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One of the basic objects of study in combinatorics are finite strings or infinite strings of symbols from some given alphabet , which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set of natural numbers can be identified with the infinite string of s and s formed by the indicator of , e.g. the even numbers can be identified with the string from the alphabet , the multiples of three can be identified with the string , and so forth. One can also consider doubly infinite strings , which among other things can be used to describe arbitrary subsets of integers.
On the other hand, the basic object of study in dynamics (and in related fields, such as ergodic theory) is that of a dynamical system , that is to say a space together with a shift map (which is often assumed to be invertible, although one can certainly study non-invertible dynamical systems as well). One often adds additional structure to this dynamical system, such as topological structure (giving rise topological dynamics), measure-theoretic structure (giving rise to ergodic theory), complex structure (giving rise to complex dynamics), and so forth. A dynamical system gives rise to an action of the natural numbers on the space by using the iterates of for ; if is invertible, we can extend this action to an action of the integers on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than or (e.g. one can consider continuous dynamical systems in which the evolution group is ), but we will restrict attention to the classical situation of or actions here.
There is a fundamental correspondence principle connecting the study of strings (or subsets of natural numbers or integers) with the study of dynamical systems. In one direction, given a dynamical system , an observable taking values in some alphabet , and some initial datum , we can first form the forward orbit of , and then observe this orbit using to obtain an infinite string . If the shift in this system is invertible, one can extend this infinite string into a doubly infinite string . Thus we see that every quadruplet consisting of a dynamical system , an observable , and an initial datum creates an infinite string.
Example 1 If is the three-element set with the shift map , is the observable that takes the value at the residue class and zero at the other two classes, and one starts with the initial datum , then the observed string becomes the indicator of the multiples of three.
In the converse direction, every infinite string in some alphabet arises (in a decidedly non-unique fashion) from a quadruple in the above fashion. This can be easily seen by the following “universal” construction: take to be the set of infinite strings in the alphabet , let be the shift map
let be the observable
and let be the initial point
Then one easily sees that the observed string is nothing more than the original string . Note also that this construction can easily be adapted to doubly infinite strings by using instead of , at which point the shift map now becomes invertible. An important variant of this construction also attaches an invariant probability measure to that is associated to the limiting density of various sets associated to the string , and leads to the Furstenberg correspondence principle, discussed for instance in these previous blog posts. Such principles allow one to rigorously pass back and forth between the combinatorics of strings and the dynamics of systems; for instance, Furstenberg famously used his correspondence principle to demonstrate the equivalence of Szemerédi’s theorem on arithmetic progressions with what is now known as the Furstenberg multiple recurrence theorem in ergodic theory.
In the case when the alphabet is the binary alphabet , and (for technical reasons related to the infamous non-injectivity of the decimal representation system) the string does not end with an infinite string of s, then one can reformulate the above universal construction by taking to be the interval , to be the doubling map , to be the observable that takes the value on and on (that is, is the first binary digit of ), and is the real number (that is, in binary).
The above universal construction is very easy to describe, and is well suited for “generic” strings that have no further obvious structure to them, but it often leads to dynamical systems that are much larger and more complicated than is actually needed to produce the desired string , and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space and a dynamics which does not obviously reflect the key features of the sequence such as its periodicity. (Using the unit interval model, the dynamics arise from the orbit of under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)
A related aesthetic objection to the universal construction is that of the four components of the quadruplet used to generate the sequence , three of the components are completely universal (in that they do not depend at all on the sequence ), leaving only the initial datum to carry all the distinctive features of the original sequence. While there is nothing wrong with this mathematically, from a conceptual point of view it would make sense to make all four components of the quadruplet to be adapted to the sequence, in order to take advantage of the accumulated intuition about various special dynamical systems (and special observables), not just special initial data.
One step in this direction can be made by restricting to the orbit of the initial datum (actually for technical reasons it is better to restrict to the topological closure of this orbit, in order to keep compact). For instance, starting with the sequence , the orbit now consists of just three points , , , bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet , because any other such representation is a factor of this representation (in the sense that there is a unique map with , , and ). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system are interpreted as infinite strings rather than elements of a more geometrically or algebraically rich object (e.g. points in a circle, torus, or other homogeneous space).
For general sequences , locating relevant geometric or algebraic structure in a dynamical system generating that sequence is an important but very difficult task (see e.g. this paper of Host and Kra, which is more or less devoted to precisely this task in the context of working out what component of a dynamical system controls the multiple recurrence behaviour of that system). However, for specific examples of sequences , one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple that generates that sequence. This is not a particularly difficult or deep operation, but I found it very helpful in internalising the intuition behind the correspondence principle. Being non-rigorous, this procedure does not seem to be emphasised in most presentations of the correspondence principle, so I thought I would describe it here.
The rectification principle in arithmetic combinatorics asserts, roughly speaking, that very small subsets (or, alternatively, small structured subsets) of an additive group or a field of large characteristic can be modeled (for the purposes of arithmetic combinatorics) by subsets of a group or field of zero characteristic, such as the integers or the complex numbers . The additive form of this principle is known as the Freiman rectification principle; it has several formulations, going back of course to the original work of Freiman. Here is one formulation as given by Bilu, Lev, and Ruzsa:
Proposition 1 (Additive rectification) Let be a subset of the additive group for some prime , and let be an integer. Suppose that . Then there exists a map into a subset of the integers which is a Freiman isomorphism of order in the sense that for any , one has
if and only if
Furthermore is a right-inverse of the obvious projection homomorphism from to .
The original version of the rectification principle allowed the sets involved to be substantially larger in size (cardinality up to a small constant multiple of ), but with the additional hypothesis of bounded doubling involved; see the above-mentioned papers, as well as this later paper of Green and Ruzsa, for further discussion.
The proof of Proposition 1 is quite short (see Theorem 3.1 of Bilu-Lev-Ruzsa); the main idea is to use Minkowski’s theorem to find a non-trivial dilate of that is contained in a small neighbourhood of the origin in , at which point the rectification map can be constructed by hand.
Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map preserves both additive and multiplicative structure:
Theorem 2 (Arithmetic rectification) Let be a subset of the finite field for some prime , and let be an integer. Suppose that . Then there exists a map into a subset of the complex numbers which is a Freiman field isomorphism of order in the sense that for any and any polynomial of degree at most and integer coefficients of magnitude summing to at most , one has
if and only if
Note that it is necessary to use an algebraically closed field such as for this theorem, in contrast to the integers used in Proposition 1, as can contain objects such as square roots of which can only map to in the complex numbers (once is at least ).
Using Theorem 2, one can transfer results in arithmetic combinatorics (e.g. sum-product or Szemerédi-Trotter type theorems) regarding finite subsets of to analogous results regarding sufficiently small subsets of ; see the paper of Grosu for several examples of this. This should be compared with the paper of Vu, Wood, and Wood, which introduces a converse principle that embeds finite subsets of (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of for arbitrarily large primes , allowing one to transfer arithmetic combinatorical facts from the latter setting to the former.
Grosu’s argument uses some quantitative elimination theory, and in particular a quantitative variant of a lemma of Chang that was discussed previously on this blog. In that previous blog post, it was observed that (an ineffective version of) Chang’s theorem could be obtained using only qualitative algebraic geometry (as opposed to quantitative algebraic geometry tools such as elimination theory results with explicit bounds) by means of nonstandard analysis (or, in what amounts to essentially the same thing in this context, the use of ultraproducts). One can then ask whether one can similarly establish an ineffective version of Grosu’s result by nonstandard means. The purpose of this post is to record that this can indeed be done without much difficulty, though the result obtained, being ineffective, is somewhat weaker than that in Theorem 2. More precisely, we obtain
Theorem 3 (Ineffective arithmetic rectification) Let . Then if is a field of characteristic at least for some depending on , and is a subset of of cardinality , then there exists a map into a subset of the complex numbers which is a Freiman field isomorphism of order .
Our arguments will not provide any effective bound on the quantity (though one could in principle eventually extract such a bound by deconstructing the proof of Proposition 4 below), making this result weaker than Theorem 2 (save for the minor generalisation that it can handle fields of prime power order as well as fields of prime order as long as the characteristic remains large).
Following the principle that ultraproducts can be used as a bridge to connect quantitative and qualitative results (as discussed in these previous blog posts), we will deduce Theorem 3 from the following (well-known) qualitative version:
This principle (first laid out in an appendix of Lefschetz’s book), among other things, often allows one to use the methods of complex analysis (e.g. Riemann surface theory) to study many other fields of characteristic zero. There are many variants and extensions of this principle; see for instance this MathOverflow post for some discussion of these. I used this baby version of the Lefschetz principle recently in a paper on expanding polynomial maps.
Proof: We give two proofs of this fact, one using transcendence bases and the other using Hilbert’s nullstellensatz.
We begin with the former proof. As is finitely generated over , it has finite transcendence degree, thus one can find algebraically independent elements of over such that is a finite extension of , and in particular by the primitive element theorem is generated by and an element which is algebraic over . (Here we use the fact that characteristic zero fields are separable.) If we then define by first mapping to generic (and thus algebraically independent) complex numbers , and then setting to be a complex root of of the minimal polynomial for over after replacing each with the complex number , we obtain a field isomorphism with the required properties.
Now we give the latter proof. Let be elements of that generate that field over , but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers with the property that, for any polynomial with rational coefficients, one has
if and only if
Let be the collection of all polynomials with rational coefficients with , and be the collection of all polynomials with rational coefficients with . The set
is the intersection of countably many algebraic sets and is thus also an algebraic set (by the Hilbert basis theorem or the Noetherian property of algebraic sets). If the desired claim failed, then could be covered by the algebraic sets for . By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over cannot be covered by countably many varieties of smaller dimension, we conclude that must in fact be covered by a finite number of such sets, thus
for some . By the nullstellensatz, we thus have an identity of the form
for some natural numbers , polynomials , and polynomials with coefficients in . In particular, this identity also holds in the algebraic closure of . Evaluating this identity at we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows.
From Proposition 4 one can now deduce Theorem 3 by a routine ultraproduct argument (the same one used in these previous blog posts). Suppose for contradiction that Theorem 3 fails. Then there exists natural numbers , a sequence of finite fields of characteristic at least , and subsets of of cardinality such that for each , there does not exist a Freiman field isomorphism of order from to the complex numbers. Now we select a non-principal ultrafilter , and construct the ultraproduct of the finite fields . This is again a field (and is a basic example of what is known as a pseudo-finite field); because the characteristic of goes to infinity as , it is easy to see (using Los’s theorem) that has characteristic zero and can thus be viewed as an extension of the rationals .
Now let be the ultralimit of the , so that is the ultraproduct of the , then is a subset of of cardinality . In particular, if is the field generated by and , then is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism from to a subfield of the complex numbers. In particular, are complex numbers, and for any polynomial with integer coefficients, one has
if and only if
By Los’s theorem, we then conclude that for all sufficiently close to , one has for all polynomials of degree at most and whose coefficients are integers whose magnitude sums up to , one has
if and only if
But this gives a Freiman field isomorphism of order between and , contradicting the construction of , and Theorem 3 follows.