition of the first norm here (the other two (semi)norms are defined similarly). The norm is defined recursively by setting
where . Equivalently, one has
Informally, the Gowers uniformity norm measures the extent to which (the phase of ) behaves like a polynomial of degree less than . Indeed, if and is compact with normalised Haar measure , it is not difficult to show that is at most , with equality if and only if takes the form almost everywhere, where is a polynomial of degree less than (which means that for all ).
Our first result is to show that this result is robust, uniformly over all choices of group :
Theorem 1 (-near extremisers) Let be a compact abelian group with normalised Haar measure , and let be such that and for some and . Then there exists a polynomial of degree at most such that , where is bounded by a quantity that goes to zero as for fixed .
The quantity can be described effectively (it is of polynomial size in ), but we did not seek to optimise it here. This result was already known in the case of vector spaces over a fixed finite field (where it is essentially equivalent to the assertion that the property of being a polynomial of degree at most is locally testable); the extension to general groups turns out to fairly routine. The basic idea is to use the recursive structure of the Gowers norms, which tells us in particular that if is close to one, then is close to one for most , which by induction implies that is close to for some polynomials of degree at most and for most . (Actually, it is not difficult to use cocycle equations such as (when ) to upgrade “for most ” to “for all “.) To finish the job, one would like to express the as derivatives of a polynomial of degree at most . This turns out to be equivalent to requiring that the obey the cocycle equation
where is the translate of by . (In the paper, the sign conventions are reversed, so that , in order to be compatible with ergodic theory notation, but this makes no substantial difference to the arguments or results.) However, one does not quite get this right away; instead, by using some separation properties of polynomials, one can show the weaker statement that
where the are small real constants. To eliminate these constants, one exploits the trivial cohomology of the real line. From (1) one soon concludes that the obey the -cocycle equation
and an averaging argument then shows that is a -coboundary in the sense that
for some small scalar depending on . Subtracting from then gives the claim.
Similar results and arguments also hold for the and norms, which we will not detail here.
Dimensional analysis reveals that the norm is not actually the most natural norm with which to compare the norms against. An application of Young’s convolution inequality in fact reveals that one has the inequality
where is the critical exponent , without any compactness or normalisation hypothesis on the group and the Haar measure . This allows us to extend the norm to all of . There is then a stronger inverse theorem available:
Theorem 2 (-near extremisers) Let be a locally compact abelian group, and let be such that and for some and . Then there exists a coset of a compact open subgroup of , and a polynomial of degree at most such that .
Conversely, it is not difficult to show that equality in (2) is attained when takes the form as above. The main idea of proof is to use an inverse theorem for Young’s inequality due to Fournier to reduce matters to the case that was already established. An analogous result is also obtained for the norm on an ergodic system; but for technical reasons, the methods do not seem to apply easily to the norm. (This norm is essentially equivalent to the norm up to constants, with comparable to , but when working with near-extremisers, norms that are only equivalent up to constants can have quite different near-extremal behaviour.)
with equality attained if and only if is a gaussian modulated by a quadratic polynomial phase. This additional gain of allows one to pinpoint the threshold for the previous near-extremiser results in the case of norms. For instance, by using the Host-Kra machinery of characteristic factors for the norm, combined with an explicit and concrete analysis of the -step nilsystems generated by that machinery, we can show that
whenever is a totally ergodic system and is orthogonal to all linear and quadratic eigenfunctions (which would otherwise form immediate counterexamples to the above inequality), with the factor being best possible. We can also establish analogous results for the and norms (using the inverse theorem of Ben Green and myself, in place of the Host-Kra machinery), although it is not clear to us whether the threshold remains best possible in this case.
Tim Austin, Tanja Eisner, and I have just uploaded to the arXiv our joint paper Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems, submitted to Pacific Journal of Mathematics. This project started with the observation that the multiple recurrence theorem of Furstenberg (and the related multiple convergence theorem of Host and Kra) could be interpreted in the language of dynamical systems of commutative finite von Neumann algebras, which naturally raised the question of the extent to which the results hold in the noncommutative setting. The short answer is “yes for small averages, but not for long ones”.
The Furstenberg multiple recurrence theorem can be phrased as follows: if is a probability space with a measure-preserving shift (which naturally induces an isomorphism by setting ), is non-negative with positive trace , and is an integer, then one has
In particular, for all in a set of positive upper density. This result is famously equivalent to Szemerédi’s theorem on arithmetic progressions.
The Host-Kra multiple convergence theorem makes the related assertion that if , then the scalar averages
converge to a limit as ; a fortiori, the function averages
converge in (say) norm.
The space is a commutative example of a von Neumann algebra: an algebra of bounded linear operators on a complex Hilbert space which is closed under the weak operator topology, and under taking adjoints. Indeed, one can take to be , and identify each element of with the multiplier operator . The operation is then a finite trace for this algebra, i.e. a linear map from the algebra to the scalars such that , , and , with equality iff . The shift is then an automorphism of this algebra (preserving shift and conjugation).
We can generalise this situation to the noncommutative setting. Define a von Neumann dynamical system to be a von Neumann algebra with a finite trace and an automorphism . In addition to the commutative examples generated by measure-preserving systems, we give three other examples here:
- (Matrices) is the algebra of complex matrices, with trace and shift , where is a fixed unitary matrix.
- (Group algebras) is the closure of the group algebra of a discrete group (i.e. the algebra of finite formal complex combinations of group elements), which acts on the Hilbert space by convolution (identifying each group element with its Kronecker delta function). A trace is given by , where is the Kronecker delta at the identity. Any automorphism of the group induces a shift .
- (Noncommutative torus) is the von Neumann algebra acting on generated by the multiplier operator and the shifted multiplier operator , where is fixed. A trace is given by , where is the constant function.
Inspired by noncommutative generalisations of other results in commutative analysis, one can then ask the following questions, for a fixed and for a fixed von Neumann dynamical system :
- (Recurrence on average) Whenever is non-negative with positive trace, is it true that
- (Recurrence on a dense set) Whenever is non-negative with positive trace, is it true thatfor all in a set of positive upper density?
- (Weak convergence) With , is it true thatconverges?
- (Strong convergence) With , is it true thatconverges in using the Hilbert-Schmidt norm ?
Note that strong convergence automatically implies weak convergence, and recurrence on average automatically implies recurrence on a dense set.
For , all four questions can trivially be answered “yes”. For , the answer to the above four questions is also “yes”, thanks to the von Neumann ergodic theorem for unitary operators. For , we were able to establish a positive answer to the “recurrence on a dense set”, “weak convergence”, and “strong convergence” results assuming that is ergodic. For general , we have a positive answer to all four questions under the assumption that is asymptotically abelian, which roughly speaking means that the commutators converges to zero (in an appropriate weak sense) as . Both of these proofs adapt the usual ergodic theory arguments; the latter result generalises some earlier work of Niculescu-Stroh-Zsido, Duvenhage, and Beyers-Duvenhage-Stroh. For the result, a key observation is that the van der Corput lemma can be used to control triple averages without requiring any commutativity; the “generalised von Neumann” trick of using multiple applications of the van der Corput trick to control higher averages, however, relies much more strongly on commutativity.
In most other situations we have counterexamples to all of these questions. In particular:
- For , recurrence on average can fail on an ergodic system; indeed, one can even make the average negative. This example is ultimately based on a Behrend example construction and a von Neumann algebra construction known as the crossed product.
- For , recurrence on a dense set can also fail if the ergodicity hypothesis is dropped. This also uses the Behrend example and the crossed product construction.
- For , weak and strong convergence can fail even assuming ergodicity. This uses a group theoretic construction, which amusingly was inspired by Grothendieck’s interpretation of a group as a sheaf of flat connections, which I blogged about recently, and which I will discuss below the fold.
- For , recurrence on a dense set fails even with the ergodicity hypothesis. This uses a fancier version of the Behrend example due to Ruzsa in this paper of Bergelson, Host, and Kra. This example only applies for ; we do not know for whether recurrence on a dense set holds for ergodic systems.