Theorem 1 (Holomorphic images of disks)Let be a disk in the complex plane, and be a holomorphic function with .

- (i) (Open mapping theorem) contains a disk for some .
- (ii) (Bloch theorem) contains a disk for some absolute constant and some . (In fact there is even a holomorphic right inverse of from to .)
- (iii) (Koebe quarter theorem) If is injective, then contains the disk .
- (iv) If is a polynomial of degree , then contains the disk .
- (v) If one has a bound of the form for all and some , then contains the disk for some absolute constant . (In fact there is holomorphic right inverse of from to .)

Parts (i), (ii), (iii) of this theorem are standard, as indicated by the given links. I found part (iv) as (a consequence of) Theorem 2 of this paper of Degot, who remarks that it “seems not already known in spite of its simplicity”. The proof is simple:

*Proof:* (Proof of (iv)) Let , then we have a lower bound for the log-derivative of at :

The constant in (iv) is completely sharp: if and is non-zero then contains the disk

but avoids the origin, thus does not contain any disk of the form . This example also shows that despite parts (ii), (iii) of the theorem, one cannot hope for a general inclusion of the form for an absolute constant .Part (v) is implicit in the standard proof of Bloch’s theorem (part (ii)), and is easy to establish:

*Proof:* (Proof of (v)) From the Cauchy inequalities one has for , hence by Taylor’s theorem with remainder for . By Rouche’s theorem, this implies that the function has a unique zero in for any , if is a sufficiently small absolute constant. The claim follows.

Note that part (v) implies part (i). A standard point picking argument also lets one deduce part (ii) from part (v):

*Proof:* (Proof of (ii)) By shrinking slightly if necessary we may assume that extends analytically to the closure of the disk . Let be the constant in (v) with ; we will prove (iii) with replaced by . If we have for all then we are done by (v), so we may assume without loss of generality that there is such that . If for all then by (v) we have

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- An “arithmetic regularity lemma” that, roughly speaking, decomposes an arbitrary bounded sequence on an interval as an “irrational nilsequence” of controlled complexity, plus some “negligible” errors (where one uses the Gowers uniformity norm as the main norm to control the neglibility of the error); and
- An “arithmetic counting lemma” that gives an asymptotic formula for counting various averages for various affine-linear forms when the functions are given by irrational nilsequences.

The combination of the two theorems is then used to address various questions in additive combinatorics.

There are no direct issues with the arithmetic regularity lemma. However, it turns out that the arithmetic counting lemma is only true if one imposes an additional property (which we call the “flag property”) on the affine-linear forms . Without this property, there does not appear to be a clean asymptotic formula for these averages if the only hypothesis one places on the underlying nilsequences is irrationality. Thus when trying to understand the asymptotics of averages involving linear forms that do not obey the flag property, the paradigm of understanding these averages via a combination of the regularity lemma and a counting lemma seems to require some significant revision (in particular, one would probably have to replace the existing regularity lemma with some variant, despite the fact that the lemma is still technically true in this setting). Fortunately, for most applications studied to date (including the important subclass of translation-invariant affine forms), the flag property holds; however our claim in the paper to have resolved a conjecture of Gowers and Wolf on the true complexity of systems of affine forms must now be narrowed, as our methods only verify this conjecture under the assumption of the flag property.

In a bit more detail: the asymptotic formula for our counting lemma involved some finite-dimensional vector spaces for various natural numbers , defined as the linear span of the vectors as ranges over the parameter space . Roughly speaking, these spaces encode some constraints one would expect to see amongst the forms . For instance, in the case of length four arithmetic progressions when , , and

for , then is spanned by the vectors and and can thus be described as the two-dimensional linear space while is spanned by the vectors , , and can be described as the hyperplane As a special case of the counting lemma, we can check that if takes the form for some irrational , some arbitrary , and some smooth , then the limiting value of the average as is equal to which reflects the constraints and These constraints follow from the descriptions (1), (2), using the containment to dispense with the lower order term (which then plays no further role in the analysis).The arguments in our paper turn out to be perfectly correct under the assumption of the “flag property” that for all . The problem is that the flag property turns out to not always hold. A counterexample, provided by Daniel Altman, involves the four linear forms

Here it turns out that and and is no longer contained in . The analogue of the asymptotic formula given previously for is then valid when vanishes, but not when is non-zero, because the identity holds in the former case but not the latter. Thus the output of any purported arithmetic regularity lemma in this case is now sensitive to the lower order terms of the nilsequence and cannot be described in a uniform fashion for all “irrational” sequences. There should still be some sort of formula for the asymptotics from the general equidistribution theory of nilsequences, but it could be considerably more complicated than what is presented in this paper.Fortunately, the flag property does hold in several key cases, most notably the translation invariant case when contains , as well as “complexity one” cases. Nevertheless non-flag property systems of affine forms do exist, thus limiting the range of applicability of the techniques in this paper. In particular, the conjecture of Gowers and Wolf (Theorem 1.13 in the paper) is now open again in the non-flag property case.

]]>The book is currently under contract with Yale University Press. My coauthor Tanya Klowden can be reached at tklowden@gmail.com.

]]>In many cases the tiling set will be periodic (by which we mean translation invariant with respect to some lattice (a finite index subgroup) of ). For instance one simple example of a tiling is when is the unit square and is the lattice . However one can modify some tilings to make them less periodic. For instance, keeping one also has the tiling set

where is an arbitrary function. This tiling set is periodic in a single direction , but is not doubly periodic. For the slightly modified tile , the set for arbitrary can be verified to be a tiling set, which in general will not exhibit any periodicity whatsoever; however, it isThe most well known conjecture in this area is the Periodic Tiling Conjecture:

Conjecture 1 (Periodic tiling conjecture)If a finite tile has at least one tiling set, then it has a tiling set which is periodic.

This conjecture was stated explicitly by Lagarias and Wang, and also appears implicitly in this text of Grunbaum and Shepard. In one dimension there is a simple pigeonhole principle argument of Newman that shows that all tiling sets are in fact periodic, which certainly implies the periodic tiling conjecture in this case. The case was settled more recently by Bhattacharya, but the higher dimensional cases remain open in general.

We are able to obtain a new proof of Bhattacharya’s result that also gives some quantitative bounds on the periodic tiling set, which are polynomial in the diameter of the set if the cardinality of the tile is bounded:

Theorem 2 (Quantitative periodic tiling in )If a finite tile has at least one tiling set, then it has a tiling set which is -periodic for some .

Among other things, this shows that the problem of deciding whether a given subset of of bounded cardinality tiles or not is in the NP complexity class with respect to the diameter . (Even the decidability of this problem was not known until the result of Bhattacharya.)

We also have a closely related structural theorem:

Theorem 3 (Quantitative weakly periodic tiling in )Every tiling set of a finite tile is weakly periodic. In fact, the tiling set is the union of at most disjoint sets, each of which is periodic in a direction of magnitude .

We also have a new bound for the periodicity of tilings in :

Theorem 4 (Universal period for tilings in )Let be finite, and normalized so that . Then every tiling set of is -periodic, where is the least common multiple of all primes up to , and is the least common multiple of the magnitudes of all .

We remark that the current best complexity bound of determining whether a subset of tiles or not is , due to Biro. It may be that the results in this paper can improve upon this bound, at least for tiles of bounded cardinality.

On the other hand, we discovered a genuine difference between level one tiling and higher level tiling, by locating a counterexample to the higher level analogue of (the qualitative version of) Theorem 3:

Theorem 5 (Counterexample)There exists an eight-element subset and a level tiling such that is not weakly periodic.

We do not know if there is a corresponding counterexample to the higher level periodic tiling conjecture (that if tiles at level , then there is a periodic tiling at the same level ). Note that it is important to keep the level fixed, since one trivially always has a periodic tiling at level from the identity .

The methods of Bhattacharya used the language of ergodic theory. Our investigations also originally used ergodic-theoretic and Fourier-analytic techniques, but we ultimately found combinatorial methods to be more effective in this problem (and in particular led to quite strong quantitative bounds). The engine powering all of our results is the following remarkable fact, valid in all dimensions:

Lemma 6 (Dilation lemma)Suppose that is a tiling of a finite tile . Then is also a tiling of the dilated tile for any coprime to , where is the least common multiple of all the primes up to .

Versions of this dilation lemma have previously appeared in work of Tijdeman and of Bhattacharya. We sketch a proof here. By the fundamental theorem of arithmetic and iteration it suffices to establish the case where is a prime . We need to show that . It suffices to show the claim , since both sides take values in . The convolution algebra (or group algebra) of finitely supported functions from to is a commutative algebra of characteristic , so we have the Frobenius identity for any . As a consequence we see that . The claim now follows by convolving the identity by further copies of .

In our paper we actually establish a more general version of the dilation lemma that can handle tilings of higher level or of a periodic set, and this stronger version is useful to get the best quantitative results, but for simplicity we focus attention just on the above simple special case of the dilation lemma.

By averaging over all in an arithmetic progression, one already gets a useful structural theorem for tilings in any dimension, which appears to be new despite being an easy consequence of Lemma 6:

Corollary 7 (Structure theorem for tilings)Suppose that is a tiling of a finite tile , where we normalize . Then we have a decomposition where each is a function that is periodic in the direction , where is the least common multiple of all the primes up to .

*Proof:* From Lemma 6 we have for any , where is the Kronecker delta at . Now average over (extracting a weak limit or generalised limit as necessary) to obtain the conclusion.

The identity (1) turns out to impose a lot of constraints on the functions , particularly in one and two dimensions. On one hand, one can work modulo to eliminate the and terms to obtain the equation

which in two dimensions in particular puts a lot of structure on each individual (roughly speaking it makes the behave in a polynomial fashion, after collecting commensurable terms). On the other hand we have the inequality which can be used to exclude “equidistributed” polynomial behavior after a certain amount of combinatorial analysis. Only a small amount of further argument is then needed to conclude Theorem 3 and Theorem 2.For level tilings the analogue of (2) becomes

which is a significantly weaker inequality and now no longer seems to prohibit “equidistributed” behavior. After some trial and error we were able to come up with a completely explicit example of a tiling that actually utilises equidistributed polynomials; indeed the tiling set we ended up with was a finite boolean combination of Bohr sets.We are currently studying what this machinery can tell us about tilings in higher dimensions, focusing initially on the three-dimensional case.

]]>In the course of this project, we ran into the obstacle that many foundational results, such as the Riesz representation theorem, often require one or more of these countability hypotheses when encountered in textbooks. Other technical issues also arise in the uncountable setting, such as the need to distinguish the Borel -algebra from the (two different types of) Baire -algebra. As such we needed to spend some time reviewing and synthesizing the known literature on some foundational results of “uncountable” measure theory, which led to this paper. As such, most of the results of this paper are already in the literature, either explicitly or implicitly, in one form or another (with perhaps the exception of the canonical disintegration, which we discuss below); we view the main contribution of this paper as presenting the results in a coherent and unified fashion. In particular we found that the language of category theory was invaluable in clarifying and organizing all the different results. In subsequent work we (and some other authors) will use the results in this paper for various applications in uncountable ergodic theory.

The foundational results covered in this paper can be divided into a number of subtopics (Gelfand duality, Baire -algebras and Riesz representation, canonical models, and canonical disintegration), which we discuss further below the fold.

** — 1. Gelfand duality — **

Given a compact Hausdorff space , one can form the space of continuous functions from to ; such functions are necessarily bounded and compactly supported, and they form a commutative unital -algebra. Conversely, given a commutative unital -algebra , one can produce the *Gelfand spectrum* , which we define here as the collection of unital -homomorphisms from to . This spectrum can be viewed as a subset of and inherits a topology from the product topology on that latter space that turns the spectrum into a compact Hausdorff space. The classical Gelfand duality between compact Hausdorff spaces and unital commutative -algebras asserts that these two operations and “essentially invert” each other: is homeomorphic to , and is isomorphic as a commutative unital -algebra to . In fact there is a more precise statement: the operations , are in fact contravariant functors that form a duality of categories between the category of compact Hausdorff spaces and the category of commutative unital -algebras. This duality of categories asserts, roughly speaking, that the operations , and the Gelfand duality isomorphisms interact in the “natural” fashion with respect to morphisms: for instance, any continuous map between compact Hausdorff spaces induces a commutative unital -homomorphism defined by the pullback map , and similarly any commutative unital -homomorphism induces a continuous map defined by , and the homeomorphisms between and and between and commute with the continuous maps and in the “natural” fashion. These sorts of properties are routine to verify and mostly consist of expanding out all the definitions.

It is natural to ask what the analogous Gelfand duality is for *locally* compact Hausdorff spaces. Somewhat confusingly, there appeared to be two such Gelfand dualities in the literature, which appeared at first glance to be incompatible with each other. Eventually we understood that there were *two* natural categories of locally compact Hausdorff spaces, and dually there were *two* natural categories of (non-unital) commutative algebras dual to them:

- The category of locally compact Hausdorff spaces, whose morphisms consist of arbitrary continuous maps between those spaces.
- The subcategory of locally compact Hausdorff spaces, whose morphisms consist of continuous proper maps between those spaces.
- The category of commutative algebras (not necessarily unital), whose morphisms were homomorphisms which were non-degenerate in the sense that is dense in (this was automatic in the unital case, but now needs to be imposed as an additional hypothesis).
- The larger category of commutative algebras (not necessarily unital), whose morphisms were non-degenerate homomorphisms into the multiplier algebra of . (It is not immediately obvious that one can compose two such morphisms together, but it can be shown that every such homomorphism has a unique extension to a homomorphism from to , which can then be used to create a composition law.)

The map that takes a locally compact space to the space of continuous functions that vanish at infinity, together with the previously mentioned Gelfand spectrum map , then forms a duality of categories between and , and between and . Furthermore, these dualities of categories interact well with the two standard compactifications of locally compact spaces: the Stone-Cech compactification , and the Alexandroff compactification (also known as the one point compactification). From a category theoretic perspective, it is most natural to interpret as a functor from to , and to interpret as a functor from to the category of pointed compact Hausdorff spaces (the notation here is a special case of comma category notation). (Note in particular that a continuous map between locally compact Hausdorff spaces needs to be proper in order to guarantee a continuous extension to the Alexandroff compactification, whereas no such properness condition is needed to obtain a continuous extension to the Stone-Cech compactification.) In our paper, we summarized relationships between these functors (and some other related functors) in the following diagram, which commutes up to natural isomorphisms:

Thus for instance the space of bounded continuous functions on a locally compact Hausdorff space is naturally isomorphic to the multiplier algebra of , and the Stone-Cech compactification is naturally identified with the Gelfand spectrum of :

The coloring conventions in this paper are that (a) categories of “algebras” are displayed in red (and tend to be dual to categories of “spaces”, displayed in black; and (b) functors displayed in blue will be considered “casting functors” (analogous to type casting operators in various computing languages), and are used as the default way to convert an object or morphism in one category to an object or morphism in another. For instance, if is a compact Hausdorff space, the associated unital commutative -algebra is defined to be by the casting convention.Almost every component of the above diagram was already stated somewhere in the literature; our main contribution here is the synthesis of all of these components together into the single unified diagram.

** — 2. Baire -algebras and Riesz duality — **

Now we add measure theory to Gelfand duality, by introducing -algebras and probability measures on the (locally) compact Hausdorff side of the duality, and by introducing traces on the -algebra side. Here we run into the issue that there are *three* natural choices of -algebra one can assign to a topological space :

- The Borel -algebra, generated by the open (or closed) subsets of .
- The -Baire -algebra, generated by the bounded continuous functions on (equivalently, one can use arbitrary continuous functions on ; these are also precisely the sets whose indicator functions are Baire functions with respect to a countable ordinal).
- The -Baire -algebra, generated by the compactly supported continuous functions on (equivalently, one can use continuous functions on going to zero at infinity; these are also precisely the sets generated by compact sets).

For compact Hausdorff spaces, the two types of Baire -algebras agree, but they can diverge for locally compact Hausdorff spaces. Similarly, the Borel and Baire algebras agree for compact metric spaces, but can diverge for more “uncountable” compact Hausdorff spaces. This is most dramatically exemplified by the Nedoma pathology, in which the Borel -algebra of the Cartesian square of a locally compact Hausdorff space need not be equal to the product of the individual Borel -algebra, in contrast to the Baire -algebra which reacts well with the product space construction (even when there are uncountably many factors). In particular, the group operations on a locally compact Hausdorff group can fail to be Borel measurable, even though they are always Baire measurable. For these reasons we found it desirable to adopt a “Baire-centric” point of view, in which one prefers to use the Baire -algebras over the Borel -algebras in compact Hausdorff or locally compact Hausdorff settings. (However, in Polish spaces it seems Borel -algebras remain the more natural -algebra to use.) It turns out that the two Baire -algebras can be divided up naturally between the two categories of locally compact Hausdorff spaces, with the -Baire -algebra naturally associated to and the -algebra naturally associated to . The situation can be summarized by the following commuting diagram of functors between the various categories of (locally) compact Hausdorff spaces and the category of concrete measurable spaces (sets equipped with a -algebra):

To each category of (locally) compact Hausdorff spaces, we can then define an associated category of (locally) compact Hausdorff spaces equipped with a Radon probability measure, where the underlying -algebra is as described above, and “Radon” means “compact -inner regular”. For instance, is the category of compact Hausdorff spaces equipped with a Baire probability measure with the inner regularity property

and one similarly defines , , . (In the compact Hausdorff case the inner regularity property is in fact automatic, but in locally compact Hausdorff categories it needs to be imposed as an explicit hypothesis.) On the -algebra side, categories such as can be augmented to tracial categories where the algebra is now equipped with a trace , that is to say a non-negative linear functional of operator norm . It then turns out that the Gelfand spectrum functors from -algebras to (locally) compact Hausdorff spaces can be augmented to “Riesz functors” from tracial -algebras to (locally) compact Hausdorff spaces with Radon probability measures, where the probability measure in question is given by some form of the Riesz representation theorem; dually, the functors can similarly be augmented using the Lebesgue integral with respect to the given measure as the trace. This leads one to a complete analogue of the previous diagram of functors, but now in tracial and probabilistic categories, giving a new set of Gelfand-like dualities that we call “Riesz duality”:

Again, each component of this diagram was essentially already in the literature in either explicit or implicit form. In the paper we also review the Riesz representation theory for traces on rather than , in which an additional “-smoothness” property is needed in order to recover a Radon probability measure on . The distinction between Baire and Borel -algebras ends up being largely elided in the Riesz representation theory, as it turns out that every Baire-Radon probability measure (using either of the two Baire algebras) on a locally compact Hausdorff has a unique extension to a Borel-Radon probability measure (where for Borel measures, the Radon property is now interpreted as “compact inner regular” rather than “compact inner regular”).

** — 3. The canonical model of opposite probability algebras — **

Given a concrete probability space – a set equipped with a -algebra and a probability measure – one can form the associated *probability algebra* by quotienting the -algebra by the null ideal and then similarly quotienting the measure to obtain an abstract countably additive probability measure . More generally, one can consider abstract probability algebras where is a -complete Boolean algebra and is a countably additive probability measure with the property that whenever . This gives a category . Actually to align things closer to the category of concrete probability spaces , it is better to work with the opposite category of opposite probability algebras where the directions of the morphisms are reversed.

Many problems and concepts in ergodic theory are best phrased “up to null sets”, which can be interpreted category-theoretically by applying a suitable functor from a concrete category such as to the opposite probability algebra category to remove all the null sets. However, it is sometimes convenient to reverse this process and *model* an opposite probability algebra by a more concrete probability space, and to also model morphisms between opposite probability algebras (-complete Boolean homomorphisms from to that preserve measure) by concrete maps. Ideally the model spaces should also be compact Hausdorff spaces, and the model morphisms continuous maps, so that methods from topological dynamics may be applied. There are various *ad hoc* ways to create such models in “countable” settings, but it turns out that there is a canonical and “universal” model of any opposite probability algebra that one can construct in a completely functorial setting (so that any dynamics on the opposite probability algebra automatically carry over to concrete dynamics on ). The quickest way to define this model is by the formula

- (Topological structure) is a compact Hausdorff space. In fact it is a Stone space with the additional property that every Baire-measurable set is equal to a clopen set modulo a Baire-meager set (such spaces we call -spaces in our paper). Furthermore a Baire set is null if and only if it is meager. Any opposite probability algebra morphism gives rise (in a functorial fashion) to a surjective continuous map with the additional property that the inverse image of Baire-meager sets are Baire-meager. Furthermore, one has the (somewhat surprising) “strong Lusin property”: every element of has a unique representative in (thus, every bounded measurable function on is equal almost everywhere to a unique continuous function).
- (Concrete model) The opposite probability algebra of is (naturally) isomorphic to .
- (Universality) There is a natural inclusion (in the category of abstract probability spaces) from (which is interpreted in ) into , which is universal amongst all inclusions of into compact Hausdorff probability spaces.
- (Canonical extension) Every abstractly measurable map from to a compact Hausdorff space has a unique extension to a continuous map from to .

This canonical model seems quite analogous to the Stone-Cech compactification . For instance the analogue of the canonical extension property for is that every continuous map from a locally compact space to a compact Hausdorff space has a unique extension to a continuous map from to . In both cases the model produced is usually too “large” to be separable, so this is a tool that is only available in “uncountable” frameworks.

There is an alternate description of the canonical model, which is basically due to Fremlin, and is based on Stone-type dualities rather than Riesz type dualities. Namely, one starts with the -complete Boolean algebra attached to the opposite probability space , and constructs its Stone dual, which is a space. Every Baire set in this space is equal modulo Baire-meager sets to a clopen set, which by Stone duality is identifiable with an element of the probability algebra. The measure on then induces a measure on the -space, and this compact Hausdorff probability space can serve as the canonical model . The functoriality of this construction is closely tied to the functoriality of the Loomis-Sikorski theorem (discussed in this previous blog post). The precise connection in terms of functors and categories is a little complicated to describe, though, as the following diagram indicates:

One quick use of the canonical model is that it allows one to perform a variety of constructions on opposite probability algebras by passing to the canonical model and performing the analogous operation there. For instance, if one wants to construct a product of some number of opposite probability algebras, one can first take the product of the concrete models (as a compact Hausdorff probability space), then extract the opposite probability algebra of that space. We will similarly use this model to construct group skew-products and homogeneous skew-products in a later paper.

** — 4. Canonical disintegration — **

Given a probability space and a probability-preserving factor map from to another probability space , it is often convenient to look for a *disintegration* of into fibre measures with the property that the conditional expectation of a function (defined as the orthogonal projection in to , viewed as a subspace in ) is given pointwise almost everywhere by

Among other things, the canonical disintegration makes it easy to construct relative products of opposite probability algebras, and we believe it will also be of use in developing uncountable versions of Host-Kra structure theory.

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In this case the theorem in question is the Mackey-Zimmer theorem, previously discussed in this blog post. This theorem gives an important classification of group and homogeneous extensions of measure-preserving systems. Let us first work in the (classical) setting of concrete measure-preserving systems. Let be a measure-preserving system for some group , thus is a (concrete) probability space and is a group homomorphism from to the automorphism group of the probability space. (Here we are abusing notation by using to refer both to the measure-preserving system and to the underlying set. In the notation of the paper we would instead distinguish these two objects as and respectively, reflecting two of the (many) categories one might wish to view as a member of, but for sake of this informal overview we will not maintain such precise distinctions.) If is a compact group, we define a *(concrete) cocycle* to be a collection of measurable functions for that obey the *cocycle equation*

- is the Cartesian product of and ;
- is the product measure of and Haar probability measure on ; and
- The action is given by the formula

This group skew-product comes with a factor map and a coordinate map , which by (2) are related to the action via the identities

and where in (4) we are implicitly working in the group of (concretely) measurable functions from to . Furthermore, the combined map is measure-preserving (using the product measure on ), indeed the way we have constructed things this map is just the identity map.
We can now generalize the notion of group skew-product by just working with the maps , and weakening the requirement that be measure-preserving. Namely, define a *group extension* of by to be a measure-preserving system equipped with a measure-preserving map obeying (3) and a measurable map obeying (4) for some cocycle , such that the -algebra of is generated by . There is also a more general notion of a *homogeneous extension* in which takes values in rather than . Then every group skew-product is a group extension of by , but not conversely. Here are some key counterexamples:

- (i) If is a closed subgroup of , and is a cocycle taking values in , then can be viewed as a group extension of by , taking to be the vertical coordinate (viewing now as an element of ). This will not be a skew-product by because pushes forward to the wrong measure on : it pushes forward to rather than .
- (ii) If one takes the same example as (i), but twists the vertical coordinate to another vertical coordinate for some measurable “gauge function” , then is still a group extension by , but now with the cocycle replaced by the cohomologous cocycle Again, this will not be a skew product by , because pushes forward to a twisted version of that is supported (at least in the case where is compact and the cocycle is continuous) on the -bundle .
- (iii) With the situation as in (i), take to be the union for some outside of , where we continue to use the action (2) and the standard vertical coordinate but now use the measure .

As it turns out, group extensions and homogeneous extensions arise naturally in the Furstenberg-Zimmer structural theory of measure-preserving systems; roughly speaking, every compact extension of is an inverse limit of group extensions. It is then of interest to classify such extensions.

Examples such as (iii) are annoying, but they can be excluded by imposing the additional condition that the system is ergodic – all invariant (or essentially invariant) sets are of measure zero or measure one. (An essentially invariant set is a measurable subset of such that is equal modulo null sets to for all .) For instance, the system in (iii) is non-ergodic because the set (or ) is invariant but has measure . We then have the following fundamental result of Mackey and Zimmer:

Theorem 1 (Countable Mackey Zimmer theorem)Let be a group, be a concrete measure-preserving system, and be a compact Hausdorff group. Assume that is at most countable, is a standard Borel space, and is metrizable. Then every (concrete) ergodic group extension of is abstractly isomorphic to a group skew-product (by some closed subgroup of ), and every (concrete) ergodic homogeneous extension of is similarly abstractly isomorphic to a homogeneous skew-product.

We will not define precisely what “abstractly isomorphic” means here, but it roughly speaking means “isomorphic after quotienting out the null sets”. A proof of this theorem can be found for instance in .

The main result of this paper is to remove the “countability” hypotheses from the above theorem, at the cost of working with opposite probability algebra systems rather than concrete systems. (We will discuss opposite probability algebras in a subsequent blog post relating to another paper in this series.)

Theorem 2 (Uncountable Mackey Zimmer theorem)Let be a group, be an opposite probability algebra measure-preserving system, and be a compact Hausdorff group. Then every (abstract) ergodic group extension of is abstractly isomorphic to a group skew-product (by some closed subgroup of ), and every (abstract) ergodic homogeneous extension of is similarly abstractly isomorphic to a homogeneous skew-product.

We plan to use this result in future work to obtain uncountable versions of the Furstenberg-Zimmer and Host-Kra structure theorems.

As one might expect, one locates a proof of Theorem 2 by finding a proof of Theorem 1 that does not rely too strongly on “countable” tools, such as disintegration or measurable selection, so that all of those tools can be replaced by “uncountable” counterparts. The proof we use is based on the one given in this previous post, and begins by comparing the system with the group extension . As the examples (i), (ii) show, these two systems need not be isomorphic even in the ergodic case, due to the different probability measures employed. However one can relate the two after performing an additional averaging in . More precisely, there is a canonical factor map given by the formula

This is a factor map not only of -systems, but actually of -systems, where the opposite group to acts (on the left) by right-multiplication of the second coordinate (this reversal of order is why we need to use the opposite group here). The key point is that the ergodicity properties of the system are closely tied the group that is “secretly” controlling the group extension. Indeed, in example (i), the invariant functions on take the form for some measurable , while in example (ii), the invariant functions on take the form . In either case, the invariant factor is isomorphic to , and can be viewed as a factor of the invariant factor of , which is isomorphic to . Pursuing this reasoning (using an abstract ergodic theorem of Alaoglu and Birkhoff, as discussed in the previous post) one obtains the]]>

- Reducing qualitative analysis results (e.g., convergence theorems or dimension bounds) to quantitative analysis estimates (e.g., variational inequalities or maximal function estimates).
- Using dyadic pigeonholing to locate good scales to work in or to apply truncations.
- Using random translations to amplify small sets (low density) into large sets (positive density).
- Combining large deviation inequalities with metric entropy bounds to control suprema of various random processes.

Each of these techniques is individually not too difficult to explain, and were certainly employed on occasion by various mathematicians prior to Bourgain’s work; but Jean had internalized them to the point where he would instinctively use them as soon as they became relevant to a given problem at hand. I illustrate this at the end of the paper with an exposition of one particular result of Jean, on the Erdős similarity problem, in which his main result (that any sum of three infinite sets of reals has the property that there exists a positive measure set that does not contain any homothetic copy of ) is basically proven by a sequential application of these tools (except for dyadic pigeonholing, which turns out not to be needed here).

I had initially intended to also cover some other basic tools in Jean’s toolkit, such as the uncertainty principle and the use of probabilistic decoupling, but was having trouble keeping the paper coherent with such a broad focus (certainly I could not identify a single paper of Jean’s that employed all of these tools at once). I hope though that the examples given in the paper gives some reasonable impression of Jean’s research style.

]]>Vaughan and I grew up in extremely culturally similar countries, worked in adjacent areas of mathematics, shared (as of this week) a coauthor in Dima Shylakhtenko, started out our career with the same postdoc position (as UCLA Hedrick Assistant Professors, sixteen years apart) and even ended up in sister campuses of the University of California, but surprisingly we only interacted occasionally, via chance meetings at conferences or emails on some committee business. I found him extremely easy to get along with when we did meet, though, perhaps because of our similar cultural upbringing.

I have not had much occasion to directly use much of Vaughan’s mathematical contributions, but I did very much enjoy reading his influential 1999 preprint on planar algebras (which, for some odd reason has never been formally published). Traditional algebra notation is one-dimensional in nature, with algebraic expressions being described by strings of mathematical symbols; a linear operator , for instance, might appear in the middle of such a string, taking in an input on the right and returning an output on its left that might then be fed into some other operation. There are a few mathematical notations which are two-dimensional, such as the commutative diagrams in homological algebra, the tree expansions of solutions to nonlinear PDE (particularly stochastic nonlinear PDE), or the Feynman diagrams and Penrose graphical notations from physics, but these are the exception rather than the rule, and the notation is often still concentrated on a one-dimensional complex of vertices and edges (or arrows) in the plane. Planar algebras, by contrast, fully exploit the topological nature of the plane; a planar “operator” (or “operad”) inhabits some punctured region of the plane, such as an annulus, with “inputs” entering from the inner boundaries of the region and “outputs” emerging from the outer boundary. These algebras arose for Vaughan in both operator theory and knot theory, and have since been used in some other areas of mathematics such as representation theory and homology. I myself have not found a direct use for this type of algebra in my own work, but nevertheless I found the mere possibility of higher dimensional notation being the natural choice for a given mathematical problem to be conceptually liberating.

]]>A basic motivating example is the question of counting the number of incidences between points and lines (or between points and other geometric objects). Suppose one has points and lines in a space. How many incidences can there be between these points and lines? The utterly trivial bound is , but by using the basic fact that two points determine a line (or two lines intersect in at most one point), a simple application of Cauchy-Schwarz improves this bound to . In graph theoretic terms, the point is that the bipartite incidence graph between points and lines does not contain a copy of (there does not exist two points and two lines that are all incident to each other). Without any other further hypotheses, this bound is basically sharp: consider for instance the collection of points and lines in a finite plane , that has incidences (one can make the situation more symmetric by working with a projective plane rather than an affine plane). If however one considers lines in the real plane , the famous Szemerédi-Trotter theorem improves the incidence bound further from to . Thus the incidence graph between real points and lines contains more structure than merely the absence of .

More generally, bounding on the size of bipartite graphs (or multipartite hypergraphs) not containing a copy of some complete bipartite subgraph (or in the hypergraph case) is known as *Zarankiewicz’s problem*. We have results for all and all orders of hypergraph, but for sake of this post I will focus on the bipartite case.

In our paper we improve the bound to a near-linear bound in the case that the incidence graph is “semilinear”. A model case occurs when one considers incidences between points and axis-parallel rectangles in the plane. Now the condition is not automatic (it is of course possible for two distinct points to both lie in two distinct rectangles), so we impose this condition by *fiat*:

Theorem 1Suppose one has points and axis-parallel rectangles in the plane, whose incidence graph contains no ‘s, for some large .

- (i) The total number of incidences is .
- (ii) If all the rectangles are dyadic, the bound can be improved to .
- (iii) The bound in (ii) is best possible (up to the choice of implied constant).

We don’t know whether the bound in (i) is similarly tight for non-dyadic boxes; the usual tricks for reducing the non-dyadic case to the dyadic case strangely fail to apply here. One can generalise to higher dimensions, replacing rectangles by polytopes with faces in some fixed finite set of orientations, at the cost of adding several more logarithmic factors; also, one can replace the reals by other ordered division rings, and replace polytopes by other sets of bounded “semilinear descriptive complexity”, e.g., unions of boundedly many polytopes, or which are cut out by boundedly many functions that enjoy coordinatewise monotonicity properties. For certain specific graphs we can remove the logarithmic factors entirely. We refer to the preprint for precise details.

The proof techniques are combinatorial. The proof of (i) relies primarily on the order structure of to implement a “divide and conquer” strategy in which one can efficiently control incidences between points and rectangles by incidences between approximately points and boxes. For (ii) there is additional order-theoretic structure one can work with: first there is an easy pruning device to reduce to the case when no rectangle is completely contained inside another, and then one can impose the “tile partial order” in which one dyadic rectangle is less than another if and . The point is that this order is “locally linear” in the sense that for any two dyadic rectangles , the set is linearly ordered, and this can be exploited by elementary double counting arguments to obtain a bound which eventually becomes after optimising certain parameters in the argument. The proof also suggests how to construct the counterexample in (iii), which is achieved by an elementary iterative construction.

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