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Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our joint paper “Multiple recurrence and convergence results associated to ${{\bf F}_{p}^{\omega}}$-actions“. This paper is primarily concerned with limit formulae in the theory of multiple recurrence in ergodic theory. Perhaps the most basic formula of this type is the mean ergodic theorem, which (among other things) asserts that if ${(X,{\mathcal X}, \mu,T)}$ is a measure-preserving ${{\bf Z}}$-system (which, in this post, means that ${(X,{\mathcal X}, \mu)}$ is a probability space and ${T: X \mapsto X}$ is measure-preserving and invertible, thus giving an action ${(T^n)_{n \in {\bf Z}}}$ of the integers), and ${f,g \in L^2(X,{\mathcal X}, \mu)}$ are functions, and ${X}$ is ergodic (which means that ${L^2(X,{\mathcal X}, \mu)}$ contains no ${T}$-invariant functions other than the constants (up to almost everywhere equivalence, of course)), then the average

$\displaystyle \frac{1}{N} \sum_{n=1}^N \int_X f(x) g(T^n x)\ d\mu \ \ \ \ \ (1)$

converges as ${N \rightarrow \infty}$ to the expression

$\displaystyle (\int_X f(x)\ d\mu) (\int_X g(x)\ d\mu);$

see e.g. this previous blog post. Informally, one can interpret this limit formula as an equidistribution result: if ${x}$ is drawn at random from ${X}$ (using the probability measure ${\mu}$), and ${n}$ is drawn at random from ${\{1,\ldots,N\}}$ for some large ${N}$, then the pair ${(x, T^n x)}$ becomes uniformly distributed in the product space ${X \times X}$ (using product measure ${\mu \times \mu}$) in the limit as ${N \rightarrow \infty}$.

If we allow ${(X,\mu)}$ to be non-ergodic, then we still have a limit formula, but it is a bit more complicated. Let ${{\mathcal X}^T}$ be the ${T}$-invariant measurable sets in ${{\mathcal X}}$; the ${{\bf Z}}$-system ${(X, {\mathcal X}^T, \mu, T)}$ can then be viewed as a factor of the original system ${(X, {\mathcal X}, \mu, T)}$, which is equivalent (in the sense of measure-preserving systems) to a trivial system ${(Z_0, {\mathcal Z}_0, \mu_{Z_0}, 1)}$ (known as the invariant factor) in which the shift is trivial. There is then a projection map ${\pi_0: X \rightarrow Z_0}$ to the invariant factor which is a factor map, and the average (1) converges in the limit to the expression

$\displaystyle \int_{Z_0} (\pi_0)_* f(z) (\pi_0)_* g(z)\ d\mu_{Z_0}(x), \ \ \ \ \ (2)$

where ${(\pi_0)_*: L^2(X,{\mathcal X},\mu) \rightarrow L^2(Z_0,{\mathcal Z}_0,\mu_{Z_0})}$ is the pushforward map associated to the map ${\pi_0: X \rightarrow Z_0}$; see e.g. this previous blog post. We can interpret this as an equidistribution result. If ${(x,T^n x)}$ is a pair as before, then we no longer expect complete equidistribution in ${X \times X}$ in the non-ergodic, because there are now non-trivial constraints relating ${x}$ with ${T^n x}$; indeed, for any ${T}$-invariant function ${f: X \rightarrow {\bf C}}$, we have the constraint ${f(x) = f(T^n x)}$; putting all these constraints together we see that ${\pi_0(x) = \pi_0(T^n x)}$ (for almost every ${x}$, at least). The limit (2) can be viewed as an assertion that this constraint ${\pi_0(x) = \pi_0(T^n x)}$ are in some sense the “only” constraints between ${x}$ and ${T^n x}$, and that the pair ${(x,T^n x)}$ is uniformly distributed relative to these constraints.

Limit formulae are known for multiple ergodic averages as well, although the statement becomes more complicated. For instance, consider the expression

$\displaystyle \frac{1}{N} \sum_{n=1}^N \int_X f(x) g(T^n x) h(T^{2n} x)\ d\mu \ \ \ \ \ (3)$

for three functions ${f,g,h \in L^\infty(X, {\mathcal X}, \mu)}$; this is analogous to the combinatorial task of counting length three progressions in various sets. For simplicity we assume the system ${(X,{\mathcal X},\mu,T)}$ to be ergodic. Naively one might expect this limit to then converge to

$\displaystyle (\int_X f\ d\mu) (\int_X g\ d\mu) (\int_X h\ d\mu)$

which would roughly speaking correspond to an assertion that the triplet ${(x,T^n x, T^{2n} x)}$ is asymptotically equidistributed in ${X \times X \times X}$. However, even in the ergodic case there can be additional constraints on this triplet that cannot be seen at the level of the individual pairs ${(x,T^n x)}$, ${(x, T^{2n} x)}$. The key obstruction here is that of eigenfunctions of the shift ${T: X \rightarrow X}$, that is to say non-trivial functions ${f: X \rightarrow S^1}$ that obey the eigenfunction equation ${Tf = \lambda f}$ almost everywhere for some constant (or ${T}$-invariant) ${\lambda}$. Each such eigenfunction generates a constraint

$\displaystyle f(x) \overline{f(T^n x)}^2 f(T^{2n} x) = 1 \ \ \ \ \ (4)$

tying together ${x}$, ${T^n x}$, and ${T^{2n} x}$. However, it turns out that these are in some sense the only constraints on ${x,T^n x, T^{2n} x}$ that are relevant for the limit (3). More precisely, if one sets ${{\mathcal X}_1}$ to be the sub-algebra of ${{\mathcal X}}$ generated by the eigenfunctions of ${T}$, then it turns out that the factor ${(X, {\mathcal X}_1, \mu, T)}$ is isomorphic to a shift system ${(Z_1, {\mathcal Z}_1, \mu_{Z_1}, x \mapsto x+\alpha)}$ known as the Kronecker factor, for some compact abelian group ${Z_1 = (Z_1,+)}$ and some (irrational) shift ${\alpha \in Z_1}$; the factor map ${\pi_1: X \rightarrow Z_1}$ pushes eigenfunctions forward to (affine) characters on ${Z_1}$. It is then known that the limit of (3) is

$\displaystyle \int_\Sigma (\pi_1)_* f(x_0) (\pi_1)_* g(x_1) (\pi_1)_* h(x_2)\ d\mu_\Sigma$

where ${\Sigma \subset Z_1^3}$ is the closed subgroup

$\displaystyle \Sigma = \{ (x_1,x_2,x_3) \in Z_1^3: x_1-2x_2+x_3=0 \}$

and ${\mu_\Sigma}$ is the Haar probability measure on ${\Sigma}$; see this previous blog post. The equation ${x_1-2x_2+x_3=0}$ defining ${\Sigma}$ corresponds to the constraint (4) mentioned earlier. Among other things, this limit formula implies Roth’s theorem, which in the context of ergodic theory is the assertion that the limit (or at least the limit inferior) of (3) is positive when ${f=g=h}$ is non-negative and not identically vanishing.

If one considers a quadruple average

$\displaystyle \frac{1}{N} \sum_{n=1}^N \int_X f(x) g(T^n x) h(T^{2n} x) k(T^{3n} x)\ d\mu \ \ \ \ \ (5)$

(analogous to counting length four progressions) then the situation becomes more complicated still, even in the ergodic case. In addition to the (linear) eigenfunctions that already showed up in the computation of the triple average (3), a new type of constraint also arises from quadratic eigenfunctions ${f: X \rightarrow S^1}$, which obey an eigenfunction equation ${Tf = \lambda f}$ in which ${\lambda}$ is no longer constant, but is now a linear eigenfunction. For such functions, ${f(T^n x)}$ behaves quadratically in ${n}$, and one can compute the existence of a constraint

$\displaystyle f(x) \overline{f(T^n x)}^3 f(T^{2n} x)^3 \overline{f(T^{3n} x)} = 1 \ \ \ \ \ (6)$

between ${x}$, ${T^n x}$, ${T^{2n} x}$, and ${T^{3n} x}$ that is not detected at the triple average level. As it turns out, this is not the only type of constraint relevant for (5); there is a more general class of constraint involving two-step nilsystems which we will not detail here, but see e.g. this previous blog post for more discussion. Nevertheless there is still a similar limit formula to previous examples, involving a special factor ${(Z_2, {\mathcal Z}_2, \mu_{Z_2}, S)}$ which turns out to be an inverse limit of two-step nilsystems; this limit theorem can be extracted from the structural theory in this paper of Host and Kra combined with a limit formula for nilsystems obtained by Lesigne, but will not be reproduced here. The pattern continues to higher averages (and higher step nilsystems); this was first done explicitly by Ziegler, and can also in principle be extracted from the structural theory of Host-Kra combined with nilsystem equidistribution results of Leibman. These sorts of limit formulae can lead to various recurrence results refining Roth’s theorem in various ways; see this paper of Bergelson, Host, and Kra for some examples of this.

The above discussion was concerned with ${{\bf Z}}$-systems, but one can adapt much of the theory to measure-preserving ${G}$-systems for other discrete countable abelian groups ${G}$, in which one now has a family ${(T_g)_{g \in G}}$ of shifts indexed by ${G}$ rather than a single shift, obeying the compatibility relation ${T_{g+h}=T_g T_h}$. The role of the intervals ${\{1,\ldots,N\}}$ in this more general setting is replaced by that of Folner sequences. For arbitrary countable abelian ${G}$, the theory for double averages (1) and triple limits (3) is essentially identical to the ${{\bf Z}}$-system case. But when one turns to quadruple and higher limits, the situation becomes more complicated (and, for arbitrary ${G}$, still not fully understood). However one model case which is now well understood is the finite field case when ${G = {\bf F}_p^\omega = \bigcup_{n=1}^\infty {\bf F}_p^n}$ is an infinite-dimensional vector space over a finite field ${{\bf F}_p}$ (with the finite subspaces ${{\bf F}_p^n}$ then being a good choice for the Folner sequence). Here, the analogue of the structural theory of Host and Kra was worked out by Vitaly, Tamar, and myself in these previous papers (treating the high characteristic and low characteristic cases respectively). In the finite field setting, it turns out that nilsystems no longer appear, and one only needs to deal with linear, quadratic, and higher order eigenfunctions (known collectively as phase polynomials). It is then natural to look for a limit formula that asserts, roughly speaking, that if ${x}$ is drawn at random from a ${{\bf F}_p^\omega}$-system and ${n}$ drawn randomly from a large subspace of ${{\bf F}_p^\omega}$, then the only constraints between ${x, T^n x, \ldots, T^{(p-1)n} x}$ are those that arise from phase polynomials. The main theorem of this paper is to establish this limit formula (which, again, is a little complicated to state explicitly and will not be done here). In particular, we establish for the first time that the limit actually exists (a result which, for ${{\bf Z}}$-systems, was one of the main results of this paper of Host and Kra).

As a consequence, we can recover finite field analogues of most of the results of Bergelson-Host-Kra, though interestingly some of the counterexamples demonstrating sharpness of their results for ${{\bf Z}}$-systems (based on Behrend set constructions) do not seem to be present in the finite field setting (cf. this previous blog post on the cap set problem). In particular, we are able to largely settle the question of when one has a Khintchine-type theorem that asserts that for any measurable set ${A}$ in an ergodic ${{\bf F}_p^\omega}$-system and any ${\epsilon>0}$, one has

$\displaystyle \mu( T_{c_1 n} A \cap \ldots \cap T_{c_k n} A ) > \mu(A)^k - \epsilon$

for a syndetic set of ${n}$, where ${c_1,\ldots,c_k \in {\bf F}_p}$ are distinct residue classes. It turns out that Khintchine-type theorems always hold for ${k=1,2,3}$ (and for ${k=1,2}$ ergodicity is not required), and for ${k=4}$ it holds whenever ${c_1,c_2,c_3,c_4}$ form a parallelogram, but not otherwise (though the counterexample here was such a painful computation that we ended up removing it from the paper, and may end up putting it online somewhere instead), and for larger ${k}$ we could show that the Khintchine property failed for generic choices of ${c_1,\ldots,c_k}$, though the problem of determining exactly the tuples for which the Khintchine property failed looked to be rather messy and we did not completely settle it.

Suppose that ${G = (G,\cdot)}$ is a finite group of even order, thus ${|G|}$ is a multiple of two. By Cauchy’s theorem, this implies that ${G}$ contains an involution: an element ${g}$ in ${G}$ of order two. (Indeed, if no such involution existed, then ${G}$ would be partitioned into doubletons ${\{g,g^{-1}\}}$ together with the identity, so that ${|G|}$ would be odd, a contradiction.) Of course, groups of odd order have no involutions ${g}$, thanks to Lagrange’s theorem (since ${G}$ cannot split into doubletons ${\{ h, hg \}}$).

The classical Brauer-Fowler theorem asserts that if a group ${G}$ has many involutions, then it must have a large non-trivial subgroup:

Theorem 1 (Brauer-Fowler theorem) Let ${G}$ be a finite group with at least ${|G|/n}$ involutions for some ${n > 1}$. Then ${G}$ contains a proper subgroup ${H}$ of index at most ${n^2}$.

This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen ${n^2}$ slightly to ${n(n+2)/2}$) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if ${g}$ is a an involution of a group ${G}$, and the centraliser ${C_G(g) := \{ h \in G: gh = hg\}}$ has order ${n}$, then clearly ${n \geq 2}$ (as ${C_G(g)}$ contains ${1}$ and ${g}$) and the conjugacy class ${\{ aga^{-1}: a \in G \}}$ has order ${|G|/n}$ (since the map ${a \mapsto aga^{-1}}$ has preimages that are cosets of ${C_G(g)}$). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem ${G}$ contains a subgroup of order at least ${\max( n, |G|/n^2)}$. In particular, we can conclude that every group ${G}$ of even order contains a proper subgroup of order at least ${|G|^{1/3}}$.

Another corollary is that the size of a simple group of even order can be controlled by the size of a centraliser of one of its involutions:

Corollary 2 (Brauer-Fowler theorem) Let ${G}$ be a finite simple group with an involution ${g}$, and suppose that ${C_G(g)}$ has order ${n}$. Then ${G}$ has order at most ${(n^2)!}$.

Indeed, by the previous discussion ${G}$ has a proper subgroup ${H}$ of index less than ${n^2}$, which then gives a non-trivial permutation action of ${G}$ on the coset space ${G/H}$. The kernel of this action is a proper normal subgroup of ${G}$ and is thus trivial, so the action is faithful, and the claim follows.

If one assumes the Feit-Thompson theorem that all groups of odd order are solvable, then Corollary 2 suggests a strategy (first proposed by Brauer himself in 1954) to prove the classification of finite simple groups (CFSG) by induction on the order of the group. Namely, assume for contradiction that the CFSG failed, so that there is a counterexample ${G}$ of minimal order ${|G|}$ to the classification. This is a non-abelian finite simple group; by the Feit-Thompson theorem, it has even order and thus has at least one involution ${g}$. Take such an involution and consider its centraliser ${C_G(g)}$; this is a proper subgroup of ${G}$ of some order ${n < |G|}$. As ${G}$ is a minimal counterexample to the classification, one can in principle describe ${C_G(g)}$ in terms of the CFSG by factoring the group into simple components (via a composition series) and applying the CFSG to each such component. Now, the “only” thing left to do is to verify, for each isomorphism class of ${C_G(g)}$, that all the possible simple groups ${G}$ that could have this type of group as a centraliser of an involution obey the CFSG; Corollary 2 tells us that for each such isomorphism class for ${C_G(g)}$, there are only finitely many ${G}$ that could generate this class for one of its centralisers, so this task should be doable in principle for any given isomorphism class for ${C_G(g)}$. That’s all one needs to do to prove the classification of finite simple groups!

Needless to say, this program turns out to be far more difficult than the above summary suggests, and the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument is based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a ${p}$-group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of ${p}$-groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups which dictates a very specific type of interaction structure between these ${p}$-groups in the key case when ${G}$ is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic ${p}$. See the text of Aschbacher, Lyons, Smith, and Solomon for a more detailed description of this strategy.

The Brauer-Fowler theorem can be proven by a nice application of character theory, of the type discussed in this recent blog post, ultimately based on analysing the alternating tensor power of representations; I reproduce a version of this argument (taken from this text of Isaacs) below the fold. (The original argument of Brauer and Fowler is more combinatorial in nature.) However, I wanted to record a variant of the argument that relies not on the fine properties of characters, but on the cruder theory of quasirandomness for groups, the modern study of which was initiated by Gowers, and is discussed for instance in this previous post. It gives the following slightly weaker version of Corollary 2:

Corollary 3 (Weak Brauer-Fowler theorem) Let ${G}$ be a finite simple group with an involution ${g}$, and suppose that ${C_G(g)}$ has order ${n}$. Then ${G}$ can be identified with a subgroup of the unitary group ${U_{4n^3}({\bf C})}$.

One can get an upper bound on ${|G|}$ from this corollary using Jordan’s theorem, but the resulting bound is a bit weaker than that in Corollary 2 (and the best bounds on Jordan’s theorem require the CFSG!).

Proof: Let ${A}$ be the set of all involutions in ${G}$, then as discussed above ${|A| \geq |G|/n}$. We may assume that ${G}$ has no non-trivial unitary representation of dimension less than ${4n^3}$ (since such representations are automatically faithful by the simplicity of ${G}$); thus, in the language of quasirandomness, ${G}$ is ${4n^3}$-quasirandom, and is also non-abelian. We have the basic convolution estimate

$\displaystyle \|1_A * 1_A * 1_A - \frac{|A|^3}{|G|} \|_{\ell^\infty(G)} \leq (4n^3)^{-1/2} |G|^{1/2} |A|^{3/2}$

(see Exercise 10 from this previous blog post). In particular,

$\displaystyle 1_A * 1_A * 1_A(0) \geq \frac{|A|^3}{|G|} - (4n^3)^{-1/2} |G|^{1/2} |A|^{3/2} \geq \frac{1}{2n^3} |G|^2$

and so there are at least ${|G|^2/2n^3}$ pairs ${(g,h) \in A \times A}$ such that ${gh \in A^{-1} = A}$, i.e. involutions ${g,h}$ whose product is also an involution. But any such involutions necessarily commute, since

$\displaystyle g (gh) h = g^2 h^2 = 1 = (gh)^2 = g (hg) h.$

Thus there are at least ${|G|^2/2n^3}$ pairs ${(g,h) \in G \times G}$ of non-identity elements that commute, so by the pigeonhole principle there is a non-identity ${g \in G}$ whose centraliser ${C_G(g)}$ has order at least ${|G|/2n^3}$. This centraliser cannot be all of ${G}$ since this would make ${g}$ central which contradicts the non-abelian simple nature of ${G}$. But then the quasiregular representation of ${G}$ on ${G/C_G(g)}$ has dimension at most ${2n^3}$, contradicting the quasirandomness. $\Box$

An abstract finite-dimensional complex Lie algebra, or Lie algebra for short, is a finite-dimensional complex vector space ${{\mathfrak g}}$ together with an anti-symmetric bilinear form ${[,] = [,]_{\mathfrak g}: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}}$ that obeys the Jacobi identity

$\displaystyle [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 \ \ \ \ \ (1)$

for all ${x,y,z \in {\mathfrak g}}$; by anti-symmetry one can also rewrite the Jacobi identity as

$\displaystyle [x,[y,z]] = [[x,y],z] + [y,[x,z]]. \ \ \ \ \ (2)$

We will usually omit the subscript from the Lie bracket ${[,]_{\mathfrak g}}$ when this will not cause ambiguity. A homomorphism ${\phi: {\mathfrak g} \rightarrow {\mathfrak h}}$ between two Lie algebras ${{\mathfrak g},{\mathfrak h}}$ is a linear map that respects the Lie bracket, thus ${\phi([x,y]_{\mathfrak g}) =[\phi(x),\phi(y)]_{\mathfrak h}}$ for all ${x,y \in {\mathfrak g}}$. As with many other classes of mathematical objects, the class of Lie algebras together with their homomorphisms then form a category. One can of course also consider Lie algebras in infinite dimension or over other fields, but we will restrict attention throughout these notes to the finite-dimensional complex case. The trivial, zero-dimensional Lie algebra is denoted ${0}$; Lie algebras of positive dimension will be called non-trivial.

Lie algebras come up in many contexts in mathematics, in particular arising as the tangent space of complex Lie groups. It is thus very profitable to think of Lie algebras as being the infinitesimal component of a Lie group, and in particular almost all of the notation and concepts that are applicable to Lie groups (e.g. nilpotence, solvability, extensions, etc.) have infinitesimal counterparts in the category of Lie algebras (often with exactly the same terminology). See this previous blog post for more discussion about the connection between Lie algebras and Lie groups (that post was focused over the reals instead of the complexes, but much of the discussion carries over to the complex case).

A particular example of a Lie algebra is the general linear Lie algebra ${{\mathfrak{gl}}(V)}$ of linear transformations ${x: V \rightarrow V}$ on a finite-dimensional complex vector space (or vector space for short) ${V}$, with the commutator Lie bracket ${[x,y] := xy-yx}$; one easily verifies that this is indeed an abstract Lie algebra. We will define a concrete Lie algebra to be a Lie algebra that is a subalgebra of ${{\mathfrak{gl}}(V)}$ for some vector space ${V}$, and similarly define a representation of a Lie algebra ${{\mathfrak g}}$ to be a homomorphism ${\rho: {\mathfrak g} \rightarrow {\mathfrak h}}$ into a concrete Lie algebra ${{\mathfrak h}}$. It is a deep theorem of Ado (discussed in this previous post) that every abstract Lie algebra is in fact isomorphic to a concrete one (or equivalently, that every abstract Lie algebra has a faithful representation), but we will not need or prove this fact here.

Even without Ado’s theorem, though, the structure of abstract Lie algebras is very well understood. As with many other objects in an algebraic category, a basic way to understand a Lie algebra ${{\mathfrak g}}$ is to factor it into two simpler algebras ${{\mathfrak h}, {\mathfrak k}}$ via a short exact sequence

$\displaystyle 0 \rightarrow {\mathfrak h} \rightarrow {\mathfrak g} \rightarrow {\mathfrak k} \rightarrow 0, \ \ \ \ \ (3)$

thus one has an injective homomorphism from ${{\mathfrak h}}$ to ${{\mathfrak g}}$ and a surjective homomorphism from ${{\mathfrak g}}$ to ${{\mathfrak k}}$ such that the image of the former homomorphism is the kernel of the latter. (To be pedantic, a short exact sequence in a general category requires these homomorphisms to be monomorphisms and epimorphisms respectively, but in the category of Lie algebras these turn out to reduce to the more familiar concepts of injectivity and surjectivity respectively.) Given such a sequence, one can (non-uniquely) identify ${{\mathfrak g}}$ with the vector space ${{\mathfrak h} \times {\mathfrak k}}$ equipped with a Lie bracket of the form

$\displaystyle [(t,x), (s,y)]_{\mathfrak g} = ([t,s]_{\mathfrak h} + A(t,y) - A(s,x) + B(x,y), [x,y]_{\mathfrak k}) \ \ \ \ \ (4)$

for some bilinear maps ${A: {\mathfrak h} \times {\mathfrak k} \rightarrow {\mathfrak h}}$ and ${B: {\mathfrak k} \times {\mathfrak k} \rightarrow {\mathfrak h}}$ that obey some Jacobi-type identities which we will not record here. Understanding exactly what maps ${A,B}$ are possible here (up to coordinate change) can be a difficult task (and is one of the key objectives of Lie algebra cohomology), but in principle at least, the problem of understanding ${{\mathfrak g}}$ can be reduced to that of understanding that of its factors ${{\mathfrak k}, {\mathfrak h}}$. To emphasise this, I will (perhaps idiosyncratically) express the existence of a short exact sequence (3) by the ATLAS-type notation

$\displaystyle {\mathfrak g} = {\mathfrak h} . {\mathfrak k} \ \ \ \ \ (5)$

although one should caution that for given ${{\mathfrak h}}$ and ${{\mathfrak k}}$, there can be multiple non-isomorphic ${{\mathfrak g}}$ that can form a short exact sequence with ${{\mathfrak h},{\mathfrak k}}$, so that ${{\mathfrak h} . {\mathfrak k}}$ is not a uniquely defined combination of ${{\mathfrak h}}$ and ${{\mathfrak k}}$; one could emphasise this by writing ${{\mathfrak h} ._{A,B} {\mathfrak k}}$ instead of ${{\mathfrak h} . {\mathfrak k}}$, though we will not do so here. We will refer to ${{\mathfrak g}}$ as an extension of ${{\mathfrak k}}$ by ${{\mathfrak h}}$, and read the notation (5) as “ ${{\mathfrak g}}$ is ${{\mathfrak h}}$-by-${{\mathfrak k}}$“; confusingly, these two notations reverse the subject and object of “by”, but unfortunately both notations are well entrenched in the literature. We caution that the operation ${.}$ is not commutative, and it is only partly associative: every Lie algebra of the form ${{\mathfrak k} . ({\mathfrak h} . \l)}$ is also of the form ${({\mathfrak k} . {\mathfrak h}) . {\mathfrak l}}$, but the converse is not true (see this previous blog post for some related discussion). As we are working in the infinitesimal world of Lie algebras (which have an additive group operation) rather than Lie groups (in which the group operation is usually written multiplicatively), it may help to think of ${{\mathfrak h} . {\mathfrak k}}$ as a (twisted) “sum” of ${{\mathfrak h}}$ and ${{\mathfrak k}}$ rather than a “product”; for instance, we have ${{\mathfrak g} = 0 . {\mathfrak g}}$ and ${{\mathfrak g} = {\mathfrak g} . 0}$, and also ${\dim {\mathfrak h} . {\mathfrak k} = \dim {\mathfrak h} + \dim {\mathfrak k}}$.

Special examples of extensions ${{\mathfrak h} .{\mathfrak k}}$ of ${{\mathfrak k}}$ by ${{\mathfrak h}}$ include the direct sum (or direct product) ${{\mathfrak h} \oplus {\mathfrak k}}$ (also denoted ${{\mathfrak h} \times {\mathfrak k}}$,) which is given by the construction (4) with ${A}$ and ${B}$ both vanishing, and the split extension (or semidirect product) ${{\mathfrak h} : {\mathfrak k} = {\mathfrak h} :_\rho {\mathfrak k}}$ (also denoted ${{\mathfrak h} \ltimes {\mathfrak k} = {\mathfrak h} \ltimes_\rho {\mathfrak k}}$), which is given by the construction (4) with ${B}$ vanishing and the bilinear map ${A: {\mathfrak h} \times {\mathfrak k} \rightarrow {\mathfrak h}}$ taking the form

$\displaystyle A( t, x ) = \rho(x)(t)$

for some representation ${\rho: {\mathfrak k} \rightarrow \hbox{Der} {\mathfrak h}}$ of ${{\mathfrak k}}$ in the concrete Lie algebra of derivations ${\hbox{Der} {\mathfrak h} \subset {\mathfrak{gl}}({\mathfrak h})}$ of ${{\mathfrak h}}$, that is to say the algebra of linear maps ${D: {\mathfrak h} \rightarrow {\mathfrak h}}$ that obey the Leibniz rule

$\displaystyle D[s,t]_{\mathfrak h} = [Ds,t]_{\mathfrak h} + [s,Dt]_{\mathfrak h}$

for all ${s,t \in {\mathfrak h}}$. (The derivation algebra ${\hbox{Der} {\mathfrak g}}$ of a Lie algebra ${{\mathfrak g}}$ is analogous to the automorphism group ${\hbox{Aut}(G)}$ of a Lie group ${G}$, with the two concepts being intertwined by the tangent space functor ${G \mapsto {\mathfrak g}}$ from Lie groups to Lie algebras (i.e. the derivation algebra is the infinitesimal version of the automorphism group). Of course, this functor also intertwines the Lie algebra and Lie group versions of most of the other concepts discussed here, such as extensions, semidirect products, etc.)

There are two general ways to factor a Lie algebra ${{\mathfrak g}}$ as an extension ${{\mathfrak h} . {\mathfrak k}}$ of a smaller Lie algebra ${{\mathfrak k}}$ by another smaller Lie algebra ${{\mathfrak h}}$. One is to locate a Lie algebra ideal (or ideal for short) ${{\mathfrak h}}$ in ${{\mathfrak g}}$, thus ${[{\mathfrak h},{\mathfrak g}] \subset {\mathfrak h}}$, where ${[{\mathfrak h},{\mathfrak g}]}$ denotes the Lie algebra generated by ${\{ [x,y]: x \in {\mathfrak h}, y \in {\mathfrak g} \}}$, and then take ${{\mathfrak k}}$ to be the quotient space ${{\mathfrak g}/{\mathfrak h}}$ in the usual manner; one can check that ${{\mathfrak h}}$, ${{\mathfrak k}}$ are also Lie algebras and that we do indeed have a short exact sequence

$\displaystyle {\mathfrak g} = {\mathfrak h} . ({\mathfrak g}/{\mathfrak h}).$

Conversely, whenever one has a factorisation ${{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}$, one can identify ${{\mathfrak h}}$ with an ideal in ${{\mathfrak g}}$, and ${{\mathfrak k}}$ with the quotient of ${{\mathfrak g}}$ by ${{\mathfrak h}}$.

The other general way to obtain such a factorisation is is to start with a homomorphism ${\rho: {\mathfrak g} \rightarrow {\mathfrak m}}$ of ${{\mathfrak g}}$ into another Lie algebra ${{\mathfrak m}}$, take ${{\mathfrak k}}$ to be the image ${\rho({\mathfrak g})}$ of ${{\mathfrak g}}$, and ${{\mathfrak h}}$ to be the kernel ${\hbox{ker} \rho := \{ x \in {\mathfrak g}: \rho(x) = 0 \}}$. Again, it is easy to see that this does indeed create a short exact sequence:

$\displaystyle {\mathfrak g} = \hbox{ker} \rho . \rho({\mathfrak g}).$

Conversely, whenever one has a factorisation ${{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}$, one can identify ${{\mathfrak k}}$ with the image of ${{\mathfrak g}}$ under some homomorphism, and ${{\mathfrak h}}$ with the kernel of that homomorphism. Note that if a representation ${\rho: {\mathfrak g} \rightarrow {\mathfrak m}}$ is faithful (i.e. injective), then the kernel is trivial and ${{\mathfrak g}}$ is isomorphic to ${\rho({\mathfrak g})}$.

Now we consider some examples of factoring some class of Lie algebras into simpler Lie algebras. The easiest examples of Lie algebras to understand are the abelian Lie algebras ${{\mathfrak g}}$, in which the Lie bracket identically vanishes. Every one-dimensional Lie algebra is automatically abelian, and thus isomorphic to the scalar algebra ${{\bf C}}$. Conversely, by using an arbitrary linear basis of ${{\mathfrak g}}$, we see that an abelian Lie algebra is isomorphic to the direct sum of one-dimensional algebras. Thus, a Lie algebra is abelian if and only if it is isomorphic to the direct sum of finitely many copies of ${{\bf C}}$.

Now consider a Lie algebra ${{\mathfrak g}}$ that is not necessarily abelian. We then form the derived algebra ${[{\mathfrak g},{\mathfrak g}]}$; this algebra is trivial if and only if ${{\mathfrak g}}$ is abelian. It is easy to see that ${[{\mathfrak h},{\mathfrak k}]}$ is an ideal whenever ${{\mathfrak h},{\mathfrak k}}$ are ideals, so in particular the derived algebra ${[{\mathfrak g},{\mathfrak g}]}$ is an ideal and we thus have the short exact sequence

$\displaystyle {\mathfrak g} = [{\mathfrak g},{\mathfrak g}] . ({\mathfrak g}/[{\mathfrak g},{\mathfrak g}]).$

The algebra ${{\mathfrak g}/[{\mathfrak g},{\mathfrak g}]}$ is the maximal abelian quotient of ${{\mathfrak g}}$, and is known as the abelianisation of ${{\mathfrak g}}$. If it is trivial, we call the Lie algebra perfect. If instead it is non-trivial, then the derived algebra has strictly smaller dimension than ${{\mathfrak g}}$. From this, it is natural to associate two series to any Lie algebra ${{\mathfrak g}}$, the lower central series

$\displaystyle {\mathfrak g}_1 = {\mathfrak g}; {\mathfrak g}_2 := [{\mathfrak g}, {\mathfrak g}_1]; {\mathfrak g}_3 := [{\mathfrak g}, {\mathfrak g}_2]; \ldots$

and the derived series

$\displaystyle {\mathfrak g}^{(1)} := {\mathfrak g}; {\mathfrak g}^{(2)} := [{\mathfrak g}^{(1)}, {\mathfrak g}^{(1)}]; {\mathfrak g}^{(3)} := [{\mathfrak g}^{(2)}, {\mathfrak g}^{(2)}]; \ldots.$

By induction we see that these are both decreasing series of ideals of ${{\mathfrak g}}$, with the derived series being slightly smaller (${{\mathfrak g}^{(k)} \subseteq {\mathfrak g}_k}$ for all ${k}$). We say that a Lie algebra is nilpotent if its lower central series is eventually trivial, and solvable if its derived series eventually becomes trivial. Thus, abelian Lie algebras are nilpotent, and nilpotent Lie algebras are solvable, but the converses are not necessarily true. For instance, in the general linear group ${{\mathfrak{gl}}_n = {\mathfrak{gl}}({\bf C}^n)}$, which can be identified with the Lie algebra of ${n \times n}$ complex matrices, the subalgebra ${{\mathfrak n}}$ of strictly upper triangular matrices is nilpotent (but not abelian for ${n \geq 3}$), while the subalgebra ${{\mathfrak n}}$ of upper triangular matrices is solvable (but not nilpotent for ${n \geq 2}$). It is also clear that any subalgebra of a nilpotent algebra is nilpotent, and similarly for solvable or abelian algebras.

From the above discussion we see that a Lie algebra is solvable if and only if it can be represented by a tower of abelian extensions, thus

$\displaystyle {\mathfrak g} = {\mathfrak a}_1 . ({\mathfrak a}_2 . \ldots ({\mathfrak a}_{k-1} . {\mathfrak a}_k) \ldots )$

for some abelian ${{\mathfrak a}_1,\ldots,{\mathfrak a}_k}$. Similarly, a Lie algebra ${{\mathfrak g}}$ is nilpotent if it is expressible as a tower of central extensions (so that in all the extensions ${{\mathfrak h} . {\mathfrak k}}$ in the above factorisation, ${{\mathfrak h}}$ is central in ${{\mathfrak h} . {\mathfrak k}}$, where we say that ${{\mathfrak h}}$ is central in ${{\mathfrak g}}$ if ${[{\mathfrak h},{\mathfrak g}]=0}$). We also see that an extension ${{\mathfrak h} . {\mathfrak k}}$ is solvable if and only of both factors ${{\mathfrak h}, {\mathfrak k}}$ are solvable.

For our next fundamental example of using short exact sequences to split a general Lie algebra into simpler objects, we observe that every abstract Lie algebra ${{\mathfrak g}}$ has an adjoint representation ${\hbox{ad}: {\mathfrak g} \rightarrow \hbox{ad} {\mathfrak g} \subset {\mathfrak{gl}}({\mathfrak g})}$, where for each ${x \in {\mathfrak g}}$, ${\hbox{ad} x \in {\mathfrak{gl}}({\mathfrak g})}$ is the linear map ${(\hbox{ad} x)(y) := [x,y]}$; one easily verifies that this is indeed a representation (indeed, (2) is equivalent to the assertion that ${\hbox{ad} [x,y] = [\hbox{ad} x, \hbox{ad} y]}$ for all ${x,y \in {\mathfrak g}}$). The kernel of this representation is the center ${Z({\mathfrak g}) := \{ x \in {\mathfrak g}: [x,{\mathfrak g}] = 0\}}$, which the maximal central subalgebra of ${{\mathfrak g}}$. We thus have the short exact sequence

$\displaystyle {\mathfrak g} = Z({\mathfrak g}) . \hbox{ad} g \ \ \ \ \ (6)$

which, among other things, shows that every abstract Lie algebra is a central extension of a concrete Lie algebra (which can serve as a cheap substitute for Ado’s theorem mentioned earlier).

For our next fundamental decomposition of Lie algebras, we need some more definitions. A Lie algebra ${{\mathfrak g}}$ is simple if it is non-abelian and has no ideals other than ${0}$ and ${{\mathfrak g}}$; thus simple Lie algebras cannot be factored ${{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}$ into strictly smaller algebras ${{\mathfrak h},{\mathfrak k}}$. In particular, simple Lie algebras are automatically perfect and centerless. We have the following fundamental theorem:

Theorem 1 (Equivalent definitions of semisimplicity) Let ${{\mathfrak g}}$ be a Lie algebra. Then the following are equivalent:

• (i) ${{\mathfrak g}}$ does not contain any non-trivial solvable ideal.
• (ii) ${{\mathfrak g}}$ does not contain any non-trivial abelian ideal.
• (iii) The Killing form ${K: {\mathfrak g} \times {\mathfrak g} \rightarrow {\bf C}}$, defined as the bilinear form ${K(x,y) := \hbox{tr}_{\mathfrak g}( (\hbox{ad} x) (\hbox{ad} y) )}$, is non-degenerate on ${{\mathfrak g}}$.
• (iv) ${{\mathfrak g}}$ is isomorphic to the direct sum of finitely many non-abelian simple Lie algebras.

We review the proof of this theorem later in these notes. A Lie algebra obeying any (and hence all) of the properties (i)-(iv) is known as a semisimple Lie algebra. The statement (iv) is usually taken as the definition of semisimplicity; the equivalence of (iv) and (i) is then known as Weyl’s complete reducibility theorem, and the equivalence of (iv) and (iii) is known as the Cartan semisimplicity criterion. (The equivalence of (i) and (ii) is easy.)

If ${{\mathfrak h}}$ and ${{\mathfrak k}}$ are solvable ideals of a Lie algebra ${{\mathfrak g}}$, then it is not difficult to see that the vector sum ${{\mathfrak h}+{\mathfrak k}}$ is also a solvable ideal (because on quotienting by ${{\mathfrak h}}$ we see that the derived series of ${{\mathfrak h}+{\mathfrak k}}$ must eventually fall inside ${{\mathfrak h}}$, and thence must eventually become trivial by the solvability of ${{\mathfrak h}}$). As our Lie algebras are finite dimensional, we conclude that ${{\mathfrak g}}$ has a unique maximal solvable ideal, known as the radical ${\hbox{rad} {\mathfrak g}}$ of ${{\mathfrak g}}$. The quotient ${{\mathfrak g}/\hbox{rad} {\mathfrak g}}$ is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the Levi decomposition

$\displaystyle {\mathfrak g} = \hbox{rad} {\mathfrak g} . ({\mathfrak g} / \hbox{rad} {\mathfrak g})$

expressing an arbitrary Lie algebra as an extension of a semisimple Lie algebra ${{\mathfrak g}/\hbox{rad}{\mathfrak g}}$ by a solvable algebra ${\hbox{rad} {\mathfrak g}}$ (and it is not hard to see that this is the only possible such extension up to isomorphism). Indeed, a deep theorem of Levi allows one to upgrade this decomposition to a split extension

$\displaystyle {\mathfrak g} = \hbox{rad} {\mathfrak g} : ({\mathfrak g} / \hbox{rad} {\mathfrak g})$

although we will not need or prove this result here.

In view of the above decompositions, we see that we can factor any Lie algebra (using a suitable combination of direct sums and extensions) into a finite number of simple Lie algebras and the scalar algebra ${{\bf C}}$. In principle, this means that one can understand an arbitrary Lie algebra once one understands all the simple Lie algebras (which, being defined over ${{\bf C}}$, are somewhat confusingly referred to as simple complex Lie algebras in the literature). Amazingly, this latter class of algebras are completely classified:

Theorem 2 (Classification of simple Lie algebras) Up to isomorphism, every simple Lie algebra is of one of the following forms:

• ${A_n = \mathfrak{sl}_{n+1}}$ for some ${n \geq 1}$.
• ${B_n = \mathfrak{so}_{2n+1}}$ for some ${n \geq 2}$.
• ${C_n = \mathfrak{sp}_{2n}}$ for some ${n \geq 3}$.
• ${D_n = \mathfrak{so}_{2n}}$ for some ${n \geq 4}$.
• ${E_6, E_7}$, or ${E_8}$.
• ${F_4}$.
• ${G_2}$.

(The precise definition of the classical Lie algebras ${A_n,B_n,C_n,D_n}$ and the exceptional Lie algebras ${E_6,E_7,E_8,F_4,G_2}$ will be recalled later.)

(One can extend the families ${A_n,B_n,C_n,D_n}$ of classical Lie algebras a little bit to smaller values of ${n}$, but the resulting algebras are either isomorphic to other algebras on this list, or cease to be simple; see this previous post for further discussion.)

This classification is a basic starting point for the classification of many other related objects, including Lie algebras and Lie groups over more general fields (e.g. the reals ${{\bf R}}$), as well as finite simple groups. Being so fundamental to the subject, this classification is covered in almost every basic textbook in Lie algebras, and I myself learned it many years ago in an honours undergraduate course back in Australia. The proof is rather lengthy, though, and I have always had difficulty keeping it straight in my head. So I have decided to write some notes on the classification in this blog post, aiming to be self-contained (though moving rapidly). There is no new material in this post, though; it is all drawn from standard reference texts (I relied particularly on Fulton and Harris’s text, which I highly recommend). In fact it seems remarkably hard to deviate from the standard routes given in the literature to the classification; I would be interested in knowing about other ways to reach the classification (or substeps in that classification) that are genuinely different from the orthodox route.

The classification of finite simple groups (CFSG), first announced in 1983 but only fully completed in 2004, is one of the monumental achievements of twentieth century mathematics. Spanning hundreds of papers and tens of thousands of pages, it has been called the “enormous theorem”. A “second generation” proof of the theorem is nearly completed which is a little shorter (estimated at about five thousand pages in length), but currently there is no reasonably sized proof of the classification.

An important precursor of the CFSG is the Feit-Thompson theorem from 1962-1963, which asserts that every finite group of odd order is solvable, or equivalently that every non-abelian finite simple group has even order. This is an immediate consequence of CFSG, and conversely the Feit-Thompson theorem is an essential starting point in the proof of the classification, since it allows one to reduce matters to groups of even order for which key additional tools (such as the Brauer-Fowler theorem) become available. The original proof of the Feit-Thompson theorem is 255 pages long, which is significantly shorter than the proof of the CFSG, but still far from short. While parts of the proof of the Feit-Thompson theorem have been simplified (and it has recently been converted, after six years of effort, into an argument that has been verified by the proof assistant Coq), the available proofs of this theorem are still extremely lengthy by any reasonable standard.

However, there is a significantly simpler special case of the Feit-Thompson theorem that was established previously by Suzuki in 1957, which was influential in the proof of the more general Feit-Thompson theorem (and thus indirectly to the proof of CFSG). Define a CA-group to be a group ${G}$ with the property that the centraliser ${C_G(x) := \{ g \in G: gx=xg \}}$ of any non-identity element ${x \in G}$ is abelian; equivalently, the commuting relation ${x \sim y}$ (defined as the relation that holds when ${x}$ commutes with ${y}$, thus ${xy=yx}$) is an equivalence relation on the non-identity elements ${G \backslash \{1\}}$ of ${G}$. Trivially, every abelian group is CA. A non-abelian example of a CA-group is the ${ax+b}$ group of invertible affine transformations ${x \mapsto ax+b}$ on a field ${F}$. A little less obviously, the special linear group ${SL_2(F_q)}$ over a finite field ${F_q}$ is a CA-group when ${q}$ is a power of two. The finite simple groups of Lie type are not, in general, CA-groups, but when the rank is bounded they tend to behave as if they were “almost CA”; the centraliser of a generic element in ${SL_d(F_q)}$, for instance, when ${d}$ is bounded and ${q}$ is large), is typically a maximal torus (because most elements in ${SL_d(F_q)}$ are regular semisimple) which is certainly abelian. In view of the CFSG, we thus see that CA or nearly CA groups form an important subclass of the simple groups, and it is thus of interest to study them separately. To this end, we have

Theorem 1 (Suzuki’s theorem on CA-groups) Every finite CA-group of odd order is solvable.

Of course, this theorem is superceded by the more general Feit-Thompson theorem, but Suzuki’s proof is substantially shorter (the original proof is nine pages) and will be given in this post. (See this survey of Solomon for some discussion of the link between Suzuki’s argument and the Feit-Thompson argument.) Suzuki’s analysis can be pushed further to give an essentially complete classification of all the finite CA-groups (of either odd or even order), but we will not pursue these matters here.

Moving even further down the ladder of simple precursors of CSFG is the following theorem of Frobenius from 1901. Define a Frobenius group to be a finite group ${G}$ which has a subgroup ${H}$ (called the Frobenius complement) with the property that all the non-trivial conjugates ${gHg^{-1}}$ of ${H}$ for ${g \in G \backslash H}$, intersect ${H}$ only at the origin. For instance the ${ax+b}$ group is also a Frobenius group (take ${H}$ to be the affine transformations that fix a specified point ${x_0 \in F}$, e.g. the origin). This example suggests that there is some overlap between the notions of a Frobenius group and a CA group. Indeed, note that if ${G}$ is a CA-group and ${H}$ is a maximal abelian subgroup of ${G}$, then any conjugate ${gHg^{-1}}$ of ${H}$ that is not identical to ${H}$ will intersect ${H}$ only at the origin (because ${H}$ and each of its conjugates consist of equivalence classes under the commuting relation ${\sim}$, together with the identity). So if a maximal abelian subgroup ${H}$ of a CA-group is its own normaliser (thus ${N(H) := \{ g \in G: gH=Hg\}}$ is equal to ${H}$), then the group is a Frobenius group.

Frobenius’ theorem places an unexpectedly strong amount of structure on a Frobenius group:

Theorem 2 (Frobenius’ theorem) Let ${G}$ be a Frobenius group with Frobenius complement ${H}$. Then there exists a normal subgroup ${K}$ of ${G}$ (called the Frobenius kernel of ${G}$) such that ${G}$ is the semi-direct product ${H \ltimes K}$ of ${H}$ and ${K}$.

Roughly speaking, this theorem indicates that all Frobenius groups “behave” like the ${ax+b}$ example (which is a quintessential example of a semi-direct product).

Note that if every CA-group of odd order was either Frobenius or abelian, then Theorem 2 would imply Theorem 1 by an induction on the order of ${G}$, since any subgroup of a CA-group is clearly again a CA-group. Indeed, the proof of Suzuki’s theorem does basically proceed by this route (Suzuki’s arguments do indeed imply that CA-groups of odd order are Frobenius or abelian, although we will not quite establish that fact here).

Frobenius’ theorem can be reformulated in the following concrete combinatorial form:

Theorem 3 (Frobenius’ theorem, equivalent version) Let ${G}$ be a group of permutations acting transitively on a finite set ${X}$, with the property that any non-identity permutation in ${G}$ fixes at most one point in ${X}$. Then the set of permutations in ${G}$ that fix no points in ${X}$, together with the identity, is closed under composition.

Again, a good example to keep in mind for this theorem is when ${G}$ is the group of affine permutations on a field ${F}$ (i.e. the ${ax+b}$ group for that field), and ${X}$ is the set of points on that field. In that case, the set of permutations in ${G}$ that do not fix any points are the non-trivial translations.

To deduce Theorem 3 from Theorem 2, one applies Theorem 2 to the stabiliser of a single point in ${X}$. Conversely, to deduce Theorem 2 from Theorem 3, set ${X := G/H = \{ gH: g \in G \}}$ to be the space of left-cosets of ${H}$, with the obvious left ${G}$-action; one easily verifies that this action is faithful, transitive, and each non-identity element ${g}$ of ${G}$ fixes at most one left-coset of ${H}$ (basically because it lies in at most one conjugate of ${H}$). If we let ${K}$ be the elements of ${G}$ that do not fix any point in ${X}$, plus the identity, then by Theorem 3 ${K}$ is closed under composition; it is also clearly closed under inverse and conjugation, and is hence a normal subgroup of ${G}$. From construction ${K}$ is the identity plus the complement of all the ${|G|/|H|}$ conjugates of ${H}$, which are all disjoint except at the identity, so by counting elements we see that

$\displaystyle |K| = |G| - \frac{|G|}{|H|}(|H|-1) = |G|/|H|.$

As ${H}$ normalises ${K}$ and is disjoint from ${K}$, we thus see that ${KH = H \ltimes K}$ is all of ${G}$, giving Theorem 2.

Despite the appealingly concrete and elementary form of Theorem 3, the only known proofs of that theorem (or equivalently, Theorem 2) in its full generality proceed via the machinery of group characters (which one can think of as a version of Fourier analysis for nonabelian groups). On the other hand, once one establishes the basic theory of these characters (reviewed below the fold), the proof of Frobenius’ theorem is very short, which gives quite a striking example of the power of character theory. The proof of Suzuki’s theorem also proceeds via character theory, and is basically a more involved version of the Frobenius argument; again, no character-free proof of Suzuki’s theorem is currently known. (The proofs of Feit-Thompson and CFSG also involve characters, but those proofs also contain many other arguments of much greater complexity than the character-based portions of the proof.)

It seems to me that the above four theorems (Frobenius, Suzuki, Feit-Thompson, and CFSG) provide a ladder of sorts (with exponentially increasing complexity at each step) to the full classification, and that any new approach to the classification might first begin by revisiting the earlier theorems on this ladder and finding new proofs of these results first (in particular, if one had a “robust” proof of Suzuki’s theorem that also gave non-trivial control on “almost CA-groups” – whatever that means – then this might lead to a new route to classifying the finite simple groups of Lie type and bounded rank). But even for the simplest two results on this ladder – Frobenius and Suzuki – it seems remarkably difficult to find any proof that is not essentially the character-based proof. (Even trying to replace character theory by its close cousin, representation theory, doesn’t seem to work unless one gives in to the temptation to take traces everywhere and put the characters back in; it seems that rather than abandon characters altogether, one needs to find some sort of “robust” generalisation of existing character-based methods.) In any case, I am recording here the standard character-based proofs of the theorems of Frobenius and Suzuki below the fold. There is nothing particularly novel here, but I wanted to collect all the relevant material in one place, largely for my own benefit.

One of the basic objects of study in combinatorics are finite strings ${(a_n)_{n=0}^N}$ or infinite strings ${(a_n)_{n=0}^\infty}$ of symbols ${a_n}$ from some given alphabet ${{\mathcal A}}$, which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set ${A}$ of natural numbers can be identified with the infinite string ${(1_A(n))_{n=0}^\infty}$ of ${0}$s and ${1}$s formed by the indicator of ${A}$, e.g. the even numbers can be identified with the string ${1010101\ldots}$ from the alphabet ${\{0,1\}}$, the multiples of three can be identified with the string ${100100100\ldots}$, and so forth. One can also consider doubly infinite strings ${(a_n)_{n \in {\bf Z}}}$, 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 ${(X,T)}$, that is to say a space ${X}$ together with a shift map ${T: X \rightarrow X}$ (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 ${{\bf N}}$ on the space ${X}$ by using the iterates ${T^n: X \rightarrow X}$ of ${T}$ for ${n=0,1,2,\ldots}$; if ${T}$ is invertible, we can extend this action to an action of the integers ${{\bf Z}}$ on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than ${{\bf N}}$ or ${{\bf Z}}$ (e.g. one can consider continuous dynamical systems in which the evolution group is ${{\bf R}}$), but we will restrict attention to the classical situation of ${{\bf N}}$ or ${{\bf Z}}$ 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 ${(X,T)}$, an observable ${c: X \rightarrow {\mathcal A}}$ taking values in some alphabet ${{\mathcal A}}$, and some initial datum ${x_0 \in X}$, we can first form the forward orbit ${(T^n x_0)_{n=0}^\infty}$ of ${x_0}$, and then observe this orbit using ${c}$ to obtain an infinite string ${(c(T^n x_0))_{n=0}^\infty}$. If the shift ${T}$ in this system is invertible, one can extend this infinite string into a doubly infinite string ${(c(T^n x_0))_{n \in {\bf Z}}}$. Thus we see that every quadruplet ${(X,T,c,x_0)}$ consisting of a dynamical system ${(X,T)}$, an observable ${c}$, and an initial datum ${x_0}$ creates an infinite string.

Example 1 If ${X}$ is the three-element set ${X = {\bf Z}/3{\bf Z}}$ with the shift map ${Tx := x+1}$, ${c: {\bf Z}/3{\bf Z} \rightarrow \{0,1\}}$ is the observable that takes the value ${1}$ at the residue class ${0 \hbox{ mod } 3}$ and zero at the other two classes, and one starts with the initial datum ${x_0 = 0 \hbox{ mod } 3}$, then the observed string ${(c(T^n x_0))_{n=0}^\infty}$ becomes the indicator ${100100100\ldots}$ of the multiples of three.

In the converse direction, every infinite string ${(a_n){n=0}^\infty}$ in some alphabet ${{\mathcal A}}$ arises (in a decidedly non-unique fashion) from a quadruple ${(X,T,c,x_0)}$ in the above fashion. This can be easily seen by the following “universal” construction: take ${X}$ to be the set ${X:= {\mathcal A}^{\bf N}}$ of infinite strings ${(b_i)_{n=0}^\infty}$ in the alphabet ${{\mathcal A}}$, let ${T: X \rightarrow X}$ be the shift map

$\displaystyle T(b_i)_{n=0}^\infty := (b_{i+1})_{n=0}^\infty,$

let ${c: X \rightarrow {\mathcal A}}$ be the observable

$\displaystyle c((b_i)_{n=0}^\infty) := b_0,$

and let ${x_0 \in X}$ be the initial point

$\displaystyle x_0 := (a_i)_{n=0}^\infty.$

Then one easily sees that the observed string ${(c(T^n x_0))_{n=0}^\infty}$ is nothing more than the original string ${(b_i)_{n=0}^\infty}$. Note also that this construction can easily be adapted to doubly infinite strings by using ${{\mathcal A}^{\bf Z}}$ instead of ${{\mathcal A}^{\bf N}}$, at which point the shift map ${T}$ now becomes invertible. An important variant of this construction also attaches an invariant probability measure to ${X}$ that is associated to the limiting density of various sets associated to the string ${(a_i)_{n=0}^\infty}$, 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 ${{\mathcal A}}$ is the binary alphabet ${\{0,1\}}$, and (for technical reasons related to the infamous non-injectivity ${0.999\ldots = 1.00\ldots}$ of the decimal representation system) the string ${(a_n)_{n=0}^\infty}$ does not end with an infinite string of ${1}$s, then one can reformulate the above universal construction by taking ${X}$ to be the interval ${[0,1)}$, ${T}$ to be the doubling map ${Tx := 2x \hbox{ mod } 1}$, ${c: X \rightarrow \{0,1\}}$ to be the observable that takes the value ${1}$ on ${[1/2,1)}$ and ${0}$ on ${[0,1/2)}$ (that is, ${c(x)}$ is the first binary digit of ${x}$), and ${x_0}$ is the real number ${x_0 := \sum_{n=0}^\infty a_n 2^{-n-1}}$ (that is, ${x_0 = 0.a_0a_1\ldots}$ in binary).

The above universal construction is very easy to describe, and is well suited for “generic” strings ${(a_n)_{n=0}^\infty}$ 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 ${(a_n)_{n=0}^\infty}$, and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator ${100100100\ldots}$ of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space ${X}$ 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 ${2/7}$ 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 ${X,T,c,x_0}$ of the quadruplet ${(X,T,c,x_0)}$ used to generate the sequence ${(a_n)_{n=0}^\infty}$, three of the components ${X,T,c}$ are completely universal (in that they do not depend at all on the sequence ${(a_n)_{n=0}^\infty}$), leaving only the initial datum ${x_0}$ 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 ${X}$ to the orbit ${\{ T^n x_0: n \in {\bf N} \}}$ of the initial datum ${x_0}$ (actually for technical reasons it is better to restrict to the topological closure ${\overline{\{ T^n x_0: n \in {\bf N} \}}}$ of this orbit, in order to keep ${X}$ compact). For instance, starting with the sequence ${100100100\ldots}$, the orbit now consists of just three points ${100100100\ldots}$, ${010010010\ldots}$, ${001001001\ldots}$, bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet ${(X,T,c,x_0)}$, because any other such representation ${(X',T',c',x'_0)}$ is a factor of this representation (in the sense that there is a unique map ${\pi: X \rightarrow X'}$ with ${T' \circ \pi = \pi \circ T}$, ${c' \circ \pi = c}$, and ${x'_0 = \pi(x_0)}$). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system ${X}$ 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 ${(a_n)_{n=0}^\infty}$, 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 ${(a_n)_{n=0}^\infty}$, one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple ${(X,T,c,x_0)}$ 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 ${{\bf Z}}$ or the complex numbers ${{\bf C}}$. 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 ${A}$ be a subset of the additive group ${{\bf Z}/p{\bf Z}}$ for some prime ${p}$, and let ${s \geq 1}$ be an integer. Suppose that ${|A| \leq \log_{2s} p}$. Then there exists a map ${\phi: A \rightarrow A'}$ into a subset ${A'}$ of the integers which is a Freiman isomorphism of order ${s}$ in the sense that for any ${x_1,\ldots,x_s,y_1,\ldots,y_s \in A}$, one has

$\displaystyle x_1+\ldots+x_s = y_1+\ldots+y_s$

if and only if

$\displaystyle \phi(x_1)+\ldots+\phi(x_s) = \phi(y_1)+\ldots+\phi(y_s).$

Furthermore ${\phi}$ is a right-inverse of the obvious projection homomorphism from ${{\bf Z}}$ to ${{\bf Z}/p{\bf Z}}$.

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 ${p}$), 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 ${aA}$ of ${A}$ that is contained in a small neighbourhood of the origin in ${{\bf Z}/p{\bf Z}}$, at which point the rectification map ${\phi}$ can be constructed by hand.

Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map ${\phi}$ preserves both additive and multiplicative structure:

Theorem 2 (Arithmetic rectification) Let ${A}$ be a subset of the finite field ${{\bf F}_p}$ for some prime ${p \geq 3}$, and let ${s \geq 1}$ be an integer. Suppose that ${|A| < \log_2 \log_{2s} \log_{2s^2} p - 1}$. Then there exists a map ${\phi: A \rightarrow A'}$ into a subset ${A'}$ of the complex numbers which is a Freiman field isomorphism of order ${s}$ in the sense that for any ${x_1,\ldots,x_n \in A}$ and any polynomial ${P(x_1,\ldots,x_n)}$ of degree at most ${s}$ and integer coefficients of magnitude summing to at most ${s}$, one has

$\displaystyle P(x_1,\ldots,x_n)=0$

if and only if

$\displaystyle P(\phi(x_1),\ldots,\phi(x_n))=0.$

Note that it is necessary to use an algebraically closed field such as ${{\bf C}}$ for this theorem, in contrast to the integers used in Proposition 1, as ${{\bf F}_p}$ can contain objects such as square roots of ${-1}$ which can only map to ${\pm i}$ in the complex numbers (once ${s}$ is at least ${2}$).

Using Theorem 2, one can transfer results in arithmetic combinatorics (e.g. sum-product or Szemerédi-Trotter type theorems) regarding finite subsets of ${{\bf C}}$ to analogous results regarding sufficiently small subsets of ${{\bf F}_p}$; 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 ${{\bf C}}$ (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of ${{\bf F}_p}$ for arbitrarily large primes ${p}$, 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 ${s, n \geq 1}$. Then if ${{\bf F}}$ is a field of characteristic at least ${C_{s,n}}$ for some ${C_{s,n}}$ depending on ${s,n}$, and ${A}$ is a subset of ${{\bf F}}$ of cardinality ${n}$, then there exists a map ${\phi: A \rightarrow A'}$ into a subset ${A'}$ of the complex numbers which is a Freiman field isomorphism of order ${s}$.

Our arguments will not provide any effective bound on the quantity ${C_{s,n}}$ (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:

Proposition 4 (Baby Lefschetz principle) Let ${k}$ be a field of characteristic zero that is finitely generated over the rationals. Then there is an isomorphism ${\phi: k \rightarrow \phi(k)}$ from ${k}$ to a subfield ${\phi(k)}$ of ${{\bf C}}$.

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 ${k}$ is finitely generated over ${{\bf Q}}$, it has finite transcendence degree, thus one can find algebraically independent elements ${x_1,\ldots,x_m}$ of ${k}$ over ${{\bf Q}}$ such that ${k}$ is a finite extension of ${{\bf Q}(x_1,\ldots,x_m)}$, and in particular by the primitive element theorem ${k}$ is generated by ${{\bf Q}(x_1,\ldots,x_m)}$ and an element ${\alpha}$ which is algebraic over ${{\bf Q}(x_1,\ldots,x_m)}$. (Here we use the fact that characteristic zero fields are separable.) If we then define ${\phi}$ by first mapping ${x_1,\ldots,x_m}$ to generic (and thus algebraically independent) complex numbers ${z_1,\ldots,z_m}$, and then setting ${\phi(\alpha)}$ to be a complex root of of the minimal polynomial for ${\alpha}$ over ${{\bf Q}(x_1,\ldots,x_m)}$ after replacing each ${x_i}$ with the complex number ${z_i}$, we obtain a field isomorphism ${\phi: k \rightarrow \phi(k)}$ with the required properties.

Now we give the latter proof. Let ${x_1,\ldots,x_m}$ be elements of ${k}$ that generate that field over ${{\bf Q}}$, but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers ${z_1,\ldots,z_m}$ with the property that, for any polynomial ${P(x_1,\ldots,x_m)}$ with rational coefficients, one has

$\displaystyle P(x_1,\ldots,x_m) = 0$

if and only if

$\displaystyle P(z_1,\ldots,z_m) = 0.$

Let ${{\mathcal P}}$ be the collection of all polynomials ${P}$ with rational coefficients with ${P(x_1,\ldots,x_m)=0}$, and ${{\mathcal Q}}$ be the collection of all polynomials ${P}$ with rational coefficients with ${P(x_1,\ldots,x_m) \neq 0}$. The set

$\displaystyle S := \{ (z_1,\ldots,z_m) \in {\bf C}^m: P(z_1,\ldots,z_m)=0 \hbox{ for all } P \in {\mathcal P} \}$

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 ${S}$ could be covered by the algebraic sets ${\{ (z_1,\ldots,z_m) \in {\bf C}^m: Q(z_1,\ldots,z_m) = 0 \}}$ for ${Q \in {\mathcal Q}}$. By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over ${{\bf C}}$ cannot be covered by countably many varieties of smaller dimension, we conclude that ${S}$ must in fact be covered by a finite number of such sets, thus

$\displaystyle S \subset \bigcup_{i=1}^n \{ (z_1,\ldots,z_m) \in {\bf C}^m: Q_i(z_1,\ldots,z_m) = 0 \}$

for some ${Q_1,\ldots,Q_n \in {\bf C}^m}$. By the nullstellensatz, we thus have an identity of the form

$\displaystyle (Q_1 \ldots Q_n)^l = P_1 R_1 + \ldots + P_r R_r$

for some natural numbers ${l,r \geq 1}$, polynomials ${P_1,\ldots,P_r \in {\mathcal P}}$, and polynomials ${R_1,\ldots,R_r}$ with coefficients in ${\overline{{\bf Q}}}$. In particular, this identity also holds in the algebraic closure ${\overline{k}}$ of ${k}$. Evaluating this identity at ${(x_1,\ldots,x_m)}$ we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows. $\Box$

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 ${s,n \geq 1}$, a sequence of finite fields ${{\bf F}_i}$ of characteristic at least ${i}$, and subsets ${A_i=\{a_{i,1},\ldots,a_{i,n}\}}$ of ${{\bf F}_i}$ of cardinality ${n}$ such that for each ${i}$, there does not exist a Freiman field isomorphism of order ${s}$ from ${A_i}$ to the complex numbers. Now we select a non-principal ultrafilter ${\alpha \in \beta {\bf N} \backslash {\bf N}}$, and construct the ultraproduct ${{\bf F} := \prod_{i \rightarrow \alpha} {\bf F}_i}$ of the finite fields ${{\bf F}_i}$. This is again a field (and is a basic example of what is known as a pseudo-finite field); because the characteristic of ${{\bf F}_i}$ goes to infinity as ${i \rightarrow \infty}$, it is easy to see (using Los’s theorem) that ${{\bf F}}$ has characteristic zero and can thus be viewed as an extension of the rationals ${{\bf Q}}$.

Now let ${a_j := \lim_{i \rightarrow \alpha} a_{i,j}}$ be the ultralimit of the ${a_{i,j}}$, so that ${A := \{a_1,\ldots,a_n\}}$ is the ultraproduct of the ${A_i}$, then ${A}$ is a subset of ${{\bf F}}$ of cardinality ${n}$. In particular, if ${k}$ is the field generated by ${{\bf Q}}$ and ${A}$, then ${k}$ is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism ${\phi: k \rightarrow \phi(k)}$ from ${k}$ to a subfield ${\phi(k)}$ of the complex numbers. In particular, ${\phi(a_1),\ldots,\phi(a_n)}$ are complex numbers, and for any polynomial ${P(x_1,\ldots,x_n)}$ with integer coefficients, one has

$\displaystyle P(a_1,\ldots,a_n) = 0$

if and only if

$\displaystyle P(\phi(a_1),\ldots,\phi(a_n)) = 0.$

By Los’s theorem, we then conclude that for all ${i}$ sufficiently close to ${\alpha}$, one has for all polynomials ${P(x_1,\ldots,x_n)}$ of degree at most ${s}$ and whose coefficients are integers whose magnitude sums up to ${s}$, one has

$\displaystyle P(a_{i,1},\ldots,a_{i,n}) = 0$

if and only if

$\displaystyle P(\phi(a_1),\ldots,\phi(a_n)) = 0.$

But this gives a Freiman field isomorphism of order ${s}$ between ${A_i}$ and ${\phi(A)}$, contradicting the construction of ${A_i}$, and Theorem 3 follows.

The following result is due independently to Furstenberg and to Sarkozy:

Theorem 1 (Furstenberg-Sarkozy theorem) Let ${\delta > 0}$, and suppose that ${N}$ is sufficiently large depending on ${\delta}$. Then every subset ${A}$ of ${[N] := \{1,\ldots,N\}}$ of density ${|A|/N}$ at least ${\delta}$ contains a pair ${n, n+r^2}$ for some natural numbers ${n, r}$ with ${r \neq 0}$.

This theorem is of course similar in spirit to results such as Roth’s theorem or Szemerédi’s theorem, in which the pattern ${n,n+r^2}$ is replaced by ${n,n+r,n+2r}$ or ${n,n+r,\ldots,n+(k-1)r}$ for some fixed ${k}$ respectively. There are by now many proofs of this theorem (see this recent paper of Lyall for a survey), but most proofs involve some form of Fourier analysis (or spectral theory). This may be compared with the standard proof of Roth’s theorem, which combines some Fourier analysis with what is now known as the density increment argument.

A few years ago, Ben Green, Tamar Ziegler, and myself observed that it is possible to prove the Furstenberg-Sarkozy theorem by just using the Cauchy-Schwarz inequality (or van der Corput lemma) and the density increment argument, removing all invocations of Fourier analysis, and instead relying on Cauchy-Schwarz to linearise the quadratic shift ${r^2}$. As such, this theorem can be considered as even more elementary than Roth’s theorem (and its proof can be viewed as a toy model for the proof of Roth’s theorem). We ended up not doing too much with this observation, so decided to share it here.

The first step is to use the density increment argument that goes back to Roth. For any ${\delta > 0}$, let ${P(\delta)}$ denote the assertion that for ${N}$ sufficiently large, all sets ${A \subset [N]}$ of density at least ${\delta}$ contain a pair ${n,n+r^2}$ with ${r}$ non-zero. Note that ${P(\delta)}$ is vacuously true for ${\delta > 1}$. We will show that for any ${0 < \delta_0 \leq 1}$, one has the implication

$\displaystyle P(\delta_0 + c \delta_0^3) \implies P(\delta_0) \ \ \ \ \ (1)$

for some absolute constant ${c>0}$. This implies that ${P(\delta)}$ is true for any ${\delta>0}$ (as can be seen by considering the infimum of all ${\delta>0}$ for which ${P(\delta)}$ holds), which gives Theorem 1.

It remains to establish the implication (1). Suppose for sake of contradiction that we can find ${0 < \delta_0 \leq 1}$ for which ${P(\delta_0+c\delta^3_0)}$ holds (for some sufficiently small absolute constant ${c>0}$), but ${P(\delta_0)}$ fails. Thus, we can find arbitrarily large ${N}$, and subsets ${A}$ of ${[N]}$ of density at least ${\delta_0}$, such that ${A}$ contains no patterns of the form ${n,n+r^2}$ with ${r}$ non-zero. In particular, we have

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} 1_A(n) 1_A(n+(r+h)^2) = 0.$

(The exact ranges of ${r}$ and ${h}$ are not too important here, and could be replaced by various other small powers of ${N}$ if desired.)

Let ${\delta := |A|/N}$ be the density of ${A}$, so that ${\delta_0 \leq \delta \leq 1}$. Observe that

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} 1_A(n) \delta 1_{[N]}(n+(r+h)^2) = \delta^2 + O(N^{-1/3})$

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} \delta 1_{[N]}(n) \delta 1_{[N]}(n+(r+h)^2) = \delta^2 + O(N^{-1/3})$

and

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} \delta 1_{[N]}(n) 1_A(n+(r+h)^2) = \delta^2 + O( N^{-1/3} ).$

If we thus set ${f := 1_A - \delta 1_{[N]}}$, then

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} f(n) f(n+(r+h)^2) = -\delta^2 + O( N^{-1/3} ).$

In particular, for ${N}$ large enough,

$\displaystyle \mathop{\bf E}_{n \in [N]} |f(n)| \mathop{\bf E}_{r \in [N^{1/3}]} |\mathop{\bf E}_{h \in [N^{1/100}]} f(n+(r+h)^2)| \gg \delta^2.$

On the other hand, one easily sees that

$\displaystyle \mathop{\bf E}_{n \in [N]} |f(n)|^2 = O(\delta)$

and hence by the Cauchy-Schwarz inequality

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} |\mathop{\bf E}_{h \in [N^{1/100}]} f(n+(r+h)^2)|^2 \gg \delta^3$

which we can rearrange as

$\displaystyle |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h,h' \in [N^{1/100}]} \mathop{\bf E}_{n \in [N]} f(n+(r+h)^2) f(n+(r+h')^2)| \gg \delta^3.$

Shifting ${n}$ by ${(r+h)^2}$ we obtain (again for ${N}$ large enough)

$\displaystyle |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h,h' \in [N^{1/100}]} \mathop{\bf E}_{n \in [N]} f(n) f(n+(h'-h)(2r+h'+h))| \gg \delta^3.$

In particular, by the pigeonhole principle (and deleting the diagonal case ${h=h'}$, which we can do for ${N}$ large enough) we can find distinct ${h,h' \in [N^{1/100}]}$ such that

$\displaystyle |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{n \in [N]} f(n) f(n+(h'-h)(2r+h'+h))| \gg \delta^3,$

so in particular

$\displaystyle \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+(h'-h)(2r+h'+h))| \gg \delta^3.$

If we set ${d := 2(h'-h)}$ and shift ${n}$ by ${(h'-h) (h'+h)}$, we can simplify this (again for ${N}$ large enough) as

$\displaystyle \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr)| \gg \delta^3. \ \ \ \ \ (2)$

On the other hand, since

$\displaystyle \mathop{\bf E}_{n \in [N]} f(n) = 0$

we have

$\displaystyle \mathop{\bf E}_{n \in [N]} f(n+dr) = O( N^{-2/3+1/100})$

for any ${r \in [N^{1/3}]}$, and thus

$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) = O( N^{-2/3+1/100}).$

Averaging this with (2) we conclude that

$\displaystyle \mathop{\bf E}_{n \in [N]} \max( \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr), 0 ) \gg \delta^3.$

In particular, by the pigeonhole principle we can find ${n \in [N]}$ such that

$\displaystyle \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) \gg \delta^3,$

or equivalently ${A}$ has density at least ${\delta+c'\delta^3}$ on the arithmetic progression ${\{ n+dr: r \in [N^{1/3}]\}}$, which has length ${\lfloor N^{1/3}\rfloor }$ and spacing ${d}$, for some absolute constant ${c'>0}$. By partitioning this progression into subprogressions of spacing ${d^2}$ and length ${\lfloor N^{1/4}\rfloor}$ (plus an error set of size ${O(N^{1/4})}$, we see from the pigeonhole principle that we can find a progression ${\{ n' + d^2 r': r' \in [N^{1/4}]\}}$ of length ${\lfloor N^{1/4}\rfloor}$ and spacing ${d^2}$ on which ${A}$ has density at least ${\delta + c\delta^3}$ (and hence at least ${\delta_0+c\delta_0^3}$) for some absolute constant ${c>0}$. If we then apply the induction hypothesis to the set

$\displaystyle A' := \{ r' \in [N^{1/4}]: n' + d^2 r' \in A \}$

we conclude (for ${N}$ large enough) that ${A'}$ contains a pair ${m, m+s^2}$ for some natural numbers ${m,s}$ with ${s}$ non-zero. This implies that ${(n'+d^2 m), (n'+d^2 m) + (|d|s)^2}$ lie in ${A}$, a contradiction, establishing the implication (1).

A more careful analysis of the above argument reveals a more quantitative version of Theorem 1: for ${N \geq 100}$ (say), any subset of ${[N]}$ of density at least ${C/(\log\log N)^{1/2}}$ for some sufficiently large absolute constant ${C}$ contains a pair ${n,n+r^2}$ with ${r}$ non-zero. This is not the best bound known; a (difficult) result of Pintz, Steiger, and Szemeredi allows the density to be as low as ${C / (\log N)^{\frac{1}{4} \log\log\log\log N}}$. On the other hand, this already improves on the (simpler) Fourier-analytic argument of Green that works for densities at least ${C/(\log\log N)^{1/11}}$ (although the original argument of Sarkozy, which is a little more intricate, works up to ${C (\log\log N)^{2/3}/(\log N)^{1/3}}$). In the other direction, a construction of Rusza gives a set of density ${\frac{1}{65} N^{-0.267}}$ without any pairs ${n,n+r^2}$.

Remark 1 A similar argument also applies with ${n,n+r^2}$ replaced by ${n,n+r^k}$ for fixed ${k}$, because this sort of pattern is preserved by affine dilations ${r' \mapsto n'+d^k r'}$ into arithmetic progressions whose spacing ${d^k}$ is a ${k^{th}}$ power. By re-introducing Fourier analysis, one can also perform an argument of this type for ${n,n+d,n+2d}$ where ${d}$ is the sum of two squares; see the above-mentioned paper of Green for details. However there seems to be some technical difficulty in extending it to patterns of the form ${n,n+P(r)}$ for polynomials ${P}$ that consist of more than a single monomial (and with the normalisation ${P(0)=0}$, to avoid local obstructions), because one no longer has this preservation property.

The fundamental notions of calculus, namely differentiation and integration, are often viewed as being the quintessential concepts in mathematical analysis, as their standard definitions involve the concept of a limit. However, it is possible to capture most of the essence of these notions by purely algebraic means (almost completely avoiding the use of limits, Riemann sums, and similar devices), which turns out to be useful when trying to generalise these concepts to more abstract situations in which it becomes convenient to permit the underlying number systems involved to be something other than the real or complex numbers, even if this makes many standard analysis constructions unavailable. For instance, the algebraic notion of a derivation often serves as a substitute for the analytic notion of a derivative in such cases, by abstracting out the key algebraic properties of differentiation, namely linearity and the Leibniz rule (also known as the product rule).

Abstract algebraic analogues of integration are less well known, but can still be developed. To motivate such an abstraction, consider the integration functional ${I: {\mathcal S}({\bf R} \rightarrow {\bf C}) \rightarrow {\bf C}}$ from the space ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ of complex-valued Schwarz functions ${f: {\bf R} \rightarrow {\bf C}}$ to the complex numbers, defined by

$\displaystyle I(f) := \int_{\bf R} f(x)\ dx$

where the integration on the right is the usual Lebesgue integral (or improper Riemann integral) from analysis. This functional obeys two obvious algebraic properties. Firstly, it is linear over ${{\bf C}}$, thus

$\displaystyle I(cf) = c I(f) \ \ \ \ \ (1)$

and

$\displaystyle I(f+g) = I(f) + I(g) \ \ \ \ \ (2)$

for all ${f,g \in {\mathcal S}({\bf R} \rightarrow {\bf C})}$ and ${c \in {\bf C}}$. Secondly, it is translation invariant, thus

$\displaystyle I(\tau_h f) = I(f) \ \ \ \ \ (3)$

for all ${h \in {\bf C}}$, where ${\tau_h f(x) := f(x-h)}$ is the translation of ${f}$ by ${h}$. Motivated by the uniqueness theory of Haar measure, one might expect that these two axioms already uniquely determine ${I}$ after one sets a normalisation, for instance by requiring that

$\displaystyle I( x \mapsto e^{-\pi x^2} ) = 1. \ \ \ \ \ (4)$

This is not quite true as stated (one can modify the proof of the Hahn-Banach theorem, after first applying a Fourier transform, to create pathological translation-invariant linear functionals on ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ that are not multiples of the standard Fourier transform), but if one adds a mild analytical axiom, such as continuity of ${I}$ (using the usual Schwartz topology on ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$), then the above axioms are enough to uniquely pin down the notion of integration. Indeed, if ${I: {\mathcal S}({\bf R} \rightarrow {\bf C}) \rightarrow {\bf C}}$ is a continuous linear functional that is translation invariant, then from the linearity and translation invariance axioms one has

$\displaystyle I( \frac{\tau_h f - f}{h} ) = 0$

for all ${f \in {\mathcal S}({\bf R} \rightarrow {\bf C})}$ and non-zero reals ${h}$. If ${f}$ is Schwartz, then as ${h \rightarrow 0}$, one can verify that the Newton quotients ${\frac{\tau_h f - f}{h}}$ converge in the Schwartz topology to the derivative ${f'}$ of ${f}$, so by the continuity axiom one has

$\displaystyle I(f') = 0.$

Next, note that any Schwartz function of integral zero has an antiderivative which is also Schwartz, and so ${I}$ annihilates all zero-integral Schwartz functions, and thus must be a scalar multiple of the usual integration functional. Using the normalisation (4), we see that ${I}$ must therefore be the usual integration functional, giving the claimed uniqueness.

Motivated by the above discussion, we can define the notion of an abstract integration functional ${I: X \rightarrow R}$ taking values in some vector space ${R}$, and applied to inputs ${f}$ in some other vector space ${X}$ that enjoys a linear action ${h \mapsto \tau_h}$ (the “translation action”) of some group ${V}$, as being a functional which is both linear and translation invariant, thus one has the axioms (1), (2), (3) for all ${f,g \in X}$, scalars ${c}$, and ${h \in V}$. The previous discussion then considered the special case when ${R = {\bf C}}$, ${X = {\mathcal S}({\bf R} \rightarrow {\bf C})}$, ${V = {\bf R}}$, and ${\tau}$ was the usual translation action.

Once we have performed this abstraction, we can now present analogues of classical integration which bear very little analytic resemblance to the classical concept, but which still have much of the algebraic structure of integration. Consider for instance the situation in which we keep the complex range ${R = {\bf C}}$, the translation group ${V = {\bf R}}$, and the usual translation action ${h \mapsto \tau_h}$, but we replace the space ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ of Schwartz functions by the space ${Poly_{\leq d}({\bf R} \rightarrow {\bf C})}$ of polynomials ${x \mapsto a_0 + a_1 x + \ldots + a_d x^d}$ of degree at most ${d}$ with complex coefficients, where ${d}$ is a fixed natural number; note that this space is translation invariant, so it makes sense to talk about an abstract integration functional ${I: Poly_{\leq d}({\bf R} \rightarrow {\bf C}) \rightarrow {\bf C}}$. Of course, one cannot apply traditional integration concepts to non-zero polynomials, as they are not absolutely integrable. But one can repeat the previous arguments to show that any abstract integration functional must annihilate derivatives of polynomials of degree at most ${d}$:

$\displaystyle I(f') = 0 \hbox{ for all } f \in Poly_{\leq d}({\bf R} \rightarrow {\bf C}). \ \ \ \ \ (5)$

Clearly, every polynomial of degree at most ${d-1}$ is thus annihilated by ${I}$, which makes ${I}$ a scalar multiple of the functional that extracts the top coefficient ${a_d}$ of a polynomial, thus if one sets a normalisation

$\displaystyle I( x \mapsto x^d ) = c$

for some constant ${c}$, then one has

$\displaystyle I( x \mapsto a_0 + a_1 x + \ldots + a_d x^d ) = c a_d \ \ \ \ \ (6)$

for any polynomial ${x \mapsto a_0 + a_1 x + \ldots + a_d x^d}$. So we see that up to a normalising constant, the operation of extracting the top order coefficient of a polynomial of fixed degree serves as the analogue of integration. In particular, despite the fact that integration is supposed to be the “opposite” of differentiation (as indicated for instance by (5)), we see in this case that integration is basically (${d}$-fold) differentiation; indeed, compare (6) with the identity

$\displaystyle (\frac{d}{dx})^d ( a_0 + a_1 x + \ldots + a_d x^d ) = d! a_d.$

In particular, we see, in contrast to the usual Lebesgue integral, the integration functional (6) can be localised to an arbitrary location: one only needs to know the germ of the polynomial ${x \mapsto a_0 + a_1 x + \ldots + a_d x^d}$ at a single point ${x_0}$ in order to determine the value of the functional (6). This localisation property may initially seem at odds with the translation invariance, but the two can be reconciled thanks to the extremely rigid nature of the class ${Poly_{\leq d}({\bf R} \rightarrow {\bf C})}$, in contrast to the Schwartz class ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ which admits bump functions and so can generate local phenomena that can only be detected in small regions of the underlying spatial domain, and which therefore forces any translation-invariant integration functional on such function classes to measure the function at every single point in space.

The reversal of the relationship between integration and differentiation is also reflected in the fact that the abstract integration operation on polynomials interacts with the scaling operation ${\delta_\lambda f(x) := f(x/\lambda)}$ in essentially the opposite way from the classical integration operation. Indeed, for classical integration on ${{\bf R}^d}$, one has

$\displaystyle \int_{{\bf R}^d} f(x/\lambda)\ dx = \lambda^d \int f(x)\ dx$

for Schwartz functions ${f \in {\mathcal S}({\bf R}^d \rightarrow {\bf C})}$, and so in this case the integration functional ${I(f) := \int_{{\bf R}^d} f(x)\ dx}$ obeys the scaling law

$\displaystyle I( \delta_\lambda f ) = \lambda^d I(f).$

In contrast, the abstract integration operation defined in (6) obeys the opposite scaling law

$\displaystyle I( \delta_\lambda f ) = \lambda^{-d} I(f). \ \ \ \ \ (7)$

Remark 1 One way to interpret what is going on is to view the integration operation (6) as a renormalised version of integration. A polynomial ${x \mapsto a_0 + a_1 + \ldots + a_d x^d}$ is, in general, not absolutely integrable, and the partial integrals

$\displaystyle \int_0^N a_0 + a_1 + \ldots + a_d x^d\ dx$

diverge as ${N \rightarrow \infty}$. But if one renormalises these integrals by the factor ${\frac{1}{N^{d+1}}}$, then one recovers convergence,

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{d+1}} \int_0^N a_0 + a_1 + \ldots + a_d x^d\ dx = \frac{1}{d+1} a_d$

thus giving an interpretation of (6) as a renormalised classical integral, with the renormalisation being responsible for the unusual scaling relationship in (7). However, this interpretation is a little artificial, and it seems that it is best to view functionals such as (6) from an abstract algebraic perspective, rather than to try to force an analytic interpretation on them.

$\displaystyle I(f) := \int_{\bf R} f(x)\ dx. \ \ \ \ \ (8)$

As noted earlier, this integration functional has a translation invariance associated to translations along the real line ${{\bf R}}$, as well as a dilation invariance by real dilation parameters ${\lambda>0}$. However, if we refine the class ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ of functions somewhat, we can obtain a stronger family of invariances, in which we allow complex translations and dilations. More precisely, let ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$ denote the space of all functions ${f: {\bf C} \rightarrow {\bf C}}$ which are entire (or equivalently, are given by a Taylor series with an infinite radius of convergence around the origin) and also admit rapid decay in a sectorial neighbourhood of the real line, or more precisely there exists an ${\epsilon>0}$ such that for every ${A > 0}$ there exists ${C_A > 0}$ such that one has the bound

$\displaystyle |f(z)| \leq C_A (1+|z|)^{-A}$

whenever ${|\hbox{Im}(z)| \leq A + \epsilon |\hbox{Re}(z)|}$. For want of a better name, we shall call elements of this space Schwartz entire functions. This is clearly a complex vector space. A typical example of a Schwartz entire function are the complex gaussians

$\displaystyle f(z) := e^{-\pi (az^2 + 2bz + c)}$

where ${a,b,c}$ are complex numbers with ${\hbox{Re}(a) > 0}$. From the Cauchy integral formula (and its derivatives) we see that if ${f}$ lies in ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$, then the restriction of ${f}$ to the real line lies in ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$; conversely, from analytic continuation we see that every function in ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ has at most one extension in ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$. Thus one can identify ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$ with a subspace of ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$, and in particular the integration functional (8) is inherited by ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$, and by abuse of notation we denote the resulting functional ${I: \mathcal{SE}({\bf C} \rightarrow {\bf C}) \rightarrow {\bf C}}$ as ${I}$ also. Note, in analogy with the situation with polynomials, that this abstract integration functional is somewhat localised; one only needs to evaluate the function ${f}$ on the real line, rather than the entire complex plane, in order to compute ${I(f)}$. This is consistent with the rigid nature of Schwartz entire functions, as one can uniquely recover the entire function from its values on the real line by analytic continuation.

Of course, the functional ${I: \mathcal{SE}({\bf C} \rightarrow {\bf C}) \rightarrow {\bf C}}$ remains translation invariant with respect to real translation:

$\displaystyle I(\tau_h f) = I(f) \hbox{ for all } h \in {\bf R}.$

However, thanks to contour shifting, we now also have translation invariance with respect to complex translation:

$\displaystyle I(\tau_h f) = I(f) \hbox{ for all } h \in {\bf C},$

where of course we continue to define the translation operator ${\tau_h}$ for complex ${h}$ by the usual formula ${\tau_h f(x) := f(x-h)}$. In a similar vein, we also have the scaling law

$\displaystyle I(\delta_\lambda f) = \lambda I(f)$

for any ${f \in \mathcal{SE}({\bf C} \rightarrow {\bf C})}$, if ${\lambda}$ is a complex number sufficiently close to ${1}$ (where “sufficiently close” depends on ${f}$, and more precisely depends on the sectoral aperture parameter ${\epsilon}$ associated to ${f}$); again, one can verify that ${\delta_\lambda f}$ lies in ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$ for ${\lambda}$ sufficiently close to ${1}$. These invariances (which relocalise the integration functional ${I}$ onto other contours than the real line ${{\bf R}}$) are very useful for computing integrals, and in particular for computing gaussian integrals. For instance, the complex translation invariance tells us (after shifting by ${b/a}$) that

$\displaystyle I( z \mapsto e^{-\pi (az^2 + 2bz + c) } ) = e^{-\pi (c-b^2/a)} I( z \mapsto e^{-\pi a z^2} )$

when ${a,b,c \in {\bf C}}$ with ${\hbox{Re}(a) > 0}$, and then an application of the complex scaling law (and a continuity argument, observing that there is a compact path connecting ${a}$ to ${1}$ in the right half plane) gives

$\displaystyle I( z \mapsto e^{-\pi (az^2 + 2bz + c) } ) = a^{-1/2} e^{-\pi (c-b^2/a)} I( z \mapsto e^{-\pi z^2} )$

using the branch of ${a^{-1/2}}$ on the right half-plane for which ${1^{-1/2} = 1}$. Using the normalisation (4) we thus have

$\displaystyle I( z \mapsto e^{-\pi (az^2 + 2bz + c) } ) = a^{-1/2} e^{-\pi (c-b^2/a)}$

giving the usual gaussian integral formula

$\displaystyle \int_{\bf R} e^{-\pi (ax^2 + 2bx + c)}\ dx = a^{-1/2} e^{-\pi (c-b^2/a)}. \ \ \ \ \ (9)$

This is a basic illustration of the power that a large symmetry group (in this case, the complex homothety group) can bring to bear on the task of computing integrals.

One can extend this sort of analysis to higher dimensions. For any natural number ${n \geq 1}$, let ${\mathcal{SE}({\bf C}^n \rightarrow {\bf C})}$ denote the space of all functions ${f: {\bf C}^n \rightarrow {\bf C}}$ which is jointly entire in the sense that ${f(z_1,\ldots,z_n)}$ can be expressed as a Taylor series in ${z_1,\ldots,z_n}$ which is absolutely convergent for all choices of ${z_1,\ldots,z_n}$, and such that there exists an ${\epsilon > 0}$ such that for any ${A>0}$ there is ${C_A>0}$ for which one has the bound

$\displaystyle |f(z)| \leq C_A (1+|z|)^{-A}$

whenever ${|\hbox{Im}(z_j)| \leq A + \epsilon |\hbox{Re}(z_j)|}$ for all ${1 \leq j \leq n}$, where ${z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}}$ and ${|z| := (|z_1|^2+\ldots+|z_n|^2)^{1/2}}$. Again, we call such functions Schwartz entire functions; a typical example is the function

$\displaystyle f(z) := e^{-\pi (z^T A z + 2b^T z + c)}$

where ${A}$ is an ${n \times n}$ complex symmetric matrix with positive definite real part, ${b}$ is a vector in ${{\bf C}^n}$, and ${c}$ is a complex number. We can then define an abstract integration functional ${I: \mathcal{SE}({\bf C}^n \rightarrow {\bf C}) \rightarrow {\bf C}}$ by integration on the real slice ${{\bf R}^n}$:

$\displaystyle I(f) := \int_{{\bf R}^n} f(x)\ dx$

where ${dx}$ is the usual Lebesgue measure on ${{\bf R}^n}$. By contour shifting in each of the ${n}$ variables ${z_1,\ldots,z_n}$ separately, we see that ${I}$ is invariant with respect to complex translations of each of the ${z_j}$ variables, and is thus invariant under translating the joint variable ${z}$ by ${{\bf C}^n}$. One can also verify the scaling law

$\displaystyle I(\delta_A f) = \hbox{det}(A) I(f)$

for ${n \times n}$ complex matrices ${A}$ sufficiently close to the origin, where ${\delta_A f(z) := f(A^{-1} z)}$. This can be seen for shear transformations ${A}$ by Fubini’s theorem and the aforementioned translation invariance, while for diagonal transformations near the origin this can be seen from ${n}$ applications of one-dimensional scaling law, and the general case then follows by composition. Among other things, these laws then easily lead to the higher-dimensional generalisation

$\displaystyle \int_{{\bf R}^n} e^{-\pi (x^T A x + 2 b^T x + c)}\ dx = \hbox{det}(A)^{-1/2} e^{-\pi (c-b^T A^{-1} b)} \ \ \ \ \ (10)$

whenever ${A}$ is a complex symmetric matrix with positive definite real part, ${b}$ is a vector in ${{\bf C}^n}$, and ${c}$ is a complex number, basically by repeating the one-dimensional argument sketched earlier. Here, we choose the branch of ${\hbox{det}(A)^{-1/2}}$ for all matrices ${A}$ in the indicated class for which ${\hbox{det}(1)^{-1/2} = 1}$.

Now we turn to an integration functional suitable for computing complex gaussian integrals such as

$\displaystyle \int_{{\bf C}^n} e^{-2\pi (z^\dagger A z + b^\dagger z + z^\dagger \tilde b + c)}\ dz d\overline{z}, \ \ \ \ \ (11)$

where ${z}$ is now a complex variable

$\displaystyle z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix},$

${z^\dagger}$ is the adjoint

$\displaystyle z^\dagger := (\overline{z_1},\ldots, \overline{z_n}),$

${A}$ is a complex ${n \times n}$ matrix with positive definite Hermitian part, ${b, \tilde b}$ are column vectors in ${{\bf C}^n}$, ${c}$ is a complex number, and ${dz d\overline{z} = \prod_{j=1}^n 2 d\hbox{Re}(z_j) d\hbox{Im}(z_j)}$ is ${2^n}$ times Lebesgue measure on ${{\bf C}^n}$. (The factors of two here turn out to be a natural normalisation, but they can be ignored on a first reading.) As we shall see later, such integrals are relevant when performing computations on the Gaussian Unitary Ensemble (GUE) in random matrix theory. Note that the integrand here is not complex analytic due to the presence of the complex conjugates. However, this can be dealt with by the trick of replacing the complex conjugate ${\overline{z}}$ by a variable ${z^*}$ which is formally conjugate to ${z}$, but which is allowed to vary independently of ${z}$. More precisely, let ${\mathcal{SA}({\bf C}^n \times {\bf C}^n \rightarrow {\bf C})}$ be the space of all functions ${f: (z,z^*) \mapsto f(z,z^*)}$ of two independent ${n}$-tuples

$\displaystyle z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}, z^* = \begin{pmatrix} z_1^* \\ \vdots \\ z_n^* \end{pmatrix}$

of complex variables, which is jointly entire in all ${2n}$ variables (in the sense defined previously, i.e. there is a joint Taylor series that is absolutely convergent for all independent choices of ${z, z^* \in {\bf C}^n}$), and such that there is an ${\epsilon>0}$ such that for every ${A>0}$ there is ${C_A>0}$ such that one has the bound

$\displaystyle |f(z,z^*)| \leq C_A (1 + |z|)^{-A}$

whenever ${|z^* - \overline{z}| \leq A + \epsilon |z|}$. We will call such functions Schwartz analytic. Note that the integrand in (11) is Schwartz analytic when ${A}$ has positive definite Hermitian part, if we reinterpret ${z^\dagger}$ as the transpose of ${z^*}$ rather than as the adjoint of ${z}$ in order to make the integrand entire in ${z}$ and ${z^*}$. We can then define an abstract integration functional ${I: \mathcal{SA}({\bf C}^n \times {\bf C}^n \rightarrow {\bf C}) \rightarrow {\bf C}}$ by the formula

$\displaystyle I(f) := \int_{{\bf C}^n} f(z,\overline{z})\ dz d\overline{z}, \ \ \ \ \ (12)$

thus ${I}$ can be localised to the slice ${\{ (z,\overline{z}): z \in {\bf C}^n\}}$ of ${{\bf C}^n \times {\bf C}^n}$ (though, as with previous functionals, one can use contour shifting to relocalise ${I}$ to other slices also.) One can also write this integral as

$\displaystyle I(f) = 2^n \int_{{\bf R}^n \times {\bf R}^n} f(x+iy, x-iy)\ dx dy$

and note that the integrand here is a Schwartz entire function on ${{\bf C}^n \times {\bf C}^n}$, thus linking the Schwartz analytic integral with the Schwartz entire integral. Using this connection, one can verify that this functional ${I}$ is invariant with respect to translating ${z}$ and ${z^*}$ by independent shifts in ${{\bf C}^n}$ (thus giving a ${{\bf C}^n \times {\bf C}^n}$ translation symmetry), and one also has the independent dilation symmetry

$\displaystyle I(\delta_{A,B} f) = \hbox{det}(A) \hbox{det}(B) I(f)$

for ${n \times n}$ complex matrices ${A,B}$ that are sufficiently close to the identity, where ${\delta_{A,B} f(z,z^*) := f(A^{-1} z, B^{-1} z^*)}$. Arguing as before, we can then compute (11) as

$\displaystyle \int_{{\bf C}^n} e^{-2\pi (z^\dagger A z + b^\dagger z + z^\dagger \tilde b + c)}\ dz d\overline{z} = \hbox{det}(A)^{-1} e^{-2\pi (c - b^\dagger A^{-1} \tilde b)}. \ \ \ \ \ (13)$

In particular, this gives an integral representation for the determinant-reciprocal ${\hbox{det}(A)^{-1}}$ of a complex ${n \times n}$ matrix with positive definite Hermitian part, in terms of gaussian expressions in which ${A}$ only appears linearly in the exponential:

$\displaystyle \hbox{det}(A)^{-1} = \int_{{\bf C}^n} e^{-2\pi z^\dagger A z}\ dz d\overline{z}.$

This formula is then convenient for computing statistics such as

$\displaystyle \mathop{\bf E} \hbox{det}(W_n-E-i\eta)^{-1}$

for random matrices ${W_n}$ drawn from the Gaussian Unitary Ensemble (GUE), and some choice of spectral parameter ${E+i\eta}$ with ${\eta>0}$; we review this computation later in this post. By the trick of matrix differentiation of the determinant (as reviewed in this recent blog post), one can also use this method to compute matrix-valued statistics such as

$\displaystyle \mathop{\bf E} \hbox{det}(W_n-E-i\eta)^{-1} (W_n-E-i\eta)^{-1}.$

However, if one restricts attention to classical integrals over real or complex (and in particular, commuting or bosonic) variables, it does not seem possible to easily eradicate the negative determinant factors in such calculations, which is unfortunate because many statistics of interest in random matrix theory, such as the expected Stieltjes transform

$\displaystyle \mathop{\bf E} \frac{1}{n} \hbox{tr} (W_n-E-i\eta)^{-1},$

which is the Stieltjes transform of the density of states. However, it turns out (as I learned recently from Peter Sarnak and Tom Spencer) that it is possible to cancel out these negative determinant factors by balancing the bosonic gaussian integrals with an equal number of fermionic gaussian integrals, in which one integrates over a family of anticommuting variables. These fermionic integrals are closer in spirit to the polynomial integral (6) than to Lebesgue type integrals, and in particular obey a scaling law which is inverse to the Lebesgue scaling (in particular, a linear change of fermionic variables ${\zeta \mapsto A \zeta}$ ends up transforming a fermionic integral by ${\hbox{det}(A)}$ rather than ${\hbox{det}(A)^{-1}}$), which conveniently cancels out the reciprocal determinants in the previous calculations. Furthermore, one can combine the bosonic and fermionic integrals into a unified integration concept, known as the Berezin integral (or Grassmann integral), in which one integrates functions of supervectors (vectors with both bosonic and fermionic components), and is of particular importance in the theory of supersymmetry in physics. (The prefix “super” in physics means, roughly speaking, that the object or concept that the prefix is attached to contains both bosonic and fermionic aspects.) When one applies this unified integration concept to gaussians, this can lead to quite compact and efficient calculations (provided that one is willing to work with “super”-analogues of various concepts in classical linear algebra, such as the supertrace or superdeterminant).

Abstract integrals of the flavour of (6) arose in quantum field theory, when physicists sought to formally compute integrals of the form

$\displaystyle \int F( x_1, \ldots, x_n, \xi_1, \ldots, \xi_m )\ dx_1 \ldots dx_n d\xi_1 \ldots d\xi_m \ \ \ \ \ (14)$

where ${x_1,\ldots,x_n}$ are familiar commuting (or bosonic) variables (which, in particular, can often be localised to be scalar variables taking values in ${{\bf R}}$ or ${{\bf C}}$), while ${\xi_1,\ldots,\xi_m}$ were more exotic anticommuting (or fermionic) variables, taking values in some vector space of fermions. (As we shall see shortly, one can formalise these concepts by working in a supercommutative algebra.) The integrand ${F(x_1,\ldots,x_n,\xi_1,\ldots,\xi_m)}$ was a formally analytic function of ${x_1,\ldots,x_n,\xi_1,\ldots,\xi_m}$, in that it could be expanded as a (formal, noncommutative) power series in the variables ${x_1,\ldots,x_n,\xi_1,\ldots,\xi_m}$. For functions ${F(x_1,\ldots,x_n)}$ that depend only on bosonic variables, it is certainly possible for such analytic functions to be in the Schwartz class and thus fall under the scope of the classical integral, as discussed previously. However, functions ${F(\xi_1,\ldots,\xi_m)}$ that depend on fermionic variables ${\xi_1,\ldots,\xi_m}$ behave rather differently. Indeed, a fermonic variable ${\xi}$ must anticommute with itself, so that ${\xi^2 = 0}$. In particular, any power series in ${\xi}$ terminates after the linear term in ${\xi}$, so that a function ${F(\xi)}$ can only be analytic in ${\xi}$ if it is a polynomial of degree at most ${1}$ in ${\xi}$; more generally, an analytic function ${F(\xi_1,\ldots,\xi_m)}$ of ${m}$ fermionic variables ${\xi_1,\ldots,\xi_m}$ must be a polynomial of degree at most ${m}$, and an analytic function ${F(x_1,\ldots,x_n,\xi_1,\ldots,\xi_m)}$ of ${n}$ bosonic and ${m}$ fermionic variables can be Schwartz in the bosonic variables but will be polynomial in the fermonic variables. As such, to interpret the integral (14), one can use classical (Lebesgue) integration (or the variants discussed above for integrating Schwartz entire or Schwartz analytic functions) for the bosonic variables, but must use abstract integrals such as (6) for the fermonic variables, leading to the concept of Berezin integration mentioned earlier.

In this post I would like to set out some of the basic algebraic formalism of Berezin integration, particularly with regards to integration of gaussian-type expressions, and then show how this formalism can be used to perform computations involving GUE (for instance, one can compute the density of states of GUE by this machinery without recourse to the theory of orthogonal polynomials). The use of supersymmetric gaussian integrals to analyse ensembles such as GUE appears in the work of Efetov (and was also proposed in the slightly earlier works of Parisi-Sourlas and McKane, with a related approach also appearing in the work of Wegner); the material here is adapted from this survey of Mirlin, as well as the later papers of Disertori-Pinson-Spencer and of Disertori.

Consider the free Schrödinger equation in ${d}$ spatial dimensions, which I will normalise as

$\displaystyle i u_t + \frac{1}{2} \Delta_{{\bf R}^d} u = 0 \ \ \ \ \ (1)$

where ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ is the unknown field and ${\Delta_{{\bf R}^{d+1}} = \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}}$ is the spatial Laplacian. To avoid irrelevant technical issues I will restrict attention to smooth (classical) solutions to this equation, and will work locally in spacetime avoiding issues of decay at infinity (or at other singularities); I will also avoid issues involving branch cuts of functions such as ${t^{d/2}}$ (if one wishes, one can restrict ${d}$ to be even in order to safely ignore all branch cut issues). The space of solutions to (1) enjoys a number of symmetries. A particularly non-obvious symmetry is the pseudoconformal symmetry: if ${u}$ solves (1), then the pseudoconformal solution ${pc(u): {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ defined by

$\displaystyle pc(u)(t,x) := \frac{1}{(it)^{d/2}} \overline{u(\frac{1}{t}, \frac{x}{t})} e^{i|x|^2/2t} \ \ \ \ \ (2)$

for ${t \neq 0}$ can be seen after some computation to also solve (1). (If ${u}$ has suitable decay at spatial infinity and one chooses a suitable branch cut for ${(it)^{d/2}}$, one can extend ${pc(u)}$ continuously to the ${t=0}$ spatial slice, whereupon it becomes essentially the spatial Fourier transform of ${u(0,\cdot)}$, but we will not need this fact for the current discussion.)

An analogous symmetry exists for the free wave equation in ${d+1}$ spatial dimensions, which I will write as

$\displaystyle u_{tt} - \Delta_{{\bf R}^{d+1}} u = 0 \ \ \ \ \ (3)$

where ${u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ is the unknown field. In analogy to pseudoconformal symmetry, we have conformal symmetry: if ${u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ solves (3), then the function ${conf(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$, defined in the interior ${\{ (t,x): |x| < |t| \}}$ of the light cone by the formula

$\displaystyle conf(u)(t,x) := (t^2-|x|^2)^{-d/2} u( \frac{t}{t^2-|x|^2}, \frac{x}{t^2-|x|^2} ), \ \ \ \ \ (4)$

also solves (3).

There are also some direct links between the Schrödinger equation in ${d}$ dimensions and the wave equation in ${d+1}$ dimensions. This can be easily seen on the spacetime Fourier side: solutions to (1) have spacetime Fourier transform (formally) supported on a ${d}$-dimensional hyperboloid, while solutions to (3) have spacetime Fourier transform formally supported on a ${d+1}$-dimensional cone. To link the two, one then observes that the ${d}$-dimensional hyperboloid can be viewed as a conic section (i.e. hyperplane slice) of the ${d+1}$-dimensional cone. In physical space, this link is manifested as follows: if ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ solves (1), then the function ${\iota_{1}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ defined by

$\displaystyle \iota_{1}(u)(t,x_1,\ldots,x_{d+1}) := e^{-i(t+x_{d+1})} u( \frac{t-x_{d+1}}{2}, x_1,\ldots,x_d)$

solves (3). More generally, for any non-zero scaling parameter ${\lambda}$, the function ${\iota_{\lambda}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ defined by

$\displaystyle \iota_{\lambda}(u)(t,x_1,\ldots,x_{d+1}) :=$

$\displaystyle \lambda^{d/2} e^{-i\lambda(t+x_{d+1})} u( \lambda \frac{t-x_{d+1}}{2}, \lambda x_1,\ldots,\lambda x_d) \ \ \ \ \ (5)$

solves (3).

As an “extra challenge” posed in an exercise in one of my books (Exercise 2.28, to be precise), I asked the reader to use the embeddings ${\iota_1}$ (or more generally ${\iota_\lambda}$) to explicitly connect together the pseudoconformal transformation ${pc}$ and the conformal transformation ${conf}$. It turns out that this connection is a little bit unusual, with the “obvious” guess (namely, that the embeddings ${\iota_\lambda}$ intertwine ${pc}$ and ${conf}$) being incorrect, and as such this particular task was perhaps too difficult even for a challenge question. I’ve been asked a couple times to provide the connection more explicitly, so I will do so below the fold.

Let ${A, B}$ be ${n \times n}$ Hermitian matrices, with eigenvalues ${\lambda_1(A) \leq \ldots \leq \lambda_n(A)}$ and ${\lambda_1(B) \leq \ldots\leq \lambda_n(B)}$. The Harish-Chandra-Itzykson-Zuber integral formula exactly computes the integral

$\displaystyle \int_{U(n)} \exp( t \hbox{tr}( A U B U^* ) )\ dU$

where ${U}$ is integrated over the Haar probability measure of the unitary group ${U(n)}$ and ${t}$ is a non-zero complex parameter, as the expression

$\displaystyle c_n \frac{ \det( \exp( t \lambda_i(A) \lambda_j(B) ) )_{1 \leq i,j \leq n} }{t^{(n^2-n)/2} \Delta(\lambda(A)) \Delta(\lambda(B))}$

when the eigenvalues of ${A,B}$ are simple, where ${\Delta}$ denotes the Vandermonde determinant

$\displaystyle \Delta(\lambda(A)) := \prod_{1 \leq i

and ${c_n}$ is the constant

$\displaystyle c_n := \prod_{i=1}^{n-1} i!.$

There are at least two standard ways to prove this formula in the literature. One way is by applying the Duistermaat-Heckman theorem to the pushforward of Liouville measure on the coadjoint orbit ${{\mathcal O}_B := \{ UBU^*: U \in U(n) \}}$ (or more precisely, a rotation of such an orbit by ${i}$) under the moment map ${M \mapsto \hbox{diag}(M)}$, and then using a stationary phase expansion. Another way, which I only learned about recently, is to use the formulae for evolution of eigenvalues under Dyson Brownian motion (as well as the closely related formulae for the GUE ensemble), which were derived in this previous blog post. Both of these approaches can be found in several places in the literature (the former being observed in the original paper of Duistermaat and Heckman, and the latter observed in the paper of Itzykson and Zuber as well as in this later paper of Johansson), but I thought I would record both of these here for my own benefit.

The Harish-Chandra-Itzykson-Zuber formula can be extended to other compact Lie groups than ${U(n)}$. At first glance, this might suggest that these formulae could be of use in the study of the GOE ensemble, but unfortunately the Lie algebra associated to ${O(n)}$ corresponds to real anti-symmetric matrices rather than real symmetric matrices. This also occurs in the ${U(n)}$ case, but there one can simply multiply by ${i}$ to rotate a complex skew-Hermitian matrix into a complex Hermitian matrix. This is consistent, though, with the fact that the (somewhat rarely studied) anti-symmetric GOE ensemble has cleaner formulae (in particular, having a determinantal structure similar to GUE) than the (much more commonly studied) symmetric GOE ensemble.