# A Host–Kra F^omega_2-system of order 5 that isn’t Abramov of order 5, and non-measurability of the inverse theorem for the U^6(F^n_2) norm; The construction of completely disconnected Host–Kra–Ziegler components, and the inverse theorem for the U^ok Gowers uniformity norms on finite abelian teams of bounded torsion

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Asgar Jamneshan, Or Shalom, and myself have simply uploaded to the arXiv our preprints “A Host–Kra -system of order 5 that isn’t Abramov of order 5, and non-measurability of the inverse theorem for the norm” and “The construction of completely disconnected Host–Kra–Ziegler components, and the inverse theorem for the Gowers uniformity norms on finite abelian teams of bounded torsion“. These two papers are each involved with advancing the inverse idea for the Gowers norms and Gowers-Host-Kra seminorms; the primary paper supplies a counterexample on this idea (particularly disproving a conjecture of Bergelson, Ziegler and myself), and the second paper provides new constructive leads to the case when the underlying group is bounded torsion, or the ergodic system is completely disconnected. I focus on the 2 papers extra beneath the fold.

** — 1. System of order which isn’t Abramov of order — **

I gave a chat on this paper not too long ago on the IAS; the slides for that discuss can be found right here.

This mission might be motivated by the inverse conjecture for the Gowers norm in finite fields, which is now a theorem:

Theorem 1 (Inverse conjecture for the Gowers norm in finite fields)Let be a major and . Suppose that is a one-bounded perform with a decrease sure on the Gowers uniformity norm. Then there exists a (non-classical) polynomial of diploma at most such that .

That is now recognized for all (see this paper of Ziegler and myself for the primary proof of the overall case, and this paper of Milicevic for the newest developments regarding quantitative bounds), though preliminary outcomes centered on both small values of , or the “excessive attribute” case when is giant in comparison with . One method to this theorem proceeds through ergodic idea. Certainly it was noticed in this earlier paper of Ziegler and myself that for a given selection of and , the above theorem follows from the next ergodic analogue:

Conjecture 2 (Inverse conjecture for the Gowers-Host-Kra semi-norm in finite fields)Let be a major and . Suppose that with an ergodic -system with constructive Gowers-Host-Kra seminorm (see as an illustration this earlier publish for a definition). Then there exists a measurable polynomial of diploma at most such that has a non-zero inside product with . (Within the language of ergodic idea: each -system of order is an Abramov system of order .)

The implication proceeds by a correspondence precept analogous to the Furstenberg correspondence precept developed in that paper (see additionally this paper of Towsner for a intently associated precept, and this paper of Jamneshan and I for a refinement). In a paper with Bergelson and Ziegler, we have been capable of set up Conjecture 2 within the “excessive attribute” case , thus additionally proving Theorem 1 on this regime, and conjectured that Conjecture 2 was actually true for all . This was not too long ago verified within the barely bigger vary by Candela, Gonzalez-Sanchez, and Szegedy.

Though Theorem 1 is now recognized in full generality by different strategies, there are nonetheless combinatorial causes for investigating Conjecture 2. Certainly one of these is that the implication of Theorem 1 from Corollary 2 actually provides further management on the polynomial produced by Theorem 1, specifically that it’s some sense “measurable within the sigma-algebra generated by ” (principally as a result of the ergodic idea polynomial produced by Conjecture 2 can also be measurable in , versus merely being measurable in an extension of ). What this implies within the finitary setting of is a bit tough to jot down down exactly (for the reason that naive sigma-algebra generated by the interprets of will largely probably be the discrete sigma-algebra), however roughly talking it implies that might be approximated to arbitrary accuracy by capabilities of boundedly many (random) interprets of . This may be interpreted in a complexity idea sense by stating that Theorem 1 might be made “algorithmic” in a “probabilistic bounded time oracle” or “native record decoding” sense which we won’t make exact right here.

The principle results of this paper is

Theorem 3Conjecture 2 fails for . The truth is the “measurable inverse theorem” alluded to above additionally fails on this case.

Informally, which means for big , we will discover -bounded “pseudo-quintic” capabilities with giant norm, which then should essentially correlate with at the least one quintic by Theorem 1, however such that none of those quintics might be approximated to excessive accuracy by capabilities of (random) shifts of . Roughly talking, which means the inverse theorem can’t be made regionally algorithmic (although it’s nonetheless doable {that a} Goldreich-Levin sort results of polynomial time algorithmic inverse idea remains to be doable, as is already recognized for for ; see this latest paper of Kim, Li and Tidor for additional dialogue).

The way in which we arrived at this theorem was by (morally) decreasing issues to understanding a sure “finite nilspace cohomology downside”. In the long run it boiled all the way down to finding a sure perform from a -element set to a two-element set which was a “strongly -homogeneous cocycle” however not a “coboundary” (these phrases are outlined exactly within the paper). This strongly -homogeneous cocycle might be expressed by way of a less complicated perform that takes values on a -element house . The duty of finding turned out to be one which was throughout the vary of our (considerably rudimentary) SAGE computation talents (largely involving computing the Smith regular type of some moderately giant integer matrices), however the counterexample capabilities this produced have been initially considerably opaque to us. After cleansing up these capabilities by hand (by subtracting off numerous “coboundaries”), we finally discovered variations of those capabilities which have been good sufficient that we may confirm all of the claims wanted in a purely human-readable trend, with none additional laptop help. As a consequence, we will now describe the pseudo-quintic explicitly, although it’s secure to say we might not have been capable of provide you with this instance with out the preliminary laptop search, and we don’t at present have a broader conceptual understanding of which may probably generate such counterexamples. The perform takes the shape

the place is a randomly chosen (classical) quadratic polynomial, is a randomly chosen (non-classical) cubic polynomial, and is a randomly chosen (non-classical) quintic polynomial. This perform correlates with and has a big norm, however this quintic is “non-measurable” within the sense that it can’t be recovered from and its shifts. The quadratic polynomial seems to be measurable, as is the double of the cubic , however with a view to get better one wants to use a “sq. root” to the quadratic to get better a candidate for the cubic which might then be used to reconstruct .

** — 2. Construction of completely disconnected methods — **

Regardless of the above unfavourable end result, in our different paper we’re capable of get a weak model of Conjecture 2, that additionally extends to actions of bounded-torsion abelian teams:

Theorem 4 (Weak inverse conjecture for the Gowers-Host-Kra semi-norm in bounded torsion teams)Let be a bounded-torsion abelian group and . Suppose that with an ergodic -system with constructive Gowers-Host-Kra seminorm . Then, after lifting to a torsion-free group , there exists a measurable polynomial of diploma at most outlined on anextensionof which has a non-zero inside product with .

Combining this with the correspondence precept and a few further instruments, we get hold of a weak model of Theorem 1 that additionally extends to bounded-torsion teams:

Theorem 5 (Inverse conjecture for the Gowers norm in bounded torsion teams)Let be a finite abelian -torsion group for some and . Suppose that is a one-bounded perform with . Then there exists a (non-classical) polynomial of diploma at most such that .

The diploma produced by our arguments is polynomial in , however we conjecture that it ought to simply be .

The way in which Theorem 4 (and therefore Theorem 5) is confirmed is as follows. The now-standard equipment of Host and Kra (as mentioned as an illustration in their e-book) permits us to cut back to a system of order , which is a sure tower of extensions of compact abelian construction teams by numerous cocycles . Within the -torsion case, customary idea permits us to indicate that these construction teams are additionally -torsion, therefore completely disconnected. So it will now suffice to grasp the motion of torsion-free teams on completely disconnected methods . For the needs of proving Theorem 4 now we have the liberty to increase as we please, and we make the most of this freedom by “extending by radicals”, within the sense that at any time when we find a polynomial within the system, we adjoin to it roots of that polynomial (i.e., options to ) which can be polynomials of the *similar* diploma as ; that is often not doable to do within the unique system , however can all the time be achieved in an appropriate extension, analogously to how roots don’t all the time exist in a given area, however can all the time be positioned in some extension of that area. After making use of this course of countably many instances it seems that we will arrive at a system which is -divisible within the sense that polynomials of any diploma have roots of any order which can be of the identical diploma. In different phrases, the group of polynomials of any fastened diploma is a divisible abelian group, and thus injective within the class of such teams. This makes a number of quick actual sequences that present up within the idea cut up robotically, and tremendously simplifies the cohomological points one encounters within the idea, to the purpose the place all of the cocycles talked about beforehand can now be “straightened” into polynomials of the anticipated diploma (or, within the language of ergodic idea, this extension is a Weyl system of order , and therefore additionally Abramov of order ). That is adequate to ascertain Theorem 4. To get Theorem 5, we ran right into a technical impediment arising from the truth that the rest map will not be a polynomial mod if will not be itself a major energy. To resolve this, we established ergodic idea analogues of the Sylow decomposition of abelian -torsion teams into -groups , in addition to the Schur-Zassenhaus theorem. Roughly talking, the upshot of those theorems is that any ergodic -system , with -torsion, might be cut up because the “direct sum” of ergodic -systems for primes dividing , the place is the subgroup of consisting of these components whose order is an influence of . This enables us to cut back to the case when is a major energy with out an excessive amount of issue.

The truth is, the above evaluation provides stronger structural classifications of completely disconnected methods (during which the performing group is torsion-free). Weyl methods can be interpreted as translational methods , the place is a nilpotent Polish group and is a closed cocompact subgroup, with the motion being given by left-translation by numerous components of . Maybe essentially the most well-known examples of such translational methods are nilmanifolds, however on this setting the place the performing group will not be finitely generated, it seems to be crucial to think about extra common translational methods, during which needn’t be a Lie group (and even regionally compact), and not discrete. Our earlier outcomes then describe completely disconnected methods as *components* of such translational methods. One pure candidate for such components are the *double coset methods* shaped by quotienting out by the motion of one other closed group that’s normalized by the motion of . We have been capable of present that each one completely disconnected methods with torsion-free performing group had this double coset construction. This turned out to be surprisingly delicate at a technical degree, for at the least two causes. Firstly, after finding the closed group (which generally is Polish, however not compact and even regionally compact), it was not instantly apparent that was itself a Polish house (this quantities to the orbits of a closed set nonetheless being closed), and likewise not apparent that this double coset house had a very good nilspace construction (particularly that the issue map from to is a nilspace fibration). This latter subject we have been capable of resolve with a instrument kindly shared to us in a forthcoming work by Candela, Gonzales-Sanchez, and Szegedy, who noticed that the nilspace fibration property was obtainable if the quotient teams obeyed an algebraic “groupable” axiom which we have been capable of confirm on this case (additionally they have counterexamples exhibiting that the nilspace construction can break down with out this axiom). There was nevertheless one additional moderately annoying complication. To be able to absolutely get hold of the identification of our system with a double coset system, we would have liked the equivalence

between bounded measurable capabilities on which have been -invariant as much as null units on one hand, and bounded measurable capabilities on on the opposite. It’s fairly simple to embed the latter house isometrically into the previous house, and we thought for some time that the other inclusion was trivial, however a lot to our shock and frustration we weren’t capable of obtain this identification by “gentle” strategies. One definitely has the topological analogue

of this identification, and is the weak closure of and the weak closure of , however this isn’t fairly sufficient to shut the argument; we additionally must have a (weakly) steady projection operator from to to make the whole lot work. When is compact (or extra usually, regionally compact amenable) one may strive to do that by averaging over the Haar measure of , or (presumably) by some averages on Folner units. In our setting, we all know that can fail to be regionally compact (it could include teams like ), however we have been capable of find a “poor man’s Haar measure” on this non-locally compact group that was a compactly supported Radon likelihood measure acted like a Haar measure when pushed ahead to particular person orbits of on , which turned out to be adequate to get the averaging we would have liked (and likewise to ascertain the Polish nature of ).

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