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Much as group theory is the study of groups, or graph theory is the study of graphs, model theory is the study of *models* (also known as structures) of some language (which, in this post, will always be a single-sorted, first-order language). A structure is a set , equipped with one or more operations, constants, and relations. This is of course an extremely general type of mathematical object, but (quite remarkably) one can still say a substantial number of interesting things about very broad classes of structures.

We will observe the common abuse of notation of using the set as a metonym for the entire structure, much as we usually refer to a group simply as , a vector space simply as , and so forth. Following another common bending of the rules, we also allow some operations on structures (such as the multiplicative inverse operation on a group or field) to only be partially defined, and we allow use of the usual simplifying conventions for mathematical formulas (e.g. writing instead of or , in cases where associativity is known). We will also deviate slightly from the usual practice in logic by emphasising individual structures, rather than the theory of general classes of structures; for instance, we will talk about the theory of a single field such as or , rather than the theory of all fields of a certain type (e.g. real closed fields or algebraically closed fields).

Once one has a structure , one can introduce the notion of a definable subset of , or more generally of a Cartesian power of , defined as a set of the form

for some formula in the language with free variables and any number of constants from (that is, is a well-formed formula built up from a finite number of constants in , the relations and operations on , logical connectives such as , , , and the quantifiers ). Thus, for instance, in the theory of the arithmetic of the natural numbers , the set of primes is a definable set, since we have

In the theory of the field of reals , the unit circle is an example of a definable set,

but so is the the complement of the circle,

and the interval :

Due to the unlimited use of constants, any finite subset of a power of any structure is, by our conventions, definable in that structure. (One can of course also consider definability without parameters (also known as -definability), in which arbitrary constants are not permitted, but we will not do so here.)

We can isolate some special subclasses of definable sets:

- An
*atomic definable set*is a set of the form (1) in which is an atomic formula (i.e. it does not contain any logical connectives or quantifiers). - A
*quantifier-free definable set*is a set of the form (1) in which is quantifier-free (i.e. it can contain logical connectives, but does not contain the quantifiers ).

Example 1In the theory of a field such as , an atomic definable set is the same thing as an affine algebraic set (also known as anaffine algebraic variety, with the understanding that varieties are not necessarily assumed to be irreducible), and a quantifier-free definable set is known as aconstructible set; thus we see that algebraic geometry can be viewed in some sense as a special case of model theory. (Conversely, it can in fact be quite profitable to think of model theory as an abstraction of algebraic geometry; for instance, the concepts of Morley rank and Morley degree in model theory (discussed in this previous blog post) directly generalises the concepts of dimension and degree in algebraic geometry.) Over , the interval is a definable set, but not a quantifier-free definable set (and certainly not an atomic definable set); and similarly for the primes over .

A quantifier-free definable set in is nothing more than a finite boolean combination of atomic definable sets; in other words, the class of quantifier-free definable sets over is the smallest class that contains the atomic definable sets and is closed under boolean operations such as complementation and union (which generate all the other boolean operations). Similarly, the class of definable sets over is the smallest class that contains the quantifier-free definable sets, and is also closed under the operation of *projection* from to for every natural number , where is the map .

Some structures have the property of enjoying quantifier elimination, which means that every definable set is in fact a quantifier-free definable set, or equivalently that the projection of a quantifier-free definable set is again quantifier-free. For instance, an algebraically closed field (with the field operations) has quantifier elimination (i.e. the projection of a constructible set is again constructible); this fact can be proven by the classical tool of resultants, and among other things can be used to give a proof of Hilbert’s nullstellensatz. (Note though that projection does not necessary preserve the property of being atomic; for instance, the projection of the atomic set is the non-atomic, but still quantifier-free definable, set .) In the converse direction, it is not difficult to use the nullstellensatz to deduce quantifier elimination. For theory of the real field , which is not algebraically closed, one does not have quantifier elimination, as one can see from the example of the unit circle (which is a quantifier-free definable set) projecting down to the interval (which is definable, but not quantifer-free definable). However, if one adds the additional operation of order to the reals, giving it the language of an ordered field rather than just a field, then quantifier elimination is recovered (the class of quantifier-free definable sets now enlarges to match the class of definable sets, which in this case is also the class of semi-algebraic sets); this is the famous Tarski-Seidenberg theorem.

On the other hand, many important structures do not have quantifier elimination; typically, the projection of a quantifier-free definable set is not, in general, quantifier-free definable. This failure of the projection property also shows up in many contexts outside of model theory; for instance, Lebesgue famously made the error of thinking that the projection of a Borel measurable set remained Borel measurable (it is merely an analytic set instead). Turing’s halting theorem can be viewed as an assertion that the projection of a decidable set (also known as a *computable* or *recursive* set) is not necessarily decidable (it is merely *semi-decidable* (or *recursively enumerable*) instead). The notorious P=NP problem can also be essentially viewed in this spirit; roughly speaking (and glossing over the placement of some quantifiers), it asks whether the projection of a polynomial-time decidable set is again polynomial-time decidable. And so forth. (See this blog post of Dick Lipton for further discussion of the subtleties of projections.)

Now we consider the status of quantifier elimination for the theory of a finite field . If interpreted naively, quantifier elimination is trivial for a finite field , since every subset of is finite and thus quantifier-free definable. However, we can recover an interesting question in one of two (essentially equivalent) ways. One is to work in the asymptotic regime in which the field is large, but the length of the formulae used to construct one’s definable sets stays bounded uniformly in the size of (where we view any constant in as contributing a unit amount to the length of a formula, no matter how large is). A simple counting argument then shows that only a small number of subsets of become definable in the asymptotic limit , since the number of definable sets clearly grows at most polynomially in for any fixed bound on the formula length, while the number of all subsets of grows exponentially in .

Another way to proceed is to work not with a single finite field , or even with a sequence of finite fields, but with the ultraproduct of a sequence of finite fields, and to study the properties of definable sets over this ultraproduct. (We will be using the notation of ultraproducts and nonstandard analysis from this previous blog post.) This approach is equivalent to the more finitary approach mentioned in the previous paragraph, at least if one does not care to track of the exact bounds on the length of the formulae involved. Indeed, thanks to Los’s theorem, a definable subset of is nothing more than the ultraproduct of definable subsets of for all sufficiently close to , with the length of the formulae used to define uniformly bounded in . In the language of nonstandard analysis, one can view as a nonstandard finite field.

The ultraproduct of finite fields is an important example of a *pseudo-finite field* – a field that obeys all the sentences in the languages of fields that finite fields do, but is not necessarily itself a finite field. The model theory of pseudo-finite fields was first studied systematically by Ax (in the same paper where the Ax-Grothendieck theorem, discussed previously on this blog, was established), with important further contributions by Kiefe, by Fried-Sacerdote, by two papers of Chatzidakis-van den Dries-Macintyre, and many other authors.

As mentioned before, quantifier elimination trivially holds for finite fields. But for infinite pseudo-finite fields, such as the ultraproduct of finite fields with going to infinity, quantifier elimination fails. For instance, in a finite field , the set of quadratic residues is a definable set, with a bounded formula length, and so in the ultraproduct , the set of nonstandard quadratic residues is also a definable set. However, in one dimension, we see from the factor theorem that the only atomic definable sets are either finite or the whole field , and so the only constructible sets (i.e. the only quantifier-free definable sets) are either finite or cofinite in . Since the quadratic residues have asymptotic density in a large finite field, they cannot form a quantifier-free definable set, despite being definable.

Nevertheless, there is a very nice *almost quantifier elimination* result for these fields, in characteristic zero at least, which we phrase here as follows:

Theorem 1 (Almost quantifier elimination)Let be a nonstandard finite field of characteristic zero, and let be a definable set over . Then is the union of finitely many sets of the formwhere is an atomic definable subset of (i.e. the -points of an algebraic variety defined over in ) and is a polynomial.

Results of this type were first obtained essentially due to Catarina Kiefe, although the formulation here is closer to that of Chatzidakis-van den Dries-Macintyre.

Informally, this theorem says that while we cannot quite eliminate *all* quantifiers from a definable set over a nonstandard finite field, we can eliminate all but one existential quantifier. Note that negation has also been eliminated in this theorem; for instance, the definable set uses a negation, but can also be described using a single existential quantifier as .) I believe that there are more complicated analogues of this result in positive characteristic, but I have not studied this case in detail (Kiefe’s result does not assume characteristic zero, but her conclusion is slightly different from the one given here). In the one-dimensional case , the only varieties are the affine line and finite sets, and we can simplify the above statement, namely that any definable subset of takes the form for some polynomial (i.e. definable sets in are nothing more than the projections of the -points of a plane curve).

There is an equivalent formulation of this theorem for standard finite fields, namely that if is a finite field and is definable using a formula of length at most , then can be expressed in the form (2) with the degree of bounded by some quantity depending on and , assuming that the characteristic of is sufficiently large depending on .

The theorem gives quite a satisfactory description of definable sets in either standard or nonstandard finite fields (at least if one does not care about effective bounds in some of the constants, and if one is willing to exclude the small characteristic case); for instance, in conjunction with the Lang-Weil bound discussed in this recent blog post, it shows that any non-empty definable subset of a nonstandard finite field has a nonstandard cardinality of for some positive standard rational and integer . Equivalently, any non-empty definable subset of for some standard finite field using a formula of length at most has a standard cardinality of for some positive rational of height and some natural number between and . (For instance, in the example of the quadratic residues given above, is equal to and equal to .) There is a more precise statement to this effect, namely that the Poincaré series of a definable set is rational; see Kiefe’s paper for details.

Below the fold I give a proof of Theorem 1, which relies primarily on the Lang-Weil bound mentioned above.

This week, Henry Towsner concluded his portion of reading seminar of the Hrushovski paper, by discussing (a weaker, simplified version of) main model-theoretic theorem (Theorem 3.4 of Hrushovski), and described how this theorem implied the combinatorial application in Corollary 1.2 of Hrushovski. The presentation here differs slightly from that in Hrushovski’s paper, for instance by avoiding mention of the more general notions of S1 ideals and forking.

Here is a collection of resources so far on the Hrushovski paper:

- Henry Towsner’s notes (which most of Notes 2-4 have been based on);
- Alex Usvyatsov’s notes on the derivation of Corollary 1.2 (broadly parallel to the notes here);
- Lou van den Dries’ notes (covering most of what we have done so far, and also material on stable theories); and
- Anand Pillay’s sketch of a simplified proof of Theorem 1.1.

One of my favorite open problems, which I have blogged about in the past, is that of establishing (or even correctly formulating) a non-commutative analogue of Freiman’s theorem. Roughly speaking, the question is this: given a finite set in a non-commutative group which is of *small doubling* in the sense that the product set is not much larger than (e.g. for some ), what does this say about the structure of ? (For various technical reasons one may wish to replace small doubling by, say, small tripling (i.e. ), and one may also wish to assume that contains the identity and is symmetric, , but these are relatively minor details.)

Sets of small doubling (or tripling), etc. can be thought of as “approximate groups”, since groups themselves have a doubling constant equal to one. Another obvious example of an approximate group is that of an arithmetic progression in an additive group, and more generally of a ball (in the word metric) in a nilpotent group of bounded rank and step. It is tentatively conjectured that in fact all examples can somehow be “generated” out of these basic examples, although it is not fully clear at present what “generated” should mean.

A weaker conjecture along the same lines is that if is a set of small doubling, then there should be some sort of “pseudo-metric” on which is left-invariant, and for which is controlled (in some suitable sense) by the unit ball in this metric. (For instance, if was a subgroup of , one would take the metric which identified all the left cosets of to a point, but was otherwise a discrete metric; if were a ball in a nilpotent group, one would use some rescaled version of the word metric, and so forth.) Actually for technical reasons one would like to work with a slightly weaker notion than a pseudo-metric, namely a *Bourgain system*, but let us again ignore this technicality here.

Recently, using some powerful tools from model theory combined with the theory of topological groups, Ehud Hrushovski has apparently achieved some breakthroughs on this problem, obtaining new structural control on sets of small doubling in arbitrary groups that was not previously accessible to the known combinatorial methods. The precise results are technical to state, but here are informal versions of two typical theorems. The first applies to sets of small tripling in an arbitrary group:

Theorem 1(Rough version of Hrushovski Theorem 1.1) Let be a set of small tripling, then one can find a long sequence of nested symmetric sets , all of size comparable to and contained in , which are somewhat closed under multiplication in the sense that for all , and which are fairly well closed under commutation in the sense that . (There are also some additional statements to the effect that the efficiently cover each other, and also cover , but I will omit those here.)

This nested sequence is somewhat analogous to a Bourgain system, though it is not quite the same notion.

If one assumes that is “perfect” in a certain sense, which roughly means that there is no non-trivial abelian quotient, then one can do significantly better:

Theorem 2(Rough version of Hrushovski Corollary 1.2) Let be a set of small tripling, let , and suppose that for almost all -tuples (where ), the conjugacy classes generate most of in the sense that . Then a large part of is contained in a subgroup of size comparable to .

Note that if one quotiented out by the commutator , then all of the conjugacy classes would collapse to points. So the hypothesis here is basically a strong quantitative assertion to the effect that the commutator is extremely large, and rapidly fills out most of itself.

Here at UCLA, a group of logicians and I (consisting of Matthias Aschenbrenner, Isaac Goldbring, Greg Hjorth, Henry Towsner, Anush Tserunyan, and possibly others) have just started a weekly reading seminar to come to grips with the various combinatorial, logical, and group-theoretic notions in Hrushovski’s paper, of which we only have a partial understanding at present. The seminar is a physical one, rather than an online one, but I am going to try to put some notes on the seminar on this blog as it progresses, as I know that there are a couple of other mathematicians who are interested in these developments.

So far there have been two meetings of the seminar. In the first, I surveyed the state of knowledge of the noncommutative Freiman theorem, covering broadly the material in my previous blog post. In the second meeting, Isaac reviewed some key notions of model theory used in Hrushovski’s paper, in particular the notions of definability and type, which I will review below. It is not yet clear how these are going to be connected with the combinatorial side of things, but this is something which we will hopefully develop in future seminars. The near-term objective is to understand the statement of the main theorem on the model-theoretic side (Theorem 3.4 of Hrushovski), and then understand some of its easier combinatorial consequences, before going back and trying to understand the *proof* of that theorem.

[*Update*, Oct 19: Given the level of interest in this paper, readers are encouraged to discuss any aspect of that paper in the comments below, even if they are not currently being covered by the UCLA seminar.]

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