A finite group 
and hence
Theorem 1 (Frobenius’ theorem) Let
of
I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group
I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra 
The main ideas are as follows. The space 
is a natural number which we will call the rank of 
In the case that 
-actions
is a measure-preserving 
-system (which, in this post, means that 
is a probability space and 
is measure-preserving and invertible, thus giving an action 
of the integers), and 
are functions, and 
is ergodic (which means that 
contains no 
-invariant functions other than the constants (up to almost everywhere equivalence, of course)), then the average 
to the expression 
is drawn at random from 
), and 
is drawn at random from 
for some large 
, then the pair 
becomes uniformly distributed in the product space 
(using product measure 
) in the limit as 
to be non-ergodic, then we still have a limit formula, but it is a bit more complicated. Let 
be the 
; the 
can then be viewed as a factor of the original system 
, which is equivalent (in the sense of measure-preserving systems) to a trivial system 
(known as the invariant factor) in which the shift is trivial. There is then a projection map 
to the invariant factor which is a factor map, and the average 
is the pushforward map associated to the map 
is a pair as before, then we no longer expect complete equidistribution in 
; indeed, for any 
, we have the constraint 
; putting all these constraints together we see that 
(for almost every 
; this is analogous to the combinatorial task of counting length three progressions in various sets. For simplicity we assume the system 
to be ergodic. Naively one might expect this limit to then converge to 
is asymptotically equidistributed in 
. 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 
. The key obstruction here is that of eigenfunctions of the shift 
, that is to say non-trivial functions 
that obey the eigenfunction equation 
almost everywhere for some constant (or 
. Each such eigenfunction generates a constraint 
. However, it turns out that these are in some sense the only constraints on 
that are relevant for the limit 
to be the sub-algebra of 
is isomorphic to a shift system 
known as the Kronecker factor, for some compact abelian group 
and some (irrational) shift 
; the factor map 
pushes eigenfunctions forward to (affine) characters on 
. It is then known that the limit of 
is the closed subgroup
is the Haar probability measure on 
defining 
is non-negative and not identically vanishing. 
behaves quadratically in 
that is not detected at the triple average level. As it turns out, this is not the only type of constraint relevant for 
which turns out to be an inverse limit of two-step nilsystems; this limit theorem can be extracted from the structural theory in 
of shifts indexed by 
. The role of the intervals 
is an infinite-dimensional vector space over a finite field 
(with the finite subspaces 
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 
-system and 
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 
in an ergodic 
, one has 
are distinct residue classes. It turns out that Khintchine-type theorems always hold for 
(and for 
ergodicity is not required), and for 
it holds whenever 
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 
we could show that the Khintchine property failed for generic choices of 
, 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.
is a finite group of even order, thus 
is a multiple of two. By 
in 
together with the identity, so that 
).
involutions for some 
. Then 
. 
) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if 
has order 
(as 
contains 
has order 
has preimages that are cosets of 
. In particular, we can conclude that every group 
.
. 
. The kernel of this action is a proper normal subgroup of 
. As 
-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 
. 
. We may assume that 
(since such representations are automatically faithful by the simplicity of 
pairs 
such that 
, i.e. involutions 
whose product is also an involution. But any such involutions necessarily commute, since
of non-identity elements that commute, so by the pigeonhole principle there is a non-identity 
whose centraliser 
. This centraliser cannot be all of 
has dimension at most 
, contradicting the quasirandomness. 
together with an anti-symmetric bilinear form ![{[,] = [,]_{\mathfrak g}: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}}](http://s0.wp.com/latex.php?latex=%7B%5B%2C%5D+%3D+%5B%2C%5D_%7B%5Cmathfrak+g%7D%3A+%7B%5Cmathfrak+g%7D+%5Ctimes+%7B%5Cmathfrak+g%7D+%5Crightarrow+%7B%5Cmathfrak+g%7D%7D&bg=ffffff&fg=000000&s=0 "{[,] = [,]_{\mathfrak g}: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}}")
that obeys the ![\displaystyle [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 \ \ \ \ \ (1)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5B%5Bx%2Cy%5D%2Cz%5D+%2B+%5B%5By%2Cz%5D%2Cx%5D+%2B+%5B%5Bz%2Cx%5D%2Cy%5D+%3D+0+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 \ \ \ \ \ (1)")
; by anti-symmetry one can also rewrite the Jacobi identity as ![\displaystyle [x,[y,z]] = [[x,y],z] + [y,[x,z]]. \ \ \ \ \ (2)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Bx%2C%5By%2Cz%5D%5D+%3D+%5B%5Bx%2Cy%5D%2Cz%5D+%2B+%5By%2C%5Bx%2Cz%5D%5D.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle [x,[y,z]] = [[x,y],z] + [y,[x,z]]. \ \ \ \ \ (2)")
![{[,]_{\mathfrak g}}](http://s0.wp.com/latex.php?latex=%7B%5B%2C%5D_%7B%5Cmathfrak+g%7D%7D&bg=ffffff&fg=000000&s=0 "{[,]_{\mathfrak g}}")
when this will not cause ambiguity. A homomorphism 
between two Lie algebras 
is a linear map that respects the Lie bracket, thus ![{\phi([x,y]_{\mathfrak g}) =[\phi(x),\phi(y)]_{\mathfrak h}}](http://s0.wp.com/latex.php?latex=%7B%5Cphi%28%5Bx%2Cy%5D_%7B%5Cmathfrak+g%7D%29+%3D%5B%5Cphi%28x%29%2C%5Cphi%28y%29%5D_%7B%5Cmathfrak+h%7D%7D&bg=ffffff&fg=000000&s=0 "{\phi([x,y]_{\mathfrak g}) =[\phi(x),\phi(y)]_{\mathfrak h}}")
for all 
. As with many other classes of mathematical objects, the class of Lie algebras together with their homomorphisms then form a 
of linear transformations 
on a finite-dimensional complex vector space (or vector space for short) 
, with the commutator Lie bracket ![{[x,y] := xy-yx}](http://s0.wp.com/latex.php?latex=%7B%5Bx%2Cy%5D+%3A%3D+xy-yx%7D&bg=ffffff&fg=000000&s=0 "{[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 
into a concrete Lie algebra 
. It is a deep 
via a 
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 
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)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5B%28t%2Cx%29%2C+%28s%2Cy%29%5D_%7B%5Cmathfrak+g%7D+%3D+%28%5Bt%2Cs%5D_%7B%5Cmathfrak+h%7D+%2B+A%28t%2Cy%29+-+A%28s%2Cx%29+%2B+B%28x%2Cy%29%2C+%5Bx%2Cy%5D_%7B%5Cmathfrak+k%7D%29+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0 "\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)")
and 
that obey some Jacobi-type identities which we will not record here. Understanding exactly what maps 
are possible here (up to coordinate change) can be a difficult task (and is one of the key objectives of 
. To emphasise this, I will (perhaps idiosyncratically) express the existence of a short exact sequence 
, so that 
is not a uniquely defined combination of 
instead of 
is not commutative, and it is only partly associative: every Lie algebra of the form 
is also of the form 
, but the converse is not true (see 
and 
, and also 
.
of 
(also denoted 
both vanishing, and the 
(also denoted 
), which is given by the construction 
of 
of 
that obey the Leibniz rule
![\displaystyle D[s,t]_{\mathfrak h} = [Ds,t]_{\mathfrak h} + [s,Dt]_{\mathfrak h}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++D%5Bs%2Ct%5D_%7B%5Cmathfrak+h%7D+%3D+%5BDs%2Ct%5D_%7B%5Cmathfrak+h%7D+%2B+%5Bs%2CDt%5D_%7B%5Cmathfrak+h%7D&bg=ffffff&fg=000000&s=0 "\displaystyle D[s,t]_{\mathfrak h} = [Ds,t]_{\mathfrak h} + [s,Dt]_{\mathfrak h}")
. (The derivation algebra 
of a Lie algebra 
of a Lie group 
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.)![{[{\mathfrak h},{\mathfrak g}] \subset {\mathfrak h}}](http://s0.wp.com/latex.php?latex=%7B%5B%7B%5Cmathfrak+h%7D%2C%7B%5Cmathfrak+g%7D%5D+%5Csubset+%7B%5Cmathfrak+h%7D%7D&bg=ffffff&fg=000000&s=0 "{[{\mathfrak h},{\mathfrak g}] \subset {\mathfrak h}}")
, where ![{[{\mathfrak h},{\mathfrak g}]}](http://s0.wp.com/latex.php?latex=%7B%5B%7B%5Cmathfrak+h%7D%2C%7B%5Cmathfrak+g%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[{\mathfrak h},{\mathfrak g}]}")
denotes the Lie algebra generated by ![{\{ [x,y]: x \in {\mathfrak h}, y \in {\mathfrak g} \}}](http://s0.wp.com/latex.php?latex=%7B%5C%7B+%5Bx%2Cy%5D%3A+x+%5Cin+%7B%5Cmathfrak+h%7D%2C+y+%5Cin+%7B%5Cmathfrak+g%7D+%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{\{ [x,y]: x \in {\mathfrak h}, y \in {\mathfrak g} \}}")
, and then take 
in the usual manner; one can check that 
, one can identify 
of 
, take 
of 
. Again, it is easy to see that this does indeed create a short exact sequence: 
. Conversely, by using an arbitrary linear basis of ![{[{\mathfrak g},{\mathfrak g}]}](http://s0.wp.com/latex.php?latex=%7B%5B%7B%5Cmathfrak+g%7D%2C%7B%5Cmathfrak+g%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[{\mathfrak g},{\mathfrak g}]}")
; this algebra is trivial if and only if ![{[{\mathfrak h},{\mathfrak k}]}](http://s0.wp.com/latex.php?latex=%7B%5B%7B%5Cmathfrak+h%7D%2C%7B%5Cmathfrak+k%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[{\mathfrak h},{\mathfrak k}]}")
is an ideal whenever ![\displaystyle {\mathfrak g} = [{\mathfrak g},{\mathfrak g}] . ({\mathfrak g}/[{\mathfrak g},{\mathfrak g}]).](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cmathfrak+g%7D+%3D+%5B%7B%5Cmathfrak+g%7D%2C%7B%5Cmathfrak+g%7D%5D+.+%28%7B%5Cmathfrak+g%7D%2F%5B%7B%5Cmathfrak+g%7D%2C%7B%5Cmathfrak+g%7D%5D%29.&bg=ffffff&fg=000000&s=0 "\displaystyle {\mathfrak g} = [{\mathfrak g},{\mathfrak g}] . ({\mathfrak g}/[{\mathfrak g},{\mathfrak g}]).")
![{{\mathfrak g}/[{\mathfrak g},{\mathfrak g}]}](http://s0.wp.com/latex.php?latex=%7B%7B%5Cmathfrak+g%7D%2F%5B%7B%5Cmathfrak+g%7D%2C%7B%5Cmathfrak+g%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{{\mathfrak g}/[{\mathfrak g},{\mathfrak g}]}")
is the maximal abelian quotient of ![\displaystyle {\mathfrak g}_1 = {\mathfrak g}; {\mathfrak g}_2 := [{\mathfrak g}, {\mathfrak g}_1]; {\mathfrak g}_3 := [{\mathfrak g}, {\mathfrak g}_2]; \ldots](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cmathfrak+g%7D_1+%3D+%7B%5Cmathfrak+g%7D%3B+%7B%5Cmathfrak+g%7D_2+%3A%3D+%5B%7B%5Cmathfrak+g%7D%2C+%7B%5Cmathfrak+g%7D_1%5D%3B+%7B%5Cmathfrak+g%7D_3+%3A%3D+%5B%7B%5Cmathfrak+g%7D%2C+%7B%5Cmathfrak+g%7D_2%5D%3B+%5Cldots&bg=ffffff&fg=000000&s=0 "\displaystyle {\mathfrak g}_1 = {\mathfrak g}; {\mathfrak g}_2 := [{\mathfrak g}, {\mathfrak g}_1]; {\mathfrak g}_3 := [{\mathfrak g}, {\mathfrak g}_2]; \ldots")
![\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.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cmathfrak+g%7D%5E%7B%281%29%7D+%3A%3D+%7B%5Cmathfrak+g%7D%3B+%7B%5Cmathfrak+g%7D%5E%7B%282%29%7D+%3A%3D+%5B%7B%5Cmathfrak+g%7D%5E%7B%281%29%7D%2C+%7B%5Cmathfrak+g%7D%5E%7B%281%29%7D%5D%3B+%7B%5Cmathfrak+g%7D%5E%7B%283%29%7D+%3A%3D+%5B%7B%5Cmathfrak+g%7D%5E%7B%282%29%7D%2C+%7B%5Cmathfrak+g%7D%5E%7B%282%29%7D%5D%3B+%5Cldots.&bg=ffffff&fg=000000&s=0 "\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.")
for all 
, which can be identified with the Lie algebra of 
complex matrices, the subalgebra 
of strictly upper triangular matrices is nilpotent (but not abelian for 
), while the subalgebra 
. Similarly, a Lie algebra ![{[{\mathfrak h},{\mathfrak g}]=0}](http://s0.wp.com/latex.php?latex=%7B%5B%7B%5Cmathfrak+h%7D%2C%7B%5Cmathfrak+g%7D%5D%3D0%7D&bg=ffffff&fg=000000&s=0 "{[{\mathfrak h},{\mathfrak g}]=0}")
). We also see that an extension 
, where for each 
, 
is the linear map ![{(\hbox{ad} x)(y) := [x,y]}](http://s0.wp.com/latex.php?latex=%7B%28%5Chbox%7Bad%7D+x%29%28y%29+%3A%3D+%5Bx%2Cy%5D%7D&bg=ffffff&fg=000000&s=0 "{(\hbox{ad} x)(y) := [x,y]}")
; one easily verifies that this is indeed a representation (indeed, ![{\hbox{ad} [x,y] = [\hbox{ad} x, \hbox{ad} y]}](http://s0.wp.com/latex.php?latex=%7B%5Chbox%7Bad%7D+%5Bx%2Cy%5D+%3D+%5B%5Chbox%7Bad%7D+x%2C+%5Chbox%7Bad%7D+y%5D%7D&bg=ffffff&fg=000000&s=0 "{\hbox{ad} [x,y] = [\hbox{ad} x, \hbox{ad} y]}")
for all ![{Z({\mathfrak g}) := \{ x \in {\mathfrak g}: [x,{\mathfrak g}] = 0\}}](http://s0.wp.com/latex.php?latex=%7BZ%28%7B%5Cmathfrak+g%7D%29+%3A%3D+%5C%7B+x+%5Cin+%7B%5Cmathfrak+g%7D%3A+%5Bx%2C%7B%5Cmathfrak+g%7D%5D+%3D+0%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{Z({\mathfrak g}) := \{ x \in {\mathfrak g}: [x,{\mathfrak g}] = 0\}}")
, which the maximal central subalgebra of 
, defined as the bilinear form 
, is non-degenerate on 
is also a solvable ideal (because on quotienting by 
of 
is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the 
by a solvable algebra 
for some 
. 
for some 
for some 
for some 
. 
, or 
. 
. 
.
and the exceptional Lie algebras 
will be recalled later.) 
), 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 
of any non-identity element 
is abelian; equivalently, the commuting relation 
(defined as the relation that holds when 
, thus 
) is an equivalence relation on the non-identity elements 
of 
group of invertible affine transformations 
on a field 
. A little less obviously, the special linear group 
over a finite field 
is a CA-group when 
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 
, for instance, when 
is bounded and 
, intersect 
, 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 
, together with the identity). So if a maximal abelian subgroup 
is equal to 
of 
to be the space of left-cosets of 
conjugates of 
is all of 
or infinite strings 
of symbols 
from some given alphabet 
, which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set 
of 
from the alphabet 
, the multiples of three can be identified with the string 
, and so forth. One can also consider doubly infinite strings 
, which among other things can be used to describe arbitrary subsets of integers.
, that is to say a space 
on the space 
of 
; if 
taking values in some alphabet 
, we can first form the forward orbit 
of 
, and then observe this orbit using 
to obtain an infinite string 
. If the shift 
. Thus we see that every quadruplet 
consisting of a dynamical system 
with the shift map 
, 
is the observable that takes the value 
and zero at the other two classes, and one starts with the initial datum 
, then the observed string 
in some alphabet 
of infinite strings 
in the alphabet 
instead of 
, at which point the shift map 
, and leads to the Furstenberg correspondence principle, discussed for instance in 
of the decimal representation system) the string 
, 
, 
to be the observable that takes the value 
and 
(that is, 
is the first binary digit of 
(that is, 
in binary).
under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)
of the quadruplet 
are completely universal (in that they do not depend at all on the sequence 
of the initial datum 
of this orbit, in order to keep 
, 
, bringing the system more in line with the example in Example 
is a factor of this representation (in the sense that there is a unique map 
with 
, 
, and 
). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system 
for some prime 
be an integer. Suppose that 
. Then there exists a map 
into a subset 
of the integers which is a 
, one has
of 
, and let 
. Then there exists a map 
and any polynomial 
of degree at most 
which can only map to 
in the complex numbers (once 
).
. Then if 
is a field of characteristic at least 
for some 
, and 
from 
of 
, it has finite 
of 
, and in particular by the 
which is algebraic over 
, and then setting 
to be a complex root of of the minimal polynomial for 
with the complex number 
, we obtain a field isomorphism 
with rational coefficients, one has 
be the collection of all polynomials 
with rational coefficients with 
, and 
be the collection of all polynomials 
. The set
for 
. By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over 
. By the 
, polynomials 
, and polynomials 
with coefficients in 
. In particular, this identity also holds in the algebraic closure 
of 
we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows. 
, a sequence of finite fields 
of characteristic at least 
, and subsets 
of 
to the complex numbers. Now we select a 
, and construct the ultraproduct 
of the finite fields 
, it is easy to see (using 
be the ultralimit of the 
, so that 
is the ultraproduct of the 
are complex numbers, and for any polynomial 
, contradicting the construction of 
, and suppose that 
. Then every subset ![{[N] := \{1,\ldots,N\}}](http://s0.wp.com/latex.php?latex=%7B%5BN%5D+%3A%3D+%5C%7B1%2C%5Cldots%2CN%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{[N] := \{1,\ldots,N\}}")
of density 
at least 
for some natural numbers 
with 
. 
is replaced by 
or 
for some fixed 
. 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.
denote the assertion that for ![{A \subset [N]}](http://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%5BN%5D%7D&bg=ffffff&fg=000000&s=0 "{A \subset [N]}")
of density at least 
non-zero. Note that 
. We will show that for any 
, one has the implication 
. This implies that 
(as can be seen by considering the infimum of all 
holds (for some sufficiently small absolute constant 
fails. Thus, we can find arbitrarily large ![{[N]}](http://s0.wp.com/latex.php?latex=%7B%5BN%5D%7D&bg=ffffff&fg=000000&s=0 "{[N]}")
of density at least 
, such 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) 1_A(n+(r+h)^2) = 0.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+1_A%28n%29+1_A%28n%2B%28r%2Bh%29%5E2%29+%3D+0.&bg=ffffff&fg=000000&s=0 "\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.")
are not too important here, and could be replaced by various other small powers of 
be the density of 
. 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})](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+1_A%28n%29+%5Cdelta+1_%7B%5BN%5D%7D%28n%2B%28r%2Bh%29%5E2%29+%3D+%5Cdelta%5E2+%2B+O%28N%5E%7B-1%2F3%7D%29&bg=ffffff&fg=000000&s=0 "\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})](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+%5Cdelta+1_%7B%5BN%5D%7D%28n%29+%5Cdelta+1_%7B%5BN%5D%7D%28n%2B%28r%2Bh%29%5E2%29+%3D+%5Cdelta%5E2+%2B+O%28N%5E%7B-1%2F3%7D%29&bg=ffffff&fg=000000&s=0 "\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})")
![\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} ).](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+%5Cdelta+1_%7B%5BN%5D%7D%28n%29+1_A%28n%2B%28r%2Bh%29%5E2%29+%3D+%5Cdelta%5E2+%2B+O%28+N%5E%7B-1%2F3%7D+%29.&bg=ffffff&fg=000000&s=0 "\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} ).")
![{f := 1_A - \delta 1_{[N]}}](http://s0.wp.com/latex.php?latex=%7Bf+%3A%3D+1_A+-+%5Cdelta+1_%7B%5BN%5D%7D%7D&bg=ffffff&fg=000000&s=0 "{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} ).](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+f%28n%29+f%28n%2B%28r%2Bh%29%5E2%29+%3D+-%5Cdelta%5E2+%2B+O%28+N%5E%7B-1%2F3%7D+%29.&bg=ffffff&fg=000000&s=0 "\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} ).")
![\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.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%7Cf%28n%29%7C+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+f%28n%2B%28r%2Bh%29%5E2%29%7C+%5Cgg+%5Cdelta%5E2.&bg=ffffff&fg=000000&s=0 "\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.")
![\displaystyle \mathop{\bf E}_{n \in [N]} |f(n)|^2 = O(\delta)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%7Cf%28n%29%7C%5E2+%3D+O%28%5Cdelta%29&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{n \in [N]} |f(n)|^2 = O(\delta)")
![\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](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+f%28n%2B%28r%2Bh%29%5E2%29%7C%5E2+%5Cgg+%5Cdelta%5E3&bg=ffffff&fg=000000&s=0 "\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")
![\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.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh%2Ch%27+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%2B%28r%2Bh%29%5E2%29+f%28n%2B%28r%2Bh%27%29%5E2%29%7C+%5Cgg+%5Cdelta%5E3.&bg=ffffff&fg=000000&s=0 "\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.")
we obtain (again for ![\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.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bh%2Ch%27+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+f%28n%2B%28h%27-h%29%282r%2Bh%27%2Bh%29%29%7C+%5Cgg+%5Cdelta%5E3.&bg=ffffff&fg=000000&s=0 "\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.")
, which we can do for ![{h,h' \in [N^{1/100}]}](http://s0.wp.com/latex.php?latex=%7Bh%2Ch%27+%5Cin+%5BN%5E%7B1%2F100%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{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,](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+f%28n%2B%28h%27-h%29%282r%2Bh%27%2Bh%29%29%7C+%5Cgg+%5Cdelta%5E3%2C&bg=ffffff&fg=000000&s=0 "\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,")
![\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.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%7C%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+f%28n%2B%28h%27-h%29%282r%2Bh%27%2Bh%29%29%7C+%5Cgg+%5Cdelta%5E3.&bg=ffffff&fg=000000&s=0 "\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.")
and shift 
, we can simplify this (again for ![\displaystyle \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr)| \gg \delta^3. \ \ \ \ \ (2)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%7C%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+f%28n%2Bdr%29%7C+%5Cgg+%5Cdelta%5E3.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr)| \gg \delta^3. \ \ \ \ \ (2)")
![\displaystyle \mathop{\bf E}_{n \in [N]} f(n) = 0](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+%3D+0&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{n \in [N]} f(n) = 0")
![\displaystyle \mathop{\bf E}_{n \in [N]} f(n+dr) = O( N^{-2/3+1/100})](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%2Bdr%29+%3D+O%28+N%5E%7B-2%2F3%2B1%2F100%7D%29&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{n \in [N]} f(n+dr) = O( N^{-2/3+1/100})")
![{r \in [N^{1/3}]}](http://s0.wp.com/latex.php?latex=%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{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}).](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+f%28n%2Bdr%29+%3D+O%28+N%5E%7B-2%2F3%2B1%2F100%7D%29.&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) = O( N^{-2/3+1/100}).")
![\displaystyle \mathop{\bf E}_{n \in [N]} \max( \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr), 0 ) \gg \delta^3.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmax%28+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+f%28n%2Bdr%29%2C+0+%29+%5Cgg+%5Cdelta%5E3.&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{n \in [N]} \max( \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr), 0 ) \gg \delta^3.")
![{n \in [N]}](http://s0.wp.com/latex.php?latex=%7Bn+%5Cin+%5BN%5D%7D&bg=ffffff&fg=000000&s=0 "{n \in [N]}")
such that
![\displaystyle \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) \gg \delta^3,](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5BN%5E%7B1%2F3%7D%5D%7D+f%28n%2Bdr%29+%5Cgg+%5Cdelta%5E3%2C&bg=ffffff&fg=000000&s=0 "\displaystyle \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) \gg \delta^3,")
on the arithmetic progression ![{\{ n+dr: r \in [N^{1/3}]\}}](http://s0.wp.com/latex.php?latex=%7B%5C%7B+n%2Bdr%3A+r+%5Cin+%5BN%5E%7B1%2F3%7D%5D%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{\{ n+dr: r \in [N^{1/3}]\}}")
, which has length 
and spacing 
. By partitioning this progression into subprogressions of spacing 
and length 
(plus an error set of size 
, we see from the pigeonhole principle that we can find a progression ![{\{ n' + d^2 r': r' \in [N^{1/4}]\}}](http://s0.wp.com/latex.php?latex=%7B%5C%7B+n%27+%2B+d%5E2+r%27%3A+r%27+%5Cin+%5BN%5E%7B1%2F4%7D%5D%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{\{ n' + d^2 r': r' \in [N^{1/4}]\}}")
of length 
(and hence at least 
) for some absolute constant ![\displaystyle A' := \{ r' \in [N^{1/4}]: n' + d^2 r' \in A \}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A%27+%3A%3D+%5C%7B+r%27+%5Cin+%5BN%5E%7B1%2F4%7D%5D%3A+n%27+%2B+d%5E2+r%27+%5Cin+A+%5C%7D&bg=ffffff&fg=000000&s=0 "\displaystyle A' := \{ r' \in [N^{1/4}]: n' + d^2 r' \in A \}")
for some natural numbers 
with 
lie in 
(say), any subset of 
for some sufficiently large absolute constant 
contains a pair 
. On the other hand, this already improves on the (simpler) Fourier-analytic argument 
(although the 
). In the other direction, a 
without any pairs 
for fixed 
into arithmetic progressions whose spacing 
is a 
power. By re-introducing Fourier analysis, one can also perform an argument of this type for 
where 
for polynomials 
, to avoid local obstructions), because one no longer has this preservation property. 
from the space 
of complex-valued Schwarz functions 
to the complex numbers, defined by 
and 
. Secondly, it is translation invariant, thus 
, where 
is the translation of 
by 
after one sets a normalisation, for instance by requiring that 
and non-zero reals 
, one can verify that the Newton quotients 
converge in the Schwartz topology to the derivative 
of 
taking values in some vector space 
, and applied to inputs 
(the “translation action”) of some group 
, scalars 
. The previous discussion then considered the special case when 
, 
, 
, and 
was the usual translation action.
of polynomials 
of degree at most 
. 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 
is thus annihilated by 
of a polynomial, thus if one sets a normalisation 
in essentially the opposite way from the classical integration operation. Indeed, for classical integration on 
, one has 
, and so in this case the integration functional 
obeys the scaling law
is, in general, not absolutely integrable, and the partial integrals
, then one recovers convergence,
. However, if we refine the class 
denote the space of all functions 
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 
there exists 
such that one has the bound 
. 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
are complex numbers with 
. From the Cauchy integral formula (and its derivatives) we see that if 
as 
. 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.
for complex 
, if 
associated to 
lies in 
) that
with 
to 
on the right half-plane for which 
. Using the normalisation 
denote the space of all functions 
which is jointly entire in the sense that 
can be expressed as a Taylor series in 
which is absolutely convergent for all choices of 
such that for any 
there is 
for which one has the bound 
for all 
, where 
and 
. Again, we call such functions Schwartz entire functions; a typical example is the function
is a vector in 
, and 
by integration on the real slice 
:
is the usual Lebesgue measure on 
variables, and is thus invariant under translating the joint variable 
by 
. This can be seen for shear transformations 
for all matrices 
.
is the adjoint
are column vectors in 
is 
times Lebesgue measure on 
by a variable 
which is formally conjugate to 
be the space of all functions 
of two independent 
variables (in the sense defined previously, i.e. there is a joint Taylor series that is absolutely convergent for all independent choices of 
), and such that there is an 
. We will call such functions Schwartz analytic. Note that the integrand in 
by the formula 
of 
(though, as with previous functionals, one can use contour shifting to relocalise 
. Arguing as before, we can then compute 
of a complex 
drawn from the Gaussian Unitary Ensemble (GUE), and some choice of spectral parameter 
with 
; we review this computation later in this post. By the trick of matrix differentiation of the determinant (as reviewed in 
ends up transforming a fermionic integral by 
rather than 
are familiar commuting (or 
were more exotic anticommuting (or 
was a formally analytic function of 
, in that it could be expanded as a (formal, noncommutative) power series in the variables 
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 
that depend on fermionic variables 
must anticommute with itself, so that 
. In particular, any power series in 
can only be analytic in 
fermionic variables 
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