I’ve just finished the first draft of my book “Expansion in finite simple groups of Lie type“, which is  based in the lecture notes for my graduate course on this topic that were previously posted on this blog.  It also contains some newer material, such as the notes on Lie algebras and Lie groups that I posted most recently here.

Let ${F}$ be a field. A definable set over ${F}$ is a set of the form

$\displaystyle \{ x \in F^n | \phi(x) \hbox{ is true} \} \ \ \ \ \ (1)$

where ${n}$ is a natural number, and ${\phi(x)}$ is a predicate involving the ring operations ${+,\times}$ of ${F}$, the equality symbol ${=}$, an arbitrary number of constants and free variables in ${F}$, the quantifiers ${\forall, \exists}$, boolean operators such as ${\vee,\wedge,\neg}$, and parentheses and colons, where the quantifiers are always understood to be over the field ${F}$. Thus, for instance, the set of quadratic residues

$\displaystyle \{ x \in F | \exists y: x = y \times y \}$

is definable over ${F}$, and any algebraic variety over ${F}$ is also a definable set over ${F}$. Henceforth we will abbreviate “definable over ${F}$” simply as “definable”.

If ${F}$ is a finite field, then every subset of ${F^n}$ is definable, since finite sets are automatically definable. However, we can obtain a more interesting notion in this case by restricting the complexity of a definable set. We say that ${E \subset F^n}$ is a definable set of complexity at most ${M}$ if ${n \leq M}$, and ${E}$ can be written in the form (1) for some predicate ${\phi}$ of length at most ${M}$ (where all operators, quantifiers, relations, variables, constants, and punctuation symbols are considered to have unit length). Thus, for instance, a hypersurface in ${n}$ dimensions of degree ${d}$ would be a definable set of complexity ${O_{n,d}(1)}$. We will then be interested in the regime where the complexity remains bounded, but the field size (or field characteristic) becomes large.

In a recent paper, I established (in the large characteristic case) the following regularity lemma for dense definable graphs, which significantly strengthens the Szemerédi regularity lemma in this context, by eliminating “bad” pairs, giving a polynomially strong regularity, and also giving definability of the cells:

Lemma 1 (Algebraic regularity lemma) Let ${F}$ be a finite field, let ${V,W}$ be definable non-empty sets of complexity at most ${M}$, and let ${E \subset V \times W}$ also be definable with complexity at most ${M}$. Assume that the characteristic of ${F}$ is sufficiently large depending on ${M}$. Then we may partition ${V = V_1 \cup \ldots \cup V_m}$ and ${W = W_1 \cup \ldots \cup W_n}$ with ${m,n = O_M(1)}$, with the following properties:

• (Definability) Each of the ${V_1,\ldots,V_m,W_1,\ldots,W_n}$ are definable of complexity ${O_M(1)}$.
• (Size) We have ${|V_i| \gg_M |V|}$ and ${|W_j| \gg_M |W|}$ for all ${i=1,\ldots,m}$ and ${j=1,\ldots,n}$.
• (Regularity) We have

$\displaystyle |E \cap (A \times B)| = d_{ij} |A| |B| + O_M( |F|^{-1/4} |V| |W| ) \ \ \ \ \ (2)$

for all ${i=1,\ldots,m}$, ${j=1,\ldots,n}$, ${A \subset V_i}$, and ${B\subset W_j}$, where ${d_{ij}}$ is a rational number in ${[0,1]}$ with numerator and denominator ${O_M(1)}$.

My original proof of this lemma was quite complicated, based on an explicit calculation of the “square”

$\displaystyle \mu(w,w') := \{ v \in V: (v,w), (v,w') \in E \}$

of ${E}$ using the Lang-Weil bound and some facts about the étale fundamental group. It was the reliance on the latter which was the main reason why the result was restricted to the large characteristic setting. (I then applied this lemma to classify expanding polynomials over finite fields of large characteristic, but I will not discuss these applications here; see this previous blog post for more discussion.)

Recently, Anand Pillay and Sergei Starchenko (and independently, Udi Hrushovski) have observed that the theory of the étale fundamental group is not necessary in the argument, and the lemma can in fact be deduced from quite general model theoretic techniques, in particular using (a local version of) the concept of stability. One of the consequences of this new proof of the lemma is that the hypothesis of large characteristic can be omitted; the lemma is now known to be valid for arbitrary finite fields ${F}$ (although its content is trivial if the field is not sufficiently large depending on the complexity at most ${M}$).

Inspired by this, I decided to see if I could find yet another proof of the algebraic regularity lemma, again avoiding the theory of the étale fundamental group. It turns out that the spectral proof of the Szemerédi regularity lemma (discussed in this previous blog post) adapts very nicely to this setting. The key fact needed about definable sets over finite fields is that their cardinality takes on an essentially discrete set of values. More precisely, we have the following fundamental result of Chatzidakis, van den Dries, and Macintyre:

Proposition 2 Let ${F}$ be a finite field, and let ${M > 0}$.

• (Discretised cardinality) If ${E}$ is a non-empty definable set of complexity at most ${M}$, then one has

$\displaystyle |E| = c |F|^d + O_M( |F|^{d-1/2} ) \ \ \ \ \ (3)$

where ${d = O_M(1)}$ is a natural number, and ${c}$ is a positive rational number with numerator and denominator ${O_M(1)}$. In particular, we have ${|F|^d \ll_M |E| \ll_M |F|^d}$.

• (Definable cardinality) Assume ${|F|}$ is sufficiently large depending on ${M}$. If ${V, W}$, and ${E \subset V \times W}$ are definable sets of complexity at most ${M}$, so that ${E_w := \{ v \in V: (v,w) \in W \}}$ can be viewed as a definable subset of ${V}$ that is definably parameterised by ${w \in W}$, then for each natural number ${d = O_M(1)}$ and each positive rational ${c}$ with numerator and denominator ${O_M(1)}$, the set

$\displaystyle \{ w \in W: |E_w| = c |F|^d + O_M( |F|^{d-1/2} ) \} \ \ \ \ \ (4)$

is definable with complexity ${O_M(1)}$, where the implied constants in the asymptotic notation used to define (4) are the same as those that appearing in (3). (Informally: the “dimension” ${d}$ and “measure” ${c}$ of ${E_w}$ depends definably on ${w}$.)

We will take this proposition as a black box; a proof can be obtained by combining the description of definable sets over pseudofinite fields (discussed in this previous post) with the Lang-Weil bound (discussed in this previous post). (The former fact is phrased using nonstandard analysis, but one can use standard compactness-and-contradiction arguments to convert such statements to statements in standard analysis, as discussed in this post.)

The above proposition places severe restrictions on the cardinality of definable sets; for instance, it shows that one cannot have a definable set of complexity at most ${M}$ and cardinality ${|F|^{1/2}}$, if ${|F|}$ is sufficiently large depending on ${M}$. If ${E \subset V}$ are definable sets of complexity at most ${M}$, it shows that ${|E| = (c+ O_M(|F|^{-1/2})) |V|}$ for some rational ${0\leq c \leq 1}$ with numerator and denominator ${O_M(1)}$; furthermore, if ${c=0}$, we may improve this bound to ${|E| = O_M( |F|^{-1} |V|)}$. In particular, we obtain the following “self-improving” properties:

• If ${E \subset V}$ are definable of complexity at most ${M}$ and ${|E| \leq \epsilon |V|}$ for some ${\epsilon>0}$, then (if ${\epsilon}$ is sufficiently small depending on ${M}$ and ${F}$ is sufficiently large depending on ${M}$) this forces ${|E| = O_M( |F|^{-1} |V| )}$.
• If ${E \subset V}$ are definable of complexity at most ${M}$ and ${||E| - c |V|| \leq \epsilon |V|}$ for some ${\epsilon>0}$ and positive rational ${c}$, then (if ${\epsilon}$ is sufficiently small depending on ${M,c}$ and ${F}$ is sufficiently large depending on ${M,c}$) this forces ${|E| = c |V| + O_M( |F|^{-1/2} |V| )}$.

It turns out that these self-improving properties can be applied to the coefficients of various matrices (basically powers of the adjacency matrix associated to ${E}$) that arise in the spectral proof of the regularity lemma to significantly improve the bounds in that lemma; we describe how this is done below the fold. We also make some connections to the stability-based proofs of Pillay-Starchenko and Hrushovski.

I’ve just uploaded to the arXiv my article “Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory“, submitted to the new journal “EMS surveys in the mathematical sciences“.  This is the first draft of a survey article on the polynomial method – a technique in combinatorics and number theory for controlling a relevant set of points by comparing it with the zero set of a suitably chosen polynomial, and then using tools from algebraic geometry (e.g. Bezout’s theorem) on that zero set. As such, the method combines algebraic geometry with combinatorial geometry, and could be viewed as the philosophy of a combined field which I dub “algebraic combinatorial geometry”.   There is also an important extension of this method when one is working overthe reals, in which methods from algebraic topology (e.g. the ham sandwich theorem and its generalisation to polynomials), and not just algebraic geometry, come into play also.

The polynomial method has been used independently many times in mathematics; for instance, it plays a key role in the proof of Baker’s theorem in transcendence theory, or Stepanov’s method in giving an elementary proof of the Riemann hypothesis for finite fields over curves; in combinatorics, the nullstellenatz of Alon is also another relatively early use of the polynomial method.  More recently, it underlies Dvir’s proof of the Kakeya conjecture over finite fields and Guth and Katz’s near-complete solution to the Erdos distance problem in the plane, and can be used to give a short proof of the Szemeredi-Trotter theorem.  One of the aims of this survey is to try to present all of these disparate applications of the polynomial method in a somewhat unified context; my hope is that there will eventually be a systematic foundation for algebraic combinatorial geometry which naturally contains all of these different instances the polynomial method (and also suggests new instances to explore); but the field is unfortunately not at that stage of maturity yet.

This is something of a first draft, so comments and suggestions are even more welcome than usual.  (For instance, I have already had my attention drawn to some additional uses of the polynomial method in the literature that I was not previously aware of.)

Once again it is time to roll over the previous discussion thread, which has become rather full with comments.  The paper is nearly finished (see also the working copy at this subdirectory, as well as the rest of the directory), but several people are carefully proofreading various sections of the paper.  Once all the people doing so have signed off on it, I think we will be ready to submit (there appears to be no objection to the plan to submit to Algebra and Number Theory).

Another thing to discuss is an invitation to Polymath8 to write a feature article (up to 8000 words or 15 pages) for the Newsletter of the European Mathematical Society on our experiences with this project.  It is perhaps premature to actually start writing this article before the main research paper is finalised, but we can at least plan how to write such an article.  One suggestion, proposed by Emmanuel, is to have individual participants each contribute a brief account of their interaction with the project, which we would compile together with some additional text summarising the project as a whole (and maybe some speculation for any lessons we can apply here for future polymath projects).   Certainly I plan to have a separate blog post collecting feedback on this project once the main writing is done.

The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments.  We are still writing the paper, but it appears to have stabilised in a near-final form (source files available here); the main remaining tasks are proofreading, checking the mathematics, and polishing the exposition.  We also have a tentative consensus to submit the paper to Algebra and Number Theory when the proofreading is all complete.

The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not  need any of the advanced exponential sum estimates).  The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.

Define a partition of ${1}$ to be a finite or infinite multiset ${\Sigma}$ of real numbers in the interval ${I \in (0,1]}$ (that is, an unordered set of real numbers in ${I}$, possibly with multiplicity) whose total sum is ${1}$: ${\sum_{t \in \Sigma}t = 1}$. For instance, ${\{1/2,1/4,1/8,1/16,\ldots\}}$ is a partition of ${1}$. Such partitions arise naturally when trying to decompose a large object into smaller ones, for instance:

1. (Prime factorisation) Given a natural number ${n}$, one can decompose it into prime factors ${n = p_1 \ldots p_k}$ (counting multiplicity), and then the multiset

$\displaystyle \Sigma_{PF}(n) := \{ \frac{\log p_1}{\log n}, \ldots,\frac{\log p_k}{\log n} \}$

is a partition of ${1}$.

2. (Cycle decomposition) Given a permutation ${\sigma \in S_n}$ on ${n}$ labels ${\{1,\ldots,n\}}$, one can decompose ${\sigma}$ into cycles ${C_1,\ldots,C_k}$, and then the multiset

$\displaystyle \Sigma_{CD}(\sigma) := \{ \frac{|C_1|}{n}, \ldots, \frac{|C_k|}{n} \}$

is a partition of ${1}$.

3. (Normalisation) Given a multiset ${\Gamma}$ of positive real numbers whose sum ${S := \sum_{x\in \Gamma}x}$ is finite and non-zero, the multiset

$\displaystyle \Sigma_N( \Gamma) := \frac{1}{S} \cdot \Gamma = \{ \frac{x}{S}: x \in \Gamma \}$

is a partition of ${1}$.

In the spirit of the universality phenomenon, one can ask what is the natural distribution for what a “typical” partition should look like; thus one seeks a natural probability distribution on the space of all partitions, analogous to (say) the gaussian distributions on the real line, or GUE distributions on point processes on the line, and so forth. It turns out that there is one natural such distribution which is related to all three examples above, known as the Poisson-Dirichlet distribution. To describe this distribution, we first have to deal with the problem that it is not immediately obvious how to cleanly parameterise the space of partitions, given that the cardinality of the partition can be finite or infinite, that multiplicity is allowed, and that we would like to identify two partitions that are permutations of each other

One way to proceed is to random partition ${\Sigma}$ as a type of point process on the interval ${I}$, with the constraint that ${\sum_{x \in \Sigma} x = 1}$, in which case one can study statistics such as the counting functions

$\displaystyle N_{[a,b]} := |\Sigma \cap [a,b]| = \sum_{x \in\Sigma} 1_{[a,b]}(x)$

(where the cardinality here counts multiplicity). This can certainly be done, although in the case of the Poisson-Dirichlet process, the formulae for the joint distribution of such counting functions is moderately complicated. Another way to proceed is to order the elements of ${\Sigma}$ in decreasing order

$\displaystyle t_1 \geq t_2 \geq t_3 \geq \ldots \geq 0,$

with the convention that one pads the sequence ${t_n}$ by an infinite number of zeroes if ${\Sigma}$ is finite; this identifies the space of partitions with an infinite dimensional simplex

$\displaystyle \{ (t_1,t_2,\ldots) \in [0,1]^{\bf N}: t_1 \geq t_2 \geq \ldots; \sum_{n=1}^\infty t_n = 1 \}.$

However, it turns out that the process of ordering the elements is not “smooth” (basically because functions such as ${(x,y) \mapsto \max(x,y)}$ and ${(x,y) \mapsto \min(x,y)}$ are not smooth) and the formulae for the joint distribution in the case of the Poisson-Dirichlet process is again complicated.

It turns out that there is a better (or at least “smoother”) way to enumerate the elements ${u_1,(1-u_1)u_2,(1-u_1)(1-u_2)u_3,\ldots}$ of a partition ${\Sigma}$ than the ordered method, although it is random rather than deterministic. This procedure (which I learned from this paper of Donnelly and Grimmett) works as follows.

1. Given a partition ${\Sigma}$, let ${u_1}$ be an element of ${\Sigma}$ chosen at random, with each element ${t\in \Sigma}$ having a probability ${t}$ of being chosen as ${u_1}$ (so if ${t \in \Sigma}$ occurs with multiplicity ${m}$, the net probability that ${t}$ is chosen as ${u_1}$ is actually ${mt}$). Note that this is well-defined since the elements of ${\Sigma}$ sum to ${1}$.
2. Now suppose ${u_1}$ is chosen. If ${\Sigma \backslash \{u_1\}}$ is empty, we set ${u_2,u_3,\ldots}$ all equal to zero and stop. Otherwise, let ${u_2}$ be an element of ${\frac{1}{1-u_1} \cdot (\Sigma \backslash \{u_1\})}$ chosen at random, with each element ${t \in \frac{1}{1-u_1} \cdot (\Sigma \backslash \{u_1\})}$ having a probability ${t}$ of being chosen as ${u_2}$. (For instance, if ${u_1}$ occurred with some multiplicity ${m>1}$ in ${\Sigma}$, then ${u_2}$ can equal ${\frac{u_1}{1-u_1}}$ with probability ${(m-1)u_1/(1-u_1)}$.)
3. Now suppose ${u_1,u_2}$ are both chosen. If ${\Sigma \backslash \{u_1,u_2\}}$ is empty, we set ${u_3, u_4, \ldots}$ all equal to zero and stop. Otherwise, let ${u_3}$ be an element of ${\frac{1}{1-u_1-u_2} \cdot (\Sigma\backslash \{u_1,u_2\})}$, with ech element ${t \in \frac{1}{1-u_1-u_2} \cdot (\Sigma\backslash \{u_1,u_2\})}$ having a probability ${t}$ of being chosen as ${u_3}$.
4. We continue this process indefinitely to create elements ${u_1,u_2,u_3,\ldots \in [0,1]}$.

We denote the random sequence ${Enum(\Sigma) := (u_1,u_2,\ldots) \in [0,1]^{\bf N}}$ formed from a partition ${\Sigma}$ in the above manner as the random normalised enumeration of ${\Sigma}$; this is a random variable in the infinite unit cube ${[0,1]^{\bf N}}$, and can be defined recursively by the formula

$\displaystyle Enum(\Sigma) = (u_1, Enum(\frac{1}{1-u_1} \cdot (\Sigma\backslash \{u_1\})))$

with ${u_1}$ drawn randomly from ${\Sigma}$, with each element ${t \in \Sigma}$ chosen with probability ${t}$, except when ${\Sigma =\{1\}}$ in which case we instead have

$\displaystyle Enum(\{1\}) = (1, 0,0,\ldots).$

Note that one can recover ${\Sigma}$ from any of its random normalised enumerations ${Enum(\Sigma) := (u_1,u_2,\ldots)}$ by the formula

$\displaystyle \Sigma = \{ u_1, (1-u_1) u_2,(1-u_1)(1-u_2)u_3,\ldots\} \ \ \ \ \ (1)$

with the convention that one discards any zero elements on the right-hand side. Thus ${Enum}$ can be viewed as a (stochastic) parameterisation of the space of partitions by the unit cube ${[0,1]^{\bf N}}$, which is a simpler domain to work with than the infinite-dimensional simplex mentioned earlier.

Note that this random enumeration procedure can also be adapted to the three models described earlier:

1. Given a natural number ${n}$, one can randomly enumerate its prime factors ${n =p'_1 p'_2 \ldots p'_k}$ by letting each prime factor ${p}$ of ${n}$ be equal to ${p'_1}$ with probability ${\frac{\log p}{\log n}}$, then once ${p'_1}$ is chosen, let each remaining prime factor ${p}$ of ${n/p'_1}$ be equal to ${p'_2}$ with probability ${\frac{\log p}{\log n/p'_1}}$, and so forth.
2. Given a permutation ${\sigma\in S_n}$, one can randomly enumerate its cycles ${C'_1,\ldots,C'_k}$ by letting each cycle ${C}$ in ${\sigma}$ be equal to ${C'_1}$ with probability ${\frac{|C|}{n}}$, and once ${C'_1}$ is chosen, let each remaining cycle ${C}$ be equalto ${C'_2}$ with probability ${\frac{|C|}{n-|C'_1|}}$, and so forth. Alternatively, one traverse the elements of ${\{1,\ldots,n\}}$ in random order, then let ${C'_1}$ be the first cycle one encounters when performing this traversal, let ${C'_2}$ be the next cycle (not equal to ${C'_1}$ one encounters when performing this traversal, and so forth.
3. Given a multiset ${\Gamma}$ of positive real numbers whose sum ${S := \sum_{x\in\Gamma} x}$ is finite, we can randomly enumerate ${x'_1,x'_2,\ldots}$ the elements of this sequence by letting each ${x \in \Gamma}$ have a ${\frac{x}{S}}$ probability of being set equal to ${x'_1}$, and then once ${x'_1}$ is chosen, let each remaining ${x \in \Gamma\backslash \{x'_1\}}$ have a ${\frac{x_i}{S-x'_1}}$ probability of being set equal to ${x'_2}$, and so forth.

We then have the following result:

Proposition 1 (Existence of the Poisson-Dirichlet process) There exists a random partition ${\Sigma}$ whose random enumeration ${Enum(\Sigma) = (u_1,u_2,\ldots)}$ has the uniform distribution on ${[0,1]^{\bf N}}$, thus ${u_1,u_2,\ldots}$ are independently and identically distributed copies of the uniform distribution on ${[0,1]}$.

A random partition ${\Sigma}$ with this property will be called the Poisson-Dirichlet process. This process, first introduced by Kingman, can be described explicitly using (1) as

$\displaystyle \Sigma = \{ u_1, (1-u_1) u_2,(1-u_1)(1-u_2)u_3,\ldots\},$

where ${u_1,u_2,\ldots}$ are iid copies of the uniform distribution of ${[0,1]}$, although it is not immediately obvious from this definition that ${Enum(\Sigma)}$ is indeed uniformly distributed on ${[0,1]^{\bf N}}$. We prove this proposition below the fold.

An equivalent definition of a Poisson-Dirichlet process is a random partition ${\Sigma}$ with the property that

$\displaystyle (u_1, \frac{1}{1-u_1} \cdot (\Sigma \backslash \{u_1\})) \equiv (U, \Sigma) \ \ \ \ \ (2)$

where ${u_1}$ is a random element of ${\Sigma}$ with each ${t \in\Sigma}$ having a probability ${t}$ of being equal to ${u_1}$, ${U}$ is a uniform variable on ${[0,1]}$ that is independent of ${\Sigma}$, and ${\equiv}$ denotes equality of distribution. This can be viewed as a sort of stochastic self-similarity property of ${\Sigma}$: if one randomly removes one element from ${\Sigma}$ and rescales, one gets a new copy of ${\Sigma}$.

It turns out that each of the three ways to generate partitions listed above can lead to the Poisson-Dirichlet process, either directly or in a suitable limit. We begin with the third way, namely by normalising a Poisson process to have sum ${1}$:

Proposition 2 (Poisson-Dirichlet processes via Poisson processes) Let ${a>0}$, and let ${\Gamma_a}$ be a Poisson process on ${(0,+\infty)}$ with intensity function ${t \mapsto \frac{1}{t} e^{-at}}$. Then the sum ${S :=\sum_{x \in \Gamma_a} x}$ is almost surely finite, and the normalisation ${\Sigma_N(\Gamma_a) = \frac{1}{S} \cdot \Gamma_a}$ is a Poisson-Dirichlet process.

Again, we prove this proposition below the fold. Now we turn to the second way (a topic, incidentally, that was briefly touched upon in this previous blog post):

Proposition 3 (Large cycles of a typical permutation) For each natural number ${n}$, let ${\sigma}$ be a permutation drawn uniformly at random from ${S_n}$. Then the random partition ${\Sigma_{CD}(\sigma)}$ converges in the limit ${n \rightarrow\infty}$ to a Poisson-Dirichlet process ${\Sigma_{PF}}$ in the following sense: given any fixed sequence of intervals ${[a_1,b_1],\ldots,[a_k,b_k] \subset I}$ (independent of ${n}$), the joint discrete random variable ${(N_{[a_1,b_1]}(\Sigma_{CD}(\sigma)),\ldots,N_{[a_k,b_k]}(\Sigma_{CD}(\sigma)))}$ converges in distribution to ${(N_{[a_1,b_1]}(\Sigma),\ldots,N_{[a_k,b_k]}(\Sigma))}$.

Finally, we turn to the first way:

Proposition 4 (Large prime factors of a typical number) Let ${x > 0}$, and let ${N_x}$ be a random natural number chosen according to one of the following three rules:

1. (Uniform distribution) ${N_x}$ is drawn uniformly at random from the natural numbers in ${[1,x]}$.
2. (Shifted uniform distribution) ${N_x}$ is drawn uniformly at random from the natural numbers in ${[x,2x]}$.
3. (Zeta distribution) Each natural number ${n}$ has a probability ${\frac{1}{\zeta(s)}\frac{1}{n^s}}$ of being equal to ${N_x}$, where ${s := 1 +\frac{1}{\log x}}$and ${\zeta(s):=\sum_{n=1}^\infty \frac{1}{n^s}}$.

Then ${\Sigma_{PF}(N_x)}$ converges as ${x \rightarrow \infty}$ to a Poisson-Dirichlet process ${\Sigma}$ in the same fashion as in Proposition 3.

The process ${\Sigma_{PF}(N_x)}$ was first studied by Billingsley (and also later by Knuth-Trabb Pardo and by Vershik, but the formulae were initially rather complicated; the proposition above is due to of Donnelly and Grimmett, although the third case of the proposition is substantially easier and appears in the earlier work of Lloyd. We prove the proposition below the fold.

The previous two propositions suggests an interesting analogy between large random integers and large random permutations; see this ICM article of Vershik and this non-technical article of Granville (which, incidentally, was once adapted into a play) for further discussion.

As a sample application, consider the problem of estimating the number ${\pi(x,x^{1/u})}$ of integers up to ${x}$ which are not divisible by any prime larger than ${x^{1/u}}$ (i.e. they are ${x^{1/u}}$-smooth), where ${u>0}$ is a fixed real number. This is essentially (modulo some inessential technicalities concerning the distinction between the intervals ${[x,2x]}$ and ${[1,x]}$) the probability that ${\Sigma}$ avoids ${[1/u,1]}$, which by the above theorem converges to the probability ${\rho(u)}$ that ${\Sigma_{PF}}$ avoids ${[1/u,1]}$. Below the fold we will show that this function is given by the Dickman function, defined by setting ${\rho(u)=1}$ for ${u < 1}$ and ${u\rho'(u) = \rho(u-1)}$ for ${u \geq 1}$, thus recovering the classical result of Dickman that ${\pi(x,x^{1/u}) = (\rho(u)+o(1))x}$.

I thank Andrew Granville and Anatoly Vershik for showing me the nice link between prime factors and the Poisson-Dirichlet process. The material here is standard, and (like many of the other notes on this blog) was primarily written for my own benefit, but it may be of interest to some readers. In preparing this article I found this exposition by Kingman to be helpful.

Note: this article will emphasise the computations rather than rigour, and in particular will rely on informal use of infinitesimals to avoid dealing with stochastic calculus or other technicalities. We adopt the convention that we will neglect higher order terms in infinitesimal calculations, e.g. if ${dt}$ is infinitesimal then we will abbreviate ${dt + o(dt)}$ simply as ${dt}$.

Emmanuel Breuillard, Ben Green, Bob Guralnick, and I have just uploaded to the arXiv our joint paper “Expansion in finite simple groups of Lie type“. This long-delayed paper (announced way back in 2010!) is a followup to our previous paper in which we showed that, with one possible exception, generic pairs of elements of a simple algebraic group (over an uncountable field) generated a free group which was strongly dense in the sense that any nonabelian subgroup of this group was Zariski dense. The main result of this paper is to establish the analogous result for finite simple groups of Lie type (as defined in the previous blog post) and bounded rank, namely that almost all pairs ${a,b}$ of elements of such a group generate a Cayley graph which is a (two-sided) expander, with expansion constant bounded below by a quantity depending on the rank of the group. (Informally, this means that the random walk generated by ${a,b}$ spreads out in logarithmic time to be essentially uniformly distributed across the group, as opposed for instance to being largely trapped in an algebraic subgroup. Thus if generic elements did not generate a strongly dense group, one would probably expect expansion to fail.)

There are also some related results established in the paper. Firstly, as we discovered after writing our first paper, there was one class of algebraic groups for which our demonstration of strongly dense subgroups broke down, namely the ${Sp_4}$ groups in characteristic three. In the current paper we provide in a pair of appendices a new argument that covers this case (or more generally, ${Sp_4}$ in odd characteristic), by first reducing to the case of affine groups ${k^2 \rtimes SL_2(k)}$ (which can be found inside ${Sp_4}$ as a subgroup) and then using a ping-pong argument (in a p-adic metric) in the latter context.

Secondly, we show that the distinction between one-sided expansion and two-sided expansion (see this set of lecture notes of mine for definitions) is erased in the context of Cayley graphs of bounded degree, in the sense that such graphs are one-sided expanders if and only if they are two-sided expanders (perhaps with slightly different expansion constants). The argument turns out to be an elementary combinatorial one, based on the “pivot” argument discussed in these lecture notes of mine.

Now to the main result of the paper, namely the expansion of random Cayley graphs. This result had previously been established for ${SL_2}$ by Bourgain and Gamburd, and Ben, Emmanuel and I had used the Bourgain-Gamburd method to achieve the same result for Suzuki groups. For the other finite simple groups of Lie type, expander graphs had been constructed by Kassabov, Lubotzky, and Nikolov, but they required more than two generators, which were placed deterministically rather than randomly. (Here, I am skipping over a large number of other results on expanding Cayley graphs; see this survey of Lubotzsky for a fairly recent summary of developments.) The current paper also uses the “Bourgain-Gamburd machine”, as discussed in these lecture notes of mine, to demonstrate expansion. This machine shows how expansion of a Cayley graph follows from three basic ingredients, which we state informally as follows:

• Non-concentration (A random walk in this graph does not concentrate in a proper subgroup);
• Product theorem (A medium-sized subset of this group which is not trapped in a proper subgroup will expand under multiplication); and
• Quasirandomness (The group has no small non-trivial linear representations).

Quasirandomness of arbitrary finite simple groups of Lie type was established many years ago (predating, in fact, the introduction of the term “quasirandomness” by Gowers for this property) by Landazuri-Seitz and Seitz-Zalesskii, and the product theorem was already established by Pyber-Szabo and independently by Breuillard, Green, and myself. So the main problem is to establish non-concentration: that for a random Cayley graph on a finite simple group ${G}$ of Lie type, random walks did not concentrate in proper subgroups.

The first step was to classify the proper subgroups of ${G}$. Fortunately, these are all known; in particular, such groups are either contained in proper algebraic subgroups of the algebraic group containing ${G}$ (or a bounded cover thereof) with bounded complexity, or are else arising (up to conjugacy) from a version ${G(F')}$ of the same group ${G =G(F)}$ associated to a proper subfield ${F'}$ of the field ${F}$ respectively; this follows for instance from the work of Larsen and Pink, but also can be deduced using the classification of finite simple groups, together with some work of Aschbacher, Liebeck-Seitz, and Nori. We refer to the two types of subgroups here as “structural subgroups” and “subfield subgroups”.

To preclude concentration in a structural subgroup, we use our previous result that generic elements of an algebraic group generate a strongly dense subgroup, and so do not concentrate in any algebraic subgroup. To translate this result from the algebraic group setting to the finite group setting, we need a Schwarz-Zippel lemma for finite simple groups of Lie type. This is straightforward for Chevalley groups, but turns out to be a bit trickier for the Steinberg and Suzuki-Ree groups, and we have to go back to the Chevalley-type parameterisation of such groups in terms of (twisted) one-parameter subgroups, that can be found for instance in the text of Carter; this “twisted Schwartz-Zippel lemma” may possibly have further application to analysis on twisted simple groups of Lie type. Unfortunately, the Schwartz-Zippel estimate becomes weaker in twisted settings, and particularly in the case of triality groups ${{}^3 D_4(q)}$, which require a somewhat ad hoc additional treatment that relies on passing to a simpler subgroup present in a triality group, namely a central product of two different ${SL_2}$‘s.

To rule out concentration in a conjugate of a subfield group, we repeat an argument we introduced in our Suzuki paper and pass to a matrix model and analyse the coefficients of the characteristic polynomial of words in this Cayley graph, to prevent them from concentrating in a subfield. (Note that these coefficients are conjugation-invariant.)

In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or Lie algebras for short), with a particular focus on those Lie algebras which were semisimple or simple. Among other things, these notes discussed the Weyl complete reducibility theorem (asserting that semisimple Lie algebras are the direct sum of simple Lie algebras) and the classification of simple Lie algebras (with all such Lie algebras being (up to isomorphism) of the form ${A_n}$, ${B_n}$, ${C_n}$, ${D_n}$, ${E_6}$, ${E_7}$, ${E_8}$, ${F_4}$, or ${G_2}$).

Among other things, the structural theory of Lie algebras can then be used to build analogous structures in nearby areas of mathematics, such as Lie groups and Lie algebras over more general fields than the complex field ${{\bf C}}$ (leading in particular to the notion of a Chevalley group), as well as finite simple groups of Lie type, which form the bulk of the classification of finite simple groups (with the exception of the alternating groups and a finite number of sporadic groups).

In the case of complex Lie groups, it turns out that every simple Lie algebra ${\mathfrak{g}}$ is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group ${G_{ad}}$ (the adjoint form of the Lie group) to a “maximal” Lie group ${\tilde G}$ (the simply connected form of the Lie group) that finitely covers ${G_{ad}}$, and occasionally also a number of intermediate forms which finitely cover ${G_{ad}}$, but are in turn finitely covered by ${\tilde G}$. For instance, ${\mathfrak{sl}_n({\bf C})}$ is associated with the projective special linear group ${\hbox{PSL}_n({\bf C}) = \hbox{PGL}_n({\bf C})}$ as its adjoint form and the special linear group ${\hbox{SL}_n({\bf C})}$ as its simply connected form, and intermediate groups can be created by quotienting out ${\hbox{SL}_n({\bf C})}$ by some subgroup of its centre (which is isomorphic to the ${n^{th}}$ roots of unity). The minimal form ${G_{ad}}$ is simple in the group-theoretic sense of having no normal subgroups, but the other forms of the Lie group are merely quasisimple, although traditionally all of the forms of a Lie group associated to a simple Lie algebra are known as simple Lie groups.

Thanks to the work of Chevalley, a very similar story holds for algebraic groups over arbitrary fields ${k}$; given any Dynkin diagram, one can define a simple Lie algebra with that diagram over that field, and also one can find a finite number of connected algebraic groups over ${k}$ (known as Chevalley groups) with that Lie algebra, ranging from an adjoint form ${G_{ad}}$ to a universal form ${G_u}$, with every form having an isogeny (the analogue of a finite cover for algebraic groups) to the adjoint form, and in turn receiving an isogeny from the universal form. Thus, for instance, one could construct the universal form ${E_7(q)_u}$ of the ${E_7}$ algebraic group over a finite field ${{\bf F}_q}$ of finite order.

When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. ${\hbox{PSL}_n({\bf F}_q)}$), one usually obtains a finite simple group (with a finite number of exceptions when the rank and the field are very small, and in some cases one also has to pass to a bounded index subgroup, such as the derived group, first). One could also use other forms than the adjoint form, but one then recovers the same finite simple group as before if one quotients out by the centre. This construction was then extended by Steinberg, Suzuki, and Ree by taking a Chevalley group over a finite field and then restricting to the fixed points of a certain automorphism of that group; after some additional minor modifications such as passing to a bounded index subgroup or quotienting out a bounded centre, this gives some additional finite simple groups of Lie type, including classical examples such as the projective special unitary groups ${\hbox{PSU}_n({\bf F}_{q^2})}$, as well as some more exotic examples such as the Suzuki groups or the Ree groups.

While I learned most of the classical structural theory of Lie algebras back when I was an undergraduate, and have interacted with Lie groups in many ways in the past (most recently in connection with Hilbert’s fifth problem, as discussed in this previous series of lectures), I have only recently had the need to understand more precisely the concepts of a Chevalley group and of a finite simple group of Lie type, as well as better understand the structural theory of simple complex Lie groups. As such, I am recording some notes here regarding these concepts, mainly for my own benefit, but perhaps they will also be of use to some other readers. The material here is standard, and was drawn from a number of sources, but primarily from Carter, Gorenstein-Lyons-Solomon, and Fulton-Harris, as well as the lecture notes on Chevalley groups by my colleague Robert Steinberg. The arrangement of material also reflects my own personal preferences; in particular, I tend to favour complex-variable or Riemannian geometry methods over algebraic ones, and this influenced a number of choices I had to make regarding how to prove certain key facts. The notes below are far from a comprehensive or fully detailed discussion of these topics, and I would refer interested readers to the references above for a properly thorough treatment.

The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments.    As with the previous thread, the main focus on the comments to this thread are concerned with writing up the results of the Polymath8 “bounded gaps between primes” project; the latest files on this writeup may be found at this directory, with the most recently compiled PDF file (clocking in at about 90 pages so far, with a few sections still to be written!) being found here.  There is also still some active discussion on improving the numerical results, with a particular focus on improving the sieving step that converts distribution estimates such as $MPZ^{(i)}[\varpi,\delta]$ into weak prime tuples results $DHL[k_0,2]$.  (For a discussion of the terminology, and for a general overview of the proof strategy, see this previous progress report on the Polymath8 project.)  This post can also contain any other discussion pertinent to any aspect of the polymath8 project, of course.

There are a few sections that still need to be written for the draft, mostly concerned with the Type I, Type II, and Type III estimates.  However, the proofs of these estimates exist already on this blog, so I hope to transcribe them to the paper fairly shortly (say by the end of this week).  Barring any unexpected surprises, or major reorganisation of the paper, it seems that the main remaining task in the writing process would be the proofreading and polishing, and turning from the technical mathematical details to expository issues.  As always, feedback from casual participants, as well as those who have been closely involved with the project, would be very valuable in this regard.  (One small comment, by the way, regarding corrections: as the draft keeps changing with time, referring to a specific line of the paper using page numbers and line numbers can become inaccurate, so if one could try to use section numbers, theorem numbers, or equation numbers as reference instead (e.g. “the third line after (5.35)” instead of “the twelfth line of page 54″) that would make it easier to track down specific portions of the paper.)

Also, we have set up a wiki page for listing the participants of the polymath8 project, their contact information, and grant information (if applicable).  We have two lists of participants; one for those who have been making significant contributions to the project (comparable to that of a co-author of a traditional mathematical research paper), and another list for those who have made auxiliary contributions (e.g. typos, stylistic suggestions, or supplying references) that would typically merit inclusion in the Acknowledgments section of a traditional paper.  It’s difficult to exactly draw the line between the two types of contributions, but we have relied in the past on self-reporting, which has worked pretty well so far.  (By the time this project concludes, I may go through the comments to previous posts and see if any further names should be added to these lists that have not already been self-reported.)

The main objectives of the polymath8 project, initiated back in June, were to understand the recent breakthrough paper of Zhang establishing an infinite number of prime gaps bounded by a fixed constant ${H}$, and then to lower that value of ${H}$ as much as possible. After a large number of refinements, optimisations, and other modifications to Zhang’s method, we have now lowered the value of ${H}$ from the initial value of ${70,000,000}$ down to (provisionally) ${4,680}$, as well as to the slightly worse value of ${14,994}$ if one wishes to avoid any reliance on the deep theorems of Deligne on the Weil conjectures.

As has often been the case with other polymath projects, the pace has settled down subtantially after the initial frenzy of activity; in particular, the values of ${H}$ (and other key parameters, such as ${k_0}$, ${\varpi}$, and ${\delta}$) have stabilised over the last few weeks. While there may still be a few small improvements in these parameters that can be wrung out of our methods, I think it is safe to say that we have cleared out most of the “low-hanging fruit” (and even some of the “medium-hanging fruit”), which means that it is time to transition to the next phase of the polymath project, namely the writing phase.

After some discussion at the previous post, we have tentatively decided on writing a single research paper, which contains (in a reasonably self-contained fashion) the details of the strongest result we have (i.e. bounded gaps with ${H = 4,680}$), together with some variants, such as the bound ${H=14,994}$ that one can obtain without invoking Deligne’s theorems. We can of course also include some discussion as to where further improvements could conceivably arise from these methods, although even if one assumes the most optimistic estimates regarding distribution of the primes, we still do not have any way to get past the barrier of ${H=16}$ identified as the limit of this method by Goldston, Pintz, and Yildirim. This research paper does not necessarily represent the only output of the polymath8 project; for instance, as part of the polymath8 project the admissible tuples page was created, which is a repository of narrow prime tuples which can automatically accept (and verify) new submissions. (At an early stage of the project, it was suggested that we set up a computing challenge for mathematically inclined programmers to try to find the narrowest prime tuples of a given width; it might be worth revisiting this idea now that our value of ${k_0}$ has stabilised and the prime tuples page is up and running.) Other potential outputs include additional expository articles, lecture notes, or perhaps the details of a “minimal proof” of bounded gaps between primes that gives a lousy value of ${H}$ but with as short and conceptual a proof as possible. But it seems to me that these projects do not need to proceed via the traditional research paper route (perhaps ending up on the blog, on the wiki, or on the admissible tuples page instead). Also, these projects might also benefit from the passage of time to lend a bit of perspective and depth, especially given that there are likely to be further advances in this field from outside of the polymath project.

I have taken the liberty of setting up a Dropbox folder containing a skeletal outline of a possible research paper, and anyone who is interested in making significant contributions to the writeup of the paper can contact me to be given write access to that folder. However, I am not firmly wedded to the organisational structure of that paper, and at this stage it is quite easy to move sections around if this would lead to a more readable or more logically organised paper.

I have tried to structure the paper so that the deepest arguments – the ones which rely on Deligne’s theorems – are placed at the end of the paper, so that a reader who wishes to read and understand a proof of bounded gaps that does not rely on Deligne’s theorems can stop reading about halfway through the paper. I have also moved the top-level structure of the argument (deducing bounded gaps from a Dickson-Hardy-Littlewood claim ${DHL[k_0,2]}$, which in turn is established from a Motohashi-Pintz-Zhang distribution estimate ${MPZ^{(i)}[\varpi,\delta]}$, which is in turn deduced from Type I, Type II, and Type III estimates) to the front of the paper.

Of course, any feedback on the draft paper is encouraged, even from (or especially from!) readers who have been following this project on a casual basis, as this would be valuable in making sure that the paper is written in as accessible as fashion as possible. (Sometimes it is possible to be so close to a project that one loses some sense of perspective, and does not realise that what one is writing might not necessarily be as clear to other mathematicians as it is to the author.)