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One of the most basic theorems in linear algebra is that every finite-dimensional vector space has a finite basis. Let us give a statement of this theorem in the case when the underlying field is the rationals:
Theorem 1 (Finite generation implies finite basis, infinitary version) Let be a vector space over the rationals , and let be a finite collection of vectors in . Then there exists a collection of vectors in , with , such that
- ( generates ) Every can be expressed as a rational linear combination of the .
- ( independent) There is no non-trivial linear relation , among the (where non-trivial means that the are not all zero).
In fact, one can take to be a subset of the .
Proof: We perform the following “rank reduction argument”. Start with initialised to (so initially we have ). Clearly generates . If the are linearly independent then we are done. Otherwise, there is a non-trivial linear relation between them; after shuffling things around, we see that one of the , say , is a rational linear combination of the . In such a case, becomes redundant, and we may delete it (reducing the rank by one). We repeat this procedure; it can only run for at most steps and so terminates with obeying both of the desired properties.
In additive combinatorics, one often wants to use results like this in finitary settings, such as that of a cyclic group where is a large prime. Now, technically speaking, is not a vector space over , because one only multiply an element of by a rational number if the denominator of that rational does not divide . But for very large, “behaves” like a vector space over , at least if one restricts attention to the rationals of “bounded height” – where the numerator and denominator of the rationals are bounded. Thus we shall refer to elements of as “vectors” over , even though strictly speaking this is not quite the case.
On the other hand, saying that one element of is a rational linear combination of another set of elements is not a very interesting statement: any non-zero element of already generates the entire space! However, if one again restricts attention to rational linear combinations of bounded height, then things become interesting again. For instance, the vector can generate elements such as or using rational linear combinations of bounded height, but will not be able to generate such elements of as without using rational numbers of unbounded height.
For similar reasons, the notion of linear independence over the rationals doesn’t initially look very interesting over : any two non-zero elements of are of course rationally dependent. But again, if one restricts attention to rational numbers of bounded height, then independence begins to emerge: for instance, and are independent in this sense.
Thus, it becomes natural to ask whether there is a “quantitative” analogue of Theorem 1, with non-trivial content in the case of “vector spaces over the bounded height rationals” such as , which asserts that given any bounded collection of elements, one can find another set which is linearly independent “over the rationals up to some height”, such that the can be generated by the “over the rationals up to some height”. Of course to make this rigorous, one needs to quantify the two heights here, the one giving the independence, and the one giving the generation. In order to be useful for applications, it turns out that one often needs the former height to be much larger than the latter; exponentially larger, for instance, is not an uncommon request. Fortunately, one can accomplish this, at the cost of making the height somewhat large:
Theorem 2 (Finite generation implies finite basis, finitary version) Let be an integer, and let be a function. Let be an abelian group which admits a well-defined division operation by any natural number of size at most for some constant depending only on ; for instance one can take for a prime larger than . Let be a finite collection of “vectors” in . Then there exists a collection of vectors in , with , as well an integer , such that
- (Complexity bound) for some depending only on .
- ( generates ) Every can be expressed as a rational linear combination of the of height at most (i.e. the numerator and denominator of the coefficients are at most ).
- ( independent) There is no non-trivial linear relation among the in which the are rational numbers of height at most .
In fact, one can take to be a subset of the .
Proof: We perform the same “rank reduction argument” as before, but translated to the finitary setting. Start with initialised to (so initially we have ), and initialise . Clearly generates at this height. If the are linearly independent up to rationals of height then we are done. Otherwise, there is a non-trivial linear relation between them; after shuffling things around, we see that one of the , say , is a rational linear combination of the , whose height is bounded by some function depending on and . In such a case, becomes redundant, and we may delete it (reducing the rank by one), but note that in order for the remaining to generate we need to raise the height upper bound for the rationals involved from to some quantity depending on . We then replace by and continue the process. We repeat this procedure; it can only run for at most steps and so terminates with and obeying all of the desired properties. (Note that the bound on is quite poor, being essentially an -fold iteration of ! Thus, for instance, if is exponential, then the bound on is tower-exponential in nature.)
(A variant of this type of approximate basis lemma was used in my paper with Van Vu on the singularity probability of random Bernoulli matrices.)
Looking at the statements and proofs of these two theorems it is clear that the two results are in some sense the “same” result, except that the latter has been made sufficiently quantitative that it is meaningful in such finitary settings as . In this note I will show how this equivalence can be made formal using the language of non-standard analysis. This is not a particularly deep (or new) observation, but it is perhaps the simplest example I know of that illustrates how nonstandard analysis can be used to transfer a quantifier-heavy finitary statement, such as Theorem 2, into a quantifier-light infinitary statement, such as Theorem 1, thus lessening the need to perform “epsilon management” duties, such as keeping track of unspecified growth functions such as . This type of transference is discussed at length in this previous blog post of mine.
In this particular case, the amount of effort needed to set up the nonstandard machinery in order to reduce Theorem 2 from Theorem 1 is too great for this transference to be particularly worthwhile, especially given that Theorem 2 has such a short proof. However, when performing a particularly intricate argument in additive combinatorics, in which one is performing a number of “rank reduction arguments”, “energy increment arguments”, “regularity lemmas”, “structure theorems”, and so forth, the purely finitary approach can become bogged down with all the epsilon management one needs to do to organise all the parameters that are flying around. The nonstandard approach can efficiently hide a large number of these parameters from view, and it can then become worthwhile to invest in the nonstandard framework in order to clean up the rest of a lengthy argument. Furthermore, an advantage of moving up to the infinitary setting is that one can then deploy all the firepower of an existing well-developed infinitary theory of mathematics (in this particular case, this would be the theory of linear algebra) out of the box, whereas in the finitary setting one would have to painstakingly finitise each aspect of such a theory that one wished to use (imagine for instance trying to finitise the rank-nullity theorem for rationals of bounded height).
The nonstandard approach is very closely related to use of compactness arguments, or of the technique of taking ultralimits and ultraproducts; indeed we will use an ultrafilter in order to create the nonstandard model in the first place.
I will also discuss a two variants of both Theorem 1 and Theorem 2 which have actually shown up in my research. The first is that of the regularity lemma for polynomials over finite fields, which came up when studying the equidistribution of such polynomials (in this paper with Ben Green). The second comes up when is dealing not with a single finite collection of vectors, but rather with a family of such vectors, where ranges over a large set; this gives rise to what we call the sunflower lemma, and came up in this recent paper of myself, Ben Green, and Tamar Ziegler.
This post is mostly concerned with nonstandard translations of the “rank reduction argument”. Nonstandard translations of the “energy increment argument” and “density increment argument” were briefly discussed in this recent post; I may return to this topic in more detail in a future post.