Let be a field. A definable set over is a set of the form
where is a natural number, and is a predicate involving the ring operations of , the equality symbol , an arbitrary number of constants and free variables in , the quantifiers , boolean operators such as , and parentheses and colons, where the quantifiers are always understood to be over the field . Thus, for instance, the set of quadratic residues
is definable over , and any algebraic variety over is also a definable set over . Henceforth we will abbreviate “definable over ” simply as “definable”.
If is a finite field, then every subset of is definable, since finite sets are automatically definable. However, we can obtain a more interesting notion in this case by restricting the complexity of a definable set. We say that is a definable set of complexity at most if , and can be written in the form (1) for some predicate of length at most (where all operators, quantifiers, relations, variables, constants, and punctuation symbols are considered to have unit length). Thus, for instance, a hypersurface in dimensions of degree would be a definable set of complexity . We will then be interested in the regime where the complexity remains bounded, but the field size (or field characteristic) becomes large.
In a recent paper, I established (in the large characteristic case) the following regularity lemma for dense definable graphs, which significantly strengthens the Szemerédi regularity lemma in this context, by eliminating “bad” pairs, giving a polynomially strong regularity, and also giving definability of the cells:
Lemma 1 (Algebraic regularity lemma) Let be a finite field, let be definable non-empty sets of complexity at most , and let also be definable with complexity at most . Assume that the characteristic of is sufficiently large depending on . Then we may partition and with , with the following properties:
My original proof of this lemma was quite complicated, based on an explicit calculation of the “square”
of using the Lang-Weil bound and some facts about the étale fundamental group. It was the reliance on the latter which was the main reason why the result was restricted to the large characteristic setting. (I then applied this lemma to classify expanding polynomials over finite fields of large characteristic, but I will not discuss these applications here; see this previous blog post for more discussion.)
Recently, Anand Pillay and Sergei Starchenko (and independently, Udi Hrushovski) have observed that the theory of the étale fundamental group is not necessary in the argument, and the lemma can in fact be deduced from quite general model theoretic techniques, in particular using (a local version of) the concept of stability. One of the consequences of this new proof of the lemma is that the hypothesis of large characteristic can be omitted; the lemma is now known to be valid for arbitrary finite fields (although its content is trivial if the field is not sufficiently large depending on the complexity at most ).
Inspired by this, I decided to see if I could find yet another proof of the algebraic regularity lemma, again avoiding the theory of the étale fundamental group. It turns out that the spectral proof of the Szemerédi regularity lemma (discussed in this previous blog post) adapts very nicely to this setting. The key fact needed about definable sets over finite fields is that their cardinality takes on an essentially discrete set of values. More precisely, we have the following fundamental result of Chatzidakis, van den Dries, and Macintyre:
- (Discretised cardinality) If is a non-empty definable set of complexity at most , then one has
where is a natural number, and is a positive rational number with numerator and denominator . In particular, we have .
- (Definable cardinality) Assume is sufficiently large depending on . If , and are definable sets of complexity at most , so that can be viewed as a definable subset of that is definably parameterised by , then for each natural number and each positive rational with numerator and denominator , the set
is definable with complexity , where the implied constants in the asymptotic notation used to define (4) are the same as those that appearing in (3). (Informally: the “dimension” and “measure” of depends definably on .)
We will take this proposition as a black box; a proof can be obtained by combining the description of definable sets over pseudofinite fields (discussed in this previous post) with the Lang-Weil bound (discussed in this previous post). (The former fact is phrased using nonstandard analysis, but one can use standard compactness-and-contradiction arguments to convert such statements to statements in standard analysis, as discussed in this post.)
The above proposition places severe restrictions on the cardinality of definable sets; for instance, it shows that one cannot have a definable set of complexity at most and cardinality , if is sufficiently large depending on . If are definable sets of complexity at most , it shows that for some rational with numerator and denominator ; furthermore, if , we may improve this bound to . In particular, we obtain the following “self-improving” properties:
- If are definable of complexity at most and for some , then (if is sufficiently small depending on and is sufficiently large depending on ) this forces .
- If are definable of complexity at most and for some and positive rational , then (if is sufficiently small depending on and is sufficiently large depending on ) this forces .
It turns out that these self-improving properties can be applied to the coefficients of various matrices (basically powers of the adjacency matrix associated to ) that arise in the spectral proof of the regularity lemma to significantly improve the bounds in that lemma; we describe how this is done below the fold. We also make some connections to the stability-based proofs of Pillay-Starchenko and Hrushovski.
— 1. The regularity lemma —
We begin with proving a slightly weaker version of the lemma, in which the bound is weakened to ; we will later use the self-improving properties mentioned previously to upgrade this bound. In other words, we begin by showing
Lemma 3 (Algebraic regularity lemma, weak version) Let be a finite field, let be definable non-empty sets of complexity at most , and let also be definable with complexity at most . Then we may partition and with , with the following properties:
- (Definability) Each of the are definable of complexity .
- (Size) We have and for all and .
- (Regularity) We have
for all , , , and , where is a rational number in with numerator and denominator .
We now prove this lemma, by combining spectral theory with an (iterated) method. We may assume that is sufficiently large depending on , as the claim is trivial otherwise (just take ). We consider the probability spaces , with the uniform probability measures on ; we abbreviate as for , and similarly define (we work exclusively with real-valued functions here). The set then defines a linear operator by the formula
thus is the operator with integral kernel . We can then compute the adjoint operator as
for all .
Now we apply the singular value decomposition (which is easily justified as , are finite dimensional) to write
for some finite sequence of singular values, the are an orthonormal system in , and the are an orthonormal system in . Observe that operator can be diagonalised as
On the other hand, direct calculation shows that
and so we have the Frobenius norm (or Hilbert-Schmidt norm) bound
Next, from the identities
We let be a small quantity (depending only on ) to be chosen later, and split
where is a low rank operator
and is the error term
for any . In other words, the integral kernel of is bounded pointwise by . As for , we use (8) to discretise , where takes on at most values (say), and is bounded in magnitude by . We can then split , where
and the error term can easily be shown (by arguments similar to before) to have a bound of the form
(with many powers of to spare). We have thus written , where has integral kernel bounded pointwise by .
Now from (7), the number of summands in the definition of is at most . If we then partition using the level sets of the corresponding (removing any empty cells to ensure the are non-empty), then we have
and the integral kernel of is constant on each . Thus, the integral kernel of fluctuates by at most on each .
Thus far, we have not used the definability of . Now we will do so in order to take advantage of self-improving properties. The integral kernel of can be evaluated explicitly as
where is the set
Clearly, is a definable set of complexity . From Proposition 2 we thus have the discretisation bounds
where is a non-negative rational number with numerator and denominator . The set of such rationals is finite (basically a portion of the Farey sequence), and thus are separated from each other by some separation . If we combine this fact with the local near-constancy of on , we can thus (if is small enough) self-improve this near-constancy to the bound
for all , all , , and some non-negative rational depending only on with numerator and denominator .
Note that as is symmetric, will be symmetric too. If two are indistinguishable in the sense that for all , then we may concatenate and (reducing by one), so we may assume that elements of are pairwise distinguishable. From Proposition 2 we see that for each and rational , the set is definable with complexity . Taking boolean combinations and using distinguishability, we conclude that each is also definable with complexity . In particular, we either have or .
Now let be supported on with mean zero for some . From (10) we see that
On the other hand, from (9) we have
while from Bessel’s inequality we have
and thus by Hölder’s inequality
The left-hand side is just , so we conclude that
whenever is supported on one of the with mean zero.
In a similar vein, we may find a partition into definable sets of complexity with
with the property that
whenever is supported on one of the with mean zero.
Let and with and . From the above estimates and Cauchy-Schwarz, we see that
whenever are supported on respectively, with at least one of having mean zero. If , , then by decomposing and into mean-zero and constant components on respectively, we conclude that
On the other hand, we have
so by Proposition 2, for some rational with numerator and denominator . Also and for some further rational with numerator and denominator . This proves Lemma 3, except that we have some “small” and with and that we have not yet dealt with. However we may simply concatenate these small cells onto one of the larger cells, as this does not significantly affect the definability or regularity properties of the large cells, and we are done.
Now we improve the bound to . Let be large cells as in the preceding argument. Let be the restriction of to and (associated to the set ), giving , uniform probability measure, then we have
whenever at least one of , have mean zero. Thus, we may split where is a constant, is the rank one operator
and has operator norm . By computing as before, we see that we may take to be a rational with numerator and denominator .
On the other hand, we have
so by Proposition 2, we have either
for each . Furthermore, the set of pairs for which the latter occurs is a definable subset of with complexity . On the other hand, from (11) and Chebyshev’s inequality, this set has cardinality . By Proposition 2, we may self-improve this to . Integrating, we self-improve (11) to
so by Cauchy-Schwarz we have
whenever has mean zero, thus
— 2. Stability —
The following stability result was established by Hrushovski (see Proposition 2.25 of this paper)
Proposition 4 (Stability) Let be real numbers, and suppose that there are measurable subsets of a probability space with the property that
for . Then we have .
In the language of stability theory, this proposition asserts that the relation is stable. This proposition was the key input used by Pillay-Starchenko and by Hrushovski (together with the discretisation results of Proposition 2) to give an alternate proof of the algebraic regularity lemma.
Let us first sketch Hrushovski’s proof of this lemma. If the claim failed, then by a compactness argument one can construct infinite sequences of measurable subsets of a probability space with the property that
for all . Next, by Ramsey theory (for directed graphs), for any given , we may pass to an infinite subsequence such that
for any and , and by further compactness we may eliminate the error, so that
for any and . More generally, (using Ramsey’s theorem for directed hypergraphs and more compactness) we may assume the order-indiscernibility property
Now we give a spectral proof of the above proposition. Let be a matrix to be chosen later, and consider the expression
We can rewrite this expression as
For any , the vectors and have an norm of at most . Thus, we can bound the preceding expression by
where is the operator norm of ; thus we have