for any fixed . Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes is by Burgess, who showed that

for any fixed . See this previous post for a proof of these bounds.

In this paper, we show that the Vinogradov conjecture is a consequence of the Elliott-Halberstam conjecture. Using a variant of the argument, we also show that the “Type II” estimates established by Zhang and numerically improved by the Polymath8a project can be used to improve a little on the Vinogradov bound (1), to

although this falls well short of the Burgess bound. However, the method is somewhat different (although in both cases it is the Weil exponential sum bounds that are the source of the gain over (1)) and it is conceivable that a numerically stronger version of the Type II estimates could obtain results that are more competitive with the Burgess bound. At any rate, this demonstrates that the equidistribution estimates introduced by Zhang may have further applications beyond the family of results related to bounded gaps between primes.

The connection between the least quadratic nonresidue problem and Elliott-Halberstam is follows. Suppose for contradiction we can find a prime with unusually large. Letting be the quadratic character modulo , this implies that the sums are also unusually large for a significant range of (e.g. ), although the sum is also quite small for large (e.g. ), due to the cancellation present in . It turns out (by a sort of “uncertainty principle” for multiplicative functions, as per this paper of Granville and Soundararajan) that these facts force to be unusually large in magnitude for some large (with for two large absolute constants ). By the periodicity of , this means that

must be unusually large also. However, because is large, one can factorise as for a fairly sparsely supported function . The Elliott-Halberstam conjecture, which controls the distribution of in arithmetic progressions on the average can then be used to show that is small, giving the required contradiction.

The implication involving Type II estimates is proven by a variant of the argument. If is large, then a character sum is unusually large for a certain . By multiplicativity of , this shows that correlates with , and then by periodicity of , this shows that correlates with for various small . By the Cauchy-Schwarz inequality (cf. this previous blog post), this implies that correlates with for some distinct . But this can be ruled out by using Type II estimates.

I’ll record here a well-known observation concerning potential counterexamples to any improvement to the Burgess bound, that is to say an infinite sequence of primes with . Suppose we let be the asymptotic mean value of the quadratic character at and the mean value of ; these quantities are defined precisely in my paper, but roughly speaking one can think of

and

Thanks to the basic Dirichlet convolution identity , one can establish the *Wirsing integral equation*

for all ; see my paper for details (actually far sharper claims than this appear in previous work of Wirsing and Granville-Soundararajan). If we have an infinite sequence of counterexamples to any improvement to the Burgess bound, then we have

while from the Burgess exponential sum estimates we have

These two constraints, together with the Wirsing integral equation, end up determining and completely. It turns out that we must have

and

and then for , evolves by the integral equation

For instance

and then oscillates in a somewhat strange fashion towards zero as . This very odd behaviour of is surely impossible, but it seems remarkably hard to exclude it without invoking a strong hypothesis, such as GRH or the Elliott-Halberstam conjecture (or weaker versions thereof).

Filed under: math.NT, paper Tagged: Elliott-Halberstam conjecture, quadratic nonresidue, Type II estimate, Vinogradov conjecture ]]>

is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula

where the second von Mangoldt function is defined by the formula

(We are avoiding the use of the symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly.) For the convenience of the reader, we give a proof of the Selberg symmetry formula below the fold. Actually, for the purposes of proving the prime number theorem, the weaker estimate

In this post I would like to record a somewhat “soft analysis” reformulation of the elementary proof of the prime number theorem in terms of Banach algebras, and specifically in Banach algebra structures on (completions of) the space of compactly supported continuous functions equipped with the convolution operation

This soft argument does not easily give any quantitative decay rate in the prime number theorem, but by the same token it avoids many of the quantitative calculations in the traditional proofs of this theorem. Ultimately, the key “soft analysis” fact used is the spectral radius formula

for any element of a unital commutative Banach algebra , where is the space of characters (i.e., continuous unital algebra homomorphisms from to ) of . This formula is due to Gelfand and may be found in any text on Banach algebras; for sake of completeness we prove it below the fold.

The connection between prime numbers and Banach algebras is given by the following consequence of the Selberg symmetry formula.

Theorem 1 (Construction of a Banach algebra norm)For any , let denote the quantityThen is a seminorm on with the bound

We prove this theorem below the fold. The prime number theorem then follows from Theorem 1 and the following two assertions. The first is an application of the spectral radius formula (6) and some basic Fourier analysis (in particular, the observation that contains a plentiful supply of local units:

Theorem 2 (Non-trivial Banach algebras with many local units have non-trivial spectrum)Let be a seminorm on obeying (7), (8). Suppose that is not identically zero. Then there exists such thatfor all . In particular, by (7), one has

whenever is a non-negative function.

The second is a consequence of the Selberg symmetry formula and the fact that is real (as well as Mertens’ theorem, in the case), and is closely related to the non-vanishing of the Riemann zeta function on the line :

Theorem 3 (Breaking the parity barrier)Let . Then there exists such that is non-negative, and

Assuming Theorems 1, 2, 3, we may now quickly establish the prime number theorem as follows. Theorem 2 and Theorem 3 imply that the seminorm constructed in Theorem 1 is trivial, and thus

as for any Schwartz function (the decay rate in may depend on ). Specialising to functions of the form for some smooth compactly supported on , we conclude that

as ; by the smooth Urysohn lemma this implies that

as for any fixed , and the prime number theorem then follows by a telescoping series argument.

The same argument also yields the prime number theorem in arithmetic progressions, or equivalently that

for any fixed Dirichlet character ; the one difference is that the use of Mertens’ theorem is replaced by the basic fact that the quantity is non-vanishing.

** — 1. Proof of Selberg symmetry formula — **

We now prove (2). From (3) we have

From the integral test we have the estimates

for some absolute constants whose exact value is unimportant for us, and for any . We conclude that

for some further absolute constants . Replacing by and inserting this into (9), one obtains

The error term can be computed to be . The main term simplifies by Möbius inversion to , and the claim follows.

** — 2. Constructing the Banach algebra — **

We now prove Theorem 1. It is convenient to transform the situation from the classical context of arithmetic functions on (such as or ) to the more Fourier-analytic context of Radon measures on the real line . Define the discrete Radon measure

and for any , let denote the left translate of the measure by , thus

for any continuous compactly supported . We note in passing that the prime number theorem (1) is equivalent to the assertion that the translates converge in the vague topology to Lebesgue measure as .

where is the convolution of the Radon measures , and is the measure multiplied by the identity function . From (4) one has

We claim that the Selberg symmetry formula (5) implies (in fact, it is equivalent to) the assertion that the translates converge in the vague topology to . Indeed, (5) implies for any fixed that

or equivalently that

which we rewrite as

Since for , we thus have

which implies that converges vaguely to , and the claim follows.

Now we begin the proof of Theorem 1. Observe that the quantity can be rewritten as

and converges vaguely to , we see that the measures are precompact in the vague topology, thanks to the Helly selection principle or Prokhorov theorem. In particular, we have

for some limit point of the translates in the vague topology. From (12) we have

Finally, we prove (8). By(11), it suffices to show that

for any , where the decay errors are allowed to depend on . Since converges vaguely to , we already have from (10) that

so it suffices to show that

Let be Lebesgue measure on the half-line . Then , so converges vaguely to . The measure is equal to times the function , so by Mertens’ theorem this function also converges vaguely to . We conclude that

converges vaguely to , and so it suffices to show that

We rewrite this as

On the support of , we have , so it suffices to show that

(The error term in can be controlled by using (15) with replaced by , and modifying the preceding arguments to replace by .)

From Fubini’s theorem we have

The integrand vanishes unless . By (11), we have

and

and the claim (15) follows.

** — 3. Non-trivial algebras with many local units have non-trivial spectrum — **

We now prove Theorem 2. Let be the Banach algebra completion of under the seminorm (thus is the space of Cauchy sequences in , quotiented out by the sequences that go to zero in the seminorm ). Since is not identically zero, is a non-trivial commutative Banach algebra (but it is not necessarily unital).

It is convenient to adjoin a unit to to create a unital commutative Banach algebra with the extended norm

for and ; one easily verifies that is a unital commutative Banach algebra.

Suppose that all elements of have zero spectral radius (as defined in (6)). Let be a Schwartz function with compactly supported Fourier transform. Then we can find another Schwartz function with compactly supported Fourier transform such that (by ensuring that on the support of ; thus is a “local unit” on the Fourier support of ). Thus for all . But has spectral radius zero, thus is zero in . By density this implies that is trivial, a contradiction.

Thus there is an element of with positive spectral radius. Then by (6), there is a character that is does not vanish identically on . Suppose that for each there exists in the kernel of whose Fourier coefficient is non-vanishing. Since the kernel of is a space closed with respect to convolutions by functions, some Fourier analysis and a smooth partition of unity then shows that the kernel of contains any Schwartz function with compactly supported Fourier transform, and thus by density is trivial, a contradiction. Thus there must exist such that contains all test functions with Fourier coefficient vanishing at . From this we conclude that on is a constant multiple of the Fourier coefficient map ; being a non-trivial algebra homomorphism on , we thus have

for all . Since characters have norm at most (as can be seen for instance from (6)), we obtain the claim.

** — 4. Breaking the parity barrier — **

We now prove Theorem 3. We divide into two cases, depending on whether or . If , we let be a continuous function that equals on and is supported on for some large . From Mertens’ theorem we have

for sufficiently large depending on , and thus

The claim then follows by taking sufficiently large.

Now suppose . In the language of Section 2, we have

for some limit point of the . We can write the right-hand side as

for some phase . From (14), is a real measure between and , so by the triangle inequality we have

Now we set , where is as before. Then

Since is periodic with period and has mean value strictly less than (in fact, it has mean ), we thus have

if is sufficiently large depending on . The claim follows.

** — 5. The prime number theorem in arithmetic progressions — **

Let be a non-principal Dirichlet character of some period . We allow all implied constants in the notation to depend on . In this section we sketch the changes to the above arguments needed to establish

which gives the prime number in arithmetic progressions by the usual Fourier expansion into Dirichlet characters.

We have the twisted versions

and

of (3), (4). Since has mean zero, a decomposition into intervals of length reveals that

from which we obtain the twisted Selberg symmetry formula

If we define the twisted measures

and

then

and hence converges weakly to zero as . Introducing the twisted norms

we may verify that obeys the conclusions of Theorem 1.

By repeating the previous arguments, it will suffice that the analogue of Theorem 3 for holds. When , we can argue as in Section 4, where the role of Mertens’ theorem is replaced by Dirichlet’s theorem

which is ultimately a consequence of the non-vanishing of .

For , the argument in Section 4 works with minimal changes if is real-valued. If is complex valued, it still takes only a finite number of values in the unit disk. Then the limit measures appearing in Section 4 are equal to Lebesgue measure times a density taking values in the convex hull of this finite set of values, which is a polygon in the unit disk. One can then modify the arguments in Section 4 to bound

for some phase . If we set as before, we again observe that the function is periodic and has mean strictly less than one, and so we can again establish the required bound if is large enough.

** — 6. Proof of Gelfand formula — **

We now prove (6).

If is a character, then it has an operator norm:

But we may eliminate this norm by using the “tensor power trick”: replacing with and then taking roots we conclude that

and then on sending we have

Replacing by again, taking roots, and sending we conclude that

(The limit exists because is submultiplicative.) This gives one direction of (6). To give the other direction, suppose for sake of contradiction that we could find an such that

There are two cases, depending on whether we can find a complex number with and non-invertible. First suppose that such a exists; then generates an ideal of , which by Zorn’s lemma is contained in a maximal ideal , whose quotient is then a field. By Neumann series, any element of sufficiently close to the identity is invertible and thus not in ; since is a field, we conclude that the complement of is open, and so is closed. This makes a Banach algebra as well as a field. If is not a multiple of the identity, then is invertible for every and so (by Neumann series) is an analytic function from to which goes to zero at infinity, contradicting Liouville’s theorem. Thus is one-dimensional (this is the Banach-Mazur theorem) and thus isomorphic to ; this gives a continuous unital algebra homomorphism with in the kernel, thus , contradicting the second inequality in (16).

Now suppose that is invertible for all . Then, as in the preceding argument, is an analytic function from to which decays to zero at infinity, so we have the Cauchy integral formula

for any natural number . From the triangle inequality we conclude in particular that

which contradicts the first inequality in (16).

Filed under: expository, math.NT, math.OA, math.SP Tagged: Banach algebra, prime number theorem, spectral theorem ]]>

converge to the integral

the triangle density

converges to the integral

the four-cycle density

converges to the integral

and so forth. One can use graph limits to prove many results in graph theory that were traditionally proven using the regularity lemma, such as the triangle removal lemma, and can also reduce many asymptotic graph theory problems to continuous problems involving multilinear integrals (although the latter problems are not necessarily easy to solve!). See this text of Lovasz for a detailed study of graph limits and their applications.

One can also express graph limits (and more generally hypergraph limits) in the language of nonstandard analysis (or of ultraproducts); see for instance this paper of Elek and Szegedy, Section 6 of this previous blog post, or this paper of Towsner. (In this post we assume some familiarity with nonstandard analysis, as reviewed for instance in the previous blog post.) Here, one starts as before with a sequence of finite graphs, and then takes an ultraproduct (with respect to some arbitrarily chosen non-principal ultrafilter ) to obtain a nonstandard graph , where is the ultraproduct of the , and similarly for the . The set can then be viewed as a symmetric subset of which is measurable with respect to the Loeb -algebra of the product (see this previous blog post for the construction of Loeb measure). A crucial point is that this -algebra is larger than the product of the Loeb -algebra of the individual vertex set . This leads to a decomposition

where the “graphon” is the orthogonal projection of onto , and the “regular error” is orthogonal to all product sets for . The graphon then captures the statistics of the nonstandard graph , in exact analogy with the more traditional graph limits: for instance, the edge density

(or equivalently, the limit of the along the ultrafilter ) is equal to the integral

where denotes Loeb measure on a nonstandard finite set ; the triangle density

(or equivalently, the limit along of the triangle densities of ) is equal to the integral

and so forth. Note that with this construction, the graphon is living on the Cartesian square of an abstract probability space , which is likely to be inseparable; but it is possible to cut down the Loeb -algebra on to minimal countable -algebra for which remains measurable (up to null sets), and then one can identify with , bringing this construction of a graphon in line with the traditional notion of a graphon. (See Remark 5 of this previous blog post for more discussion of this point.)

Additive combinatorics, which studies things like the additive structure of finite subsets of an abelian group , has many analogies and connections with asymptotic graph theory; in particular, there is the arithmetic regularity lemma of Green which is analogous to the graph regularity lemma of Szemerédi. (There is also a higher order arithmetic regularity lemma analogous to hypergraph regularity lemmas, but this is not the focus of the discussion here.) Given this, it is natural to suspect that there is a theory of “additive limits” for large additive sets of bounded doubling, analogous to the theory of graph limits for large dense graphs. The purpose of this post is to record a candidate for such an additive limit. This limit can be used as a substitute for the arithmetic regularity lemma in certain results in additive combinatorics, at least if one is willing to settle for qualitative results rather than quantitative ones; I give a few examples of this below the fold.

It seems that to allow for the most flexible and powerful manifestation of this theory, it is convenient to use the nonstandard formulation (among other things, it allows for full use of the transfer principle, whereas a more traditional limit formulation would only allow for a transfer of those quantities continuous with respect to the notion of convergence). Here, the analogue of a nonstandard graph is an *ultra approximate group* in a nonstandard group , defined as the ultraproduct of finite -approximate groups for some standard . (A -approximate group is a symmetric set containing the origin such that can be covered by or fewer translates of .) We then let be the external subgroup of generated by ; equivalently, is the union of over all standard . This space has a Loeb measure , defined by setting

whenever is an internal subset of for any standard , and extended to a countably additive measure; the arguments in Section 6 of this previous blog post can be easily modified to give a construction of this measure.

The Loeb measure is a translation invariant measure on , normalised so that has Loeb measure one. As such, one should think of as being analogous to a locally compact abelian group equipped with a Haar measure. It should be noted though that is not *actually* a locally compact group with Haar measure, for two reasons:

- There is not an obvious topology on that makes it simultaneously locally compact, Hausdorff, and -compact. (One can get one or two out of three without difficulty, though.)
- The addition operation is not measurable from the product Loeb algebra to . Instead, it is measurable from the coarser Loeb algebra to (compare with the analogous situation for nonstandard graphs).

Nevertheless, the analogy is a useful guide for the arguments that follow.

Let denote the space of bounded Loeb measurable functions (modulo almost everywhere equivalence) that are supported on for some standard ; this is a complex algebra with respect to pointwise multiplication. There is also a convolution operation , defined by setting

whenever , are bounded nonstandard functions (extended by zero to all of ), and then extending to arbitrary elements of by density. Equivalently, is the pushforward of the -measurable function under the map .

The basic structural theorem is then as follows.

Theorem 1 (Kronecker factor)Let be an ultra approximate group. Then there exists a (standard) locally compact abelian group of the formfor some standard and some compact abelian group , equipped with a Haar measure and a measurable homomorphism (using the Loeb -algebra on and the Borel -algebra on ), with the following properties:

- (i) has dense image, and is the pushforward of Loeb measure by .
- (ii) There exists sets with open and compact, such that
- (iii) Whenever with compact and open, there exists a nonstandard finite set such that
- (iv) If , then we have the convolution formula
where are the pushforwards of to , the convolution on the right-hand side is convolution using , and is the pullback map from to . In particular, if , then for all .

One can view the locally compact abelian group as a “model “or “Kronecker factor” for the ultra approximate group (in close analogy with the Kronecker factor from ergodic theory). In the case that is a genuine nonstandard finite group rather than an ultra approximate group, the non-compact components of the Kronecker group are trivial, and this theorem was implicitly established by Szegedy. The compact group is quite large, and in particular is likely to be inseparable; but as with the case of graphons, when one is only studying at most countably many functions , one can cut down the size of this group to be separable (or equivalently, second countable or metrisable) if desired, so one often works with a “reduced Kronecker factor” which is a quotient of the full Kronecker factor .

Given any sequence of uniformly bounded functions for some fixed , we can view the function defined by

as an “additive limit” of the , in much the same way that graphons are limits of the indicator functions . The additive limits capture some of the statistics of the , for instance the normalised means

converge (along the ultrafilter ) to the mean

and for three sequences of functions, the normalised correlation

converges along to the correlation

the normalised Gowers norm

converges along to the Gowers norm

and so forth. We caution however that some correlations that involve evaluating more than one function at the same point will not necessarily be preserved in the additive limit; for instance the normalised norm

does not necessarily converge to the norm

but can converge instead to a larger quantity, due to the presence of the orthogonal projection in the definition (4) of .

An important special case of an additive limit occurs when the functions involved are indicator functions of some subsets of . The additive limit does not necessarily remain an indicator function, but instead takes values in (much as a graphon takes values in even though the original indicators take values in ). The convolution is then the ultralimit of the normalised convolutions ; in particular, the measure of the support of provides a lower bound on the limiting normalised cardinality of a sumset. In many situations this lower bound is an equality, but this is not necessarily the case, because the sumset could contain a large number of elements which have very few () representations as the sum of two elements of , and in the limit these portions of the sumset fall outside of the support of . (One can think of the support of as describing the “essential” sumset of , discarding those elements that have only very few representations.) Similarly for higher convolutions of . Thus one can use additive limits to partially control the growth of iterated sumsets of subsets of approximate groups , in the regime where stays bounded and goes to infinity.

Theorem 1 can be proven by Fourier-analytic means (combined with Freiman’s theorem from additive combinatorics), and we will do so below the fold. For now, we give some illustrative examples of additive limits.

Example 1 (Bohr sets)We take to be the intervals , where is a sequence going to infinity; these are -approximate groups for all . Let be an irrational real number, let be an interval in , and for each natural number let be the Bohr setIn this case, the (reduced) Kronecker factor can be taken to be the infinite cylinder with the usual Lebesgue measure . The additive limits of and end up being and , where is the finite cylinder

and is the rectangle

Geometrically, one should think of and as being wrapped around the cylinder via the homomorphism , and then one sees that is converging in some normalised weak sense to , and similarly for and . In particular, the additive limit predicts the growth rate of the iterated sumsets to be quadratic in until becomes comparable to , at which point the growth transitions to linear growth, in the regime where is bounded and is large.

If were rational instead of irrational, then one would need to replace by the finite subgroup here.

Example 2 (Structured subsets of progressions)We take be the rank two progressionwhere is a sequence going to infinity; these are -approximate groups for all . Let be the subset

Then the (reduced) Kronecker factor can be taken to be with Lebesgue measure , and the additive limits of the and are then and , where is the square

and is the circle

Geometrically, the picture is similar to the Bohr set one, except now one uses a Freiman homomorphism for to embed the original sets into the plane . In particular, one now expects the growth rate of the iterated sumsets and to be quadratic in , in the regime where is bounded and is large.

Example 3 (Dissociated sets)Let be a fixed natural number, and takewhere are randomly chosen elements of a large cyclic group , where is a sequence of primes going to infinity. These are -approximate groups. The (reduced) Kronecker factor can (almost surely) then be taken to be with counting measure, and the additive limit of is , where and is the standard basis of . In particular, the growth rates of should grow approximately like for bounded and large.

Example 4 (Random subsets of groups)Let be a sequence of finite additive groups whose order is going to infinity. Let be a random subset of of some fixed density . Then (almost surely) the Kronecker factor here can be reduced all the way to the trivial group , and the additive limit of the is the constant function . The convolutions then converge in the ultralimit (modulo almost everywhere equivalence) to the pullback of ; this reflects the fact that of the elements of can be represented as the sum of two elements of in ways. In particular, occupies a proportion of .

Example 5 (Trigonometric series)Take for a sequence of primes going to infinity, and for each let be an infinite sequence of frequencies chosen uniformly and independently from . Let denote the random trigonometric seriesThen (almost surely) we can take the reduced Kronecker factor to be the infinite torus (with the Haar probability measure ), and the additive limit of the then becomes the function defined by the formula

In fact, the pullback is the ultralimit of the . As such, for any standard exponent , the normalised norm

can be seen to converge to the limit

The reader is invited to consider combinations of the above examples, e.g. random subsets of Bohr sets, to get a sense of the general case of Theorem 1.

It is likely that this theorem can be extended to the noncommutative setting, using the noncommutative Freiman theorem of Emmanuel Breuillard, Ben Green, and myself, but I have not attempted to do so here (see though this recent preprint of Anush Tserunyan for some related explorations); in a separate direction, there should be extensions that can control higher Gowers norms, in the spirit of the work of Szegedy.

Note: the arguments below will presume some familiarity with additive combinatorics and with nonstandard analysis, and will be a little sketchy in places.

** — 1. Proof of theorem — **

By Freiman’s theorem for arbitrary abelian groups (see this paper of Green and Ruzsa), we can find an *ultra coset progression* such that

for some standard ; we abbreviate the latter inclusion as . By an ultra coset progression, we mean the sumset of a nonstandard finite group and a nonstandard generalised arithmetic progression

with (known as the *rank*) standard, the *generators* in and the *dimensions* being nonstandard natural numbers. (To get the containment , one can first use the Bogulybov lemma to get a large ultra coset progression inside , so that can be covered by translates of ; one can then add these translates to the generators of to obtain an ultra coset progression with the required properties.

We call the ultra coset progression *-proper* if the sums for and for are all distinct. If fails to be -proper, then we can find a containment

where the coset progression has strictly smaller rank than ; see e.g. Lemma 5.1 of this paper of Van Vu and myself). Iterating this fact, we see that we may assume without loss of generality that is -proper. In particular, the group can now be parameterised by the sums with for , with each element of having exactly one representation of this form.

The dimensions are either bounded (and thus standard natural numbers) or unbounded. After permuting the generators if necessary, we may assume that are unbounded and are bounded for some with . We then have an external surjective group homomorphism defined by

this will end up being the non-compact portion of the projection map that we will eventually construct. The image is precompact in (in fact it is compact, thanks to countable saturation).

Now we perform some Fourier analysis on (analogous to the usual theory of Fourier analysis on locally compact abelian groups). Define a *frequency* to be a measurable homomorphism from to , and let denote the space of such frequencies; this is an additive group, which should be thought of as a “Pontryagin dual” to (even though is not a locally compact group). Meanwhile, we have the (genuine) Pontryagin dual of , using the identification

The homomorphism then induces a dual homomorphism , defined by the formula

for all and . This homomorphism is easily seen to be injective. If we let denote the cokernel of this map, then is an abelian group (which we will view as a discrete group) and we have the short exact sequence

Observe that is a divisible group. From this and a Zorn’s lemma argument we can split this short exact sequence, lifting up to a subgroup of , so that the latter group can be viewed as the direct sum of and .

Let be the Pontryagin dual of , that is to say the space of all homomorphisms from to (with no measurability or continuity hypotheses imposed). This is a compact abelian group (it is a closed subset of , which is compact by Tychonoff’s theorem). Set . We have a homomorphism , defined by

We claim that has dense image. Since is surjective, it suffices to show that the map from to has dense image from to , where

is the kernel of . The closure of the image of is a compact subgroup of , so this map did not have dense image, there would exist a non-trivial in the Pontryagin dual of which annihilates all of . The map then factors through and thus can be identified with an element of ; but and only intersect at , a contradiction. Thus has dense image.

It is a routine matter to verify that is measurable, that is precompact, and that the inverse image of any compact set is contained in for some standard . From this and the Riesz representation theorem, we can define a Haar measure on by defining

for all continuous, compactly supported functions ; the translation invariance of this measure follows from the surjectivity of . From Urysohn’s lemma and the inner and outer regularity of Haar measures, one can then show that is the pushforward of Loeb measure under .

Now we show the convolution property (3). First suppose that , which in particular implies that

for all , since the function factors through . By the Loeb measure construction, we can write as the limit (in ) of functions , where are uniformly bounded nonstandard functions and is some standard natural number. Then we have

which in particular implies that

where ranges over all nonstandard maps of the form

for some and nonstandard homomorphism . From (nonstandard) Fourier analysis, we conclude that

for any bounded nonstandard function , or equivalently that

and thus on taking limits we see that , and on taking further limits we see that for any , as required. This proves (3) when ; similarly when . To finish off the general case of (3), it suffices to show that

for bounded measurable . By Fourier decomposition, we may assume that takes the form

for some and some continuous compactly supported , and similarly

for some and continuous compactly supported .

If , then for some , and one can use this to show that and both vanish. Thus we may assume that ; using modulation symmetry we may then assume that . It thus suffices to show that

A direct calculation shows that the left and right hand sides agree up to constants; but both sides also have integral when integrated against , so they must agree identically.

Now, we prove the inclusions (1). The outer inclusion comes from the compactness (or precompactness) of . For the inner inclusion, we note from (3) and the positive measure and symmetry of that is the pullback of a continuous function on that is positive at the origin, and thus also bounded away from zero on a neighbourhood of the origin. This implies that the set has full measure in . We then let be a smaller symmetric neighbourhood of the origin such that . We then see that for any , the sets and both have full measure in , and hence lies in . This gives the inner inclusion of (1).

Finally, we show the regularity claim (2). Given , we may apply Urysohn’s lemma to find non-negative bounded continuous functions such that is supported in and is at least on . Letting be the pullbacks of by , we conclude using (3) that is at least on and vanishes outside of . Approximating in by bounded nonstandard functions supported in , we conclude that is at least (say) on and less than (say) outside of . If one then sets to be the non-standard set where , we obtain the claim.

** — 2. Sample applications of theorem — **

In this section we illustrate how this theorem can be used to reprove some existing results in additive combinatorics, reducing them to various statements in continuous harmonic analysis. We begin with a qualitative version of a result of Croot and Sisask on almost periods, which reduces to the classical fact that the convolution of two square-integrable functions is continuous.

Proposition 2 (Croot-Sisask)Let be a -approximate group in an additive group , let be subsets of , and let . Then there exists a subset of with such thatfor all (using the non-normalised convolution ).

The Croot-Sisask argument in fact gives a quantitative lower bound of exponential type on , but such bounds are not available through the qualitative limiting arguments given here. The Croot-Sisask argument also works in non-commutative groups ; it is likely the arguments here would also extend to that setting once one developed a non-commutative version of Theorem 1, but we have not investigated that here.

*Proof:* By the usual transfer arguments, it suffices to show that when is an ultra approximate group, are non-standard subsets of , and , there exists a nonstandard subset of with such that

for all (using the normalised convolution ). But by (3), is the pullback via of a continuous compactly supported function, so (5) holds for in for some neighbourhood of the identity, and thus by (2) it also holds for all in some nonstandard of positive Loeb measure. The claim follows.

Now we give a proof of Roth’s theorem (in the averaged form of Varnavides), at least for groups of odd order.

Proposition 3 (Roth’s theorem)Let , let be a finite group of odd order, and let be such that . Then there are pairs such that .

*Proof:* By the usual transfer arguments, it suffices to show that when is a nonstandard finite group of odd order, and is a nonstandard subset of with , then there are pairs such that ; equivalently, we need to show that

where . As has positive measure, is not identically zero. By a version of the Lebesgue differentiation theorem, we can then find a point in the Kronecker factor group such that has positive density on every precompact neighbourhood of , and is bounded away from zero on a subset of a symmetric open precompact neighbourhood of of density greater than . From this and (3) we see that is bounded away from zero on almost all of for some neighbourhood of . Also, as has odd order, the map is a measure-preserving map on , it must also be so on , and so we conclude that has positive measure in , and (6) follows.

Finally, we give a more advanced application of additive limits, namely reproving a lemma of Eberhard, Green, and Manners.

Proposition 4For every there is such that if is such that , then there is an arithmetic progression such that and .

*Proof:* Here, we perform the transfer step more explicitly, as it is slightly trickier here. Suppose for contradiction that the claim failed, then there exists an and a sequence with

but such that there is no arithmetic progression with such that . Note that must go to infinity (otherwise one could take to consist of a single element of , which must be non-empty from (7). Taking ultraproducts, we arrive at a nonstandard subset of for some unbounded natural number , such that

and such that there is no nonstandard arithmetic progression with and (say). Here is the Loeb measure associated with the approximate group and the group that it generates. By inspection of the proof of Theorem 1, the Kronecker factor of this group can be taken to be for some compact group , with projection given by for some measurable map , and Haar measure given by the product of Lebesgue measure on and the Haar probability measure on . If we let , then is supported in and takes values in , and from (3) and (8) we see that the set

is such that

We can view as a measurable function , defined by , and similarly the indicator function can be viewed as a measurable function defined by . Being measurable, may be approximated in by piecewise constant functions. One can then adapt the proof of the Lebesgue differentiation theorem to show that almost all are a Lebesgue point of , in the sense that

Similarly, almost all is a Lebesgue point of in the sense that

From this, we see that for almost all , we have the inclusion

up to null sets in , where the convolution is now with respect to the Haar probability measure . On the other hand, from Fubini’s theorem we have

and

Also is supported on . Thus by the pigeonhole principle, we may find an such that

and such that (9) holds up to null sets, and such that is a Lebesgue point for . If we fix this and now set and , we thus have

At this point it is convenient to split the compact abelian group as

where is the connected component of the identity, thus is a totally disconnected group. Let be the pushforward of to via the projection map , thus is a measurable function of total integral . We claim that

To see this, suppose for contradiction that

We may disintegrate

where is a measurable map from to finite measures on , such that for almost every , is supported on and has total mass . For almost every and , we then have that

is supported in . By Fubini’s theorem we have

where is the Haar probability measure on . From (10), we conclude that for almost every , there is a positive measure set of such that

for a positive measure set of (in particular ), and that is supported in . On the other hand, and are supported on sets of measure at least and . Applying Kemperman’s theorem (see this previous post) , we conclude that

for almost every with , and for a positive measure set of . But this leads to a contradiction if we take to be within of the essential supremum of . This proves (11).

As is totally disconnected, we can express the origin as the intersection of open subgroups. From this, (11), and a Lebesgue differentiation argument, we may find a coset of an open subgroup of such that

Letting be a pullback of to , we thus have

Since is a Lebesgue point for , we may thus find a neighbourhood of in such that

or equivalently

To finish the proof of the claim, it then suffices to show that differs from a nonstandard arithmetic progression by a set of arbitrarily small Loeb measure.

Consider the quotient homomorphism formed by first using to project to , then projecting to , then to . This is a Loeb measurable map, and thus the pointwise limit (up to null sets) by a nonstandard function. But observe that for , one has if and only if for almost every . In particuar, if is a nonstandard function which is sufficiently close to , then if and only if is the most common value of for . Using this, one can find a representative of that is precisely a nonstandard function on (say). Thus is now a nonstandard map from to the standard finite group , and from construction one can check that for all (and not merely almost all) . From this it is easy to see that is periodic with some bounded period, and that the level sets of are infinite nonstandard arithmetic progressions of bounded spacing. The claim then follows.

Filed under: expository, math.CO, math.GR, math.LO Tagged: additive combinatorics, locally compact abelian groups, Loeb measure, nonstandard analysis ]]>

Theorem 1 (Cayley’s theorem)Let be a group of some finite order . Then is isomorphic to a subgroup of the symmetric group on elements . Furthermore, this subgroup is simply transitive: given two elements of , there is precisely one element of such that .

One can therefore think of as a sort of “universal” group that contains (up to isomorphism) all the possible groups of order .

*Proof:* The group acts on itself by multiplication on the left, thus each element may be identified with a permutation on given by the map . This can be easily verified to identify with a simply transitive permutation group on . The claim then follows by arbitrarily identifying with .

More explicitly, the permutation group arises by arbitrarily enumerating as and then associating to each group element the permutation defined by the formula

The simply transitive group given by Cayley’s theorem is not unique, due to the arbitrary choice of identification of with , but is unique up to conjugation by an element of . On the other hand, it is easy to see that every simply transitive subgroup of is of order , and that two such groups are isomorphic if and only if they are conjugate by an element of . Thus Cayley’s theorem in fact identifies the moduli space of groups of order (up to isomorphism) with the simply transitive subgroups of (up to conjugacy by elements of ).

One can generalise Cayley’s theorem to groups of infinite order without much difficulty. But in this post, I would like to note an (easy) generalisation of Cayley’s theorem in a different direction, in which the group is no longer assumed to be of order , but rather to have an index subgroup that is isomorphic to a fixed group . The generalisation is:

Theorem 2 (Cayley’s theorem for -sets)Let be a group, and let be a group that contains an index subgroup isomorphic to . Then is isomorphic to a subgroup of the semidirect product , defined explicitly as the set of tuples with productand inverse

(This group is a wreath product of with , and is sometimes denoted , or more precisely .) Furthermore, is simply transitive in the following sense: given any two elements of and , there is precisely one in such that and .

Of course, Theorem 1 is the special case of Theorem 2 when is trivial. This theorem allows one to view as a “universal” group for modeling all groups containing a copy of as an index subgroup, in exactly the same way that is a universal group for modeling groups of order . This observation is not at all deep, but I had not seen it before, so I thought I would record it here. (EDIT: as pointed out in comments, this is a slight variant of the universal embedding theorem of Krasner and Kaloujnine, which covers the case when is normal, in which case one can embed into the wreath product , which is a subgroup of .)

*Proof:* The basic idea here is to replace the category of sets in Theorem 1 by the category of -sets, by which we mean sets with a right-action of the group . A morphism between two -sets is a function which respects the right action of , thus for all and .

Observe that if contains a copy of as a subgroup, then one can view as an -set, using the right-action of (which we identify with the indicated subgroup of ). The left action of on itself commutes with the right-action of , and so we can represent by -set automorphisms on the -set .

As has index in , we see that is (non-canonically) isomorphic (as an -set) to the -set with the obvious right action of : . It is easy to see that the group of -set automorphisms of can be identified with , with the latter group acting on the former -set by the rule

(it is routine to verify that this is indeed an action of by -set automorphisms. It is then a routine matter to verify the claims (the simple transitivity of follows from the simple transitivity of the action of on itself).

More explicitly, the group arises by arbitrarily enumerating the left-cosets of in as and then associating to each group element the element , where the permutation and the elements are defined by the formula

By noting that is an index normal subgroup of , we recover the classical result of Poincaré that any group that contains as an index subgroup, contains a normal subgroup of index dividing that is contained in . (Quotienting out the right-action, we recover also the classical *proof* of this result, as the action of on itself then collapses to the action of on the quotient space , the stabiliser of which is .)

Exercise 1Show that a simply transitive subgroup of contains a copy of as an index subgroup; in particular, there is a canonical embedding of into , and can be viewed as an -set.

Exercise 2Show that any two simply transitive subgroups of are isomorphic simultaneously as groups and as -sets (that is, there is a bijection that is simultaneously a group isomorphism and an -set isomorphism) if and only if they are conjugate by an element of .

[UPDATE: Exercises corrected; thanks to Keith Conrad for some additional corrections and comments.]

Filed under: expository, math.GR Tagged: Cayley's theorem ]]>

This post will also serve as the latest (and probably last) of the Polymath8 threads (rolling over this previous post), to wrap up any remaining discussion about any aspect of this project.

Filed under: polymath Tagged: polymath8 ]]>

Regardless of what commutative ring is in used here, we observe that Dirichlet convolution is commutative, associative, and bilinear over .

An important class of arithmetic functions in analytic number theory are the multiplicative functions, that is to say the arithmetic functions such that and

for all coprime . A subclass of these functions are the completely multiplicative functions, in which the restriction that be coprime is dropped. Basic examples of completely multiplicative functions (in the classical setting ) include

- the Kronecker delta , defined by setting for and otherwise;
- the constant function and the linear function (which by abuse of notation we denote by );
- more generally monomials for any fixed complex number (in particular, the “Archimedean characters” for any fixed ), which by abuse of notation we denote by ;
- Dirichlet characters ;
- the Liouville function ;
- the indicator function of the -smooth numbers (numbers whose prime factors are all at most ), for some given ; and
- the indicator function of the -rough numbers (numbers whose prime factors are all greater than ), for some given .

Examples of multiplicative functions that are not completely multiplicative include

- the Möbius function ;
- the divisor function (also referred to as );
- more generally, the higher order divisor functions for ;
- the Euler totient function ;
- the number of roots of a given polynomial defined over ;
- more generally, the point counting function of a given algebraic variety defined over (closely tied to the Hasse-Weil zeta function of );
- the function that counts the number of representations of as the sum of two squares;
- more generally, the function that maps a natural number to the number of ideals in a given number field of absolute norm (closely tied to the Dedekind zeta function of ).

These multiplicative functions interact well with the multiplication and convolution operations: if are multiplicative, then so are and , and if is completely multiplicative, then we also have

Finally, the product of completely multiplicative functions is again completely multiplicative. On the other hand, the sum of two multiplicative functions will never be multiplicative (just look at what happens at ), and the convolution of two completely multiplicative functions will usually just be multiplicative rather than completley multiplicative.

The specific multiplicative functions listed above are also related to each other by various important identities, for instance

where is an arbitrary arithmetic function.

On the other hand, analytic number theory also is very interested in certain arithmetic functions that are *not* exactly multiplicative (and certainly not completely multiplicative). One particularly important such function is the von Mangoldt function . This function is certainly not multiplicative, but is clearly closely related to such functions via such identities as and , where is the natural logarithm function. The purpose of this post is to point out that functions such as the von Mangoldt function lie in a class closely related to multiplicative functions, which I will call the *derived multiplicative functions*. More precisely:

Definition 1Aderived multiplicative functionis an arithmetic function that can be expressed as the formal derivativeat the origin of a family of multiplicative functions parameterised by a formal parameter . Equivalently, is a derived multiplicative function if it is the coefficient of a multiplicative function in the extension of by a nilpotent infinitesimal ; in other words, there exists an arithmetic function such that the arithmetic function is multiplicative, or equivalently that is multiplicative and one has the Leibniz rule

More generally, for any , a

-derived multiplicative functionis an arithmetic function that can be expressed as the formal derivativeat the origin of a family of multiplicative functions parameterised by formal parameters . Equivalently, is the coefficient of a multiplicative function in the extension of by nilpotent infinitesimals .

We define the notion of a -derived completely multiplicative function similarly by replacing “multiplicative” with “completely multiplicative” in the above discussion.

There are Leibniz rules similar to (2) but they are harder to state; for instance, a doubly derived multiplicative function comes with singly derived multiplicative functions and a multiplicative function such that

for all coprime .

One can then check that the von Mangoldt function is a derived multiplicative function, because is multiplicative in the ring with one infinitesimal . Similarly, the logarithm function is derived completely multiplicative because is completely multiplicative in . More generally, any additive function is derived multiplicative because it is the top order coefficient of .

Remark 1One can also phrase these concepts in terms of the formal Dirichlet series associated to an arithmetic function . A function is multiplicative if admits a (formal) Euler product; is derived multiplicative if is the (formal) first derivative of an Euler product with respect to some parameter (not necessarily , although this is certainly an option); and so forth.

Using the definition of a -derived multiplicative function as the top order coefficient of a multiplicative function of a ring with infinitesimals, it is easy to see that the product or convolution of a -derived multiplicative function and a -derived multiplicative function is necessarily a -derived multiplicative function (again taking values in ). Thus, for instance, the higher-order von Mangoldt functions are -derived multiplicative functions, because is a -derived completely multiplicative function. More explicitly, is the top order coeffiicent of the completely multiplicative function , and is the top order coefficient of the multiplicative function , with both functions taking values in the ring of complex numbers with infinitesimals attached.

It then turns out that most (if not all) of the basic identities used by analytic number theorists concerning derived multiplicative functions, can in fact be viewed as coefficients of identities involving purely multiplicative functions, with the latter identities being provable primarily from multiplicative identities, such as (1). This phenomenon is analogous to the one in linear algebra discussed in this previous blog post, in which many of the trace identities used there are derivatives of determinant identities. For instance, the Leibniz rule

for any arithmetic functions can be viewed as the top order term in

in the ring with one infinitesimal , and then we see that the Leibniz rule is a special case (or a derivative) of (1), since is completely multiplicative. Similarly, the formulae

are top order terms of

and the variant formula is the top order term of

which can then be deduced from the previous identities by noting that the completely multiplicative function inverts multiplicatively, and also noting that annihilates . The Selberg symmetry formula

which plays a key role in the Erdös-Selberg elementary proof of the prime number theorem (as discussed in this previous blog post), is the top order term of the identity

involving the multiplicative functions , , , with two infinitesimals , and this identity can be proven while staying purely within the realm of multiplicative functions, by using the identities

and (1). Similarly for higher identities such as

which arise from expanding out using (1) and the above identities; we leave this as an exercise to the interested reader.

An analogous phenomenon arises for identities that are not purely multiplicative in nature due to the presence of truncations, such as the Vaughan identity

for any , where is the restriction of a multiplicative function to the natural numbers greater than , and similarly for , , . In this particular case, (4) is the top order coefficient of the identity

which can be easily derived from the identities and . Similarly for the Heath-Brown identity

valid for natural numbers up to , where and are arbitrary parameters and denotes the -fold convolution of , and discussed in this previous blog post; this is the top order coefficient of

and arises by first observing that

vanishes up to , and then expanding the left-hand side using the binomial formula and the identity .

One consequence of this phenomenon is that identities involving derived multiplicative functions tend to have a dimensional consistency property: all terms in the identity have the same order of derivation in them. For instance, all the terms in the Selberg symmetry formula (3) are doubly derived functions, all the terms in the Vaughan identity (4) or the Heath-Brown identity (5) are singly derived functions, and so forth. One can then use dimensional analysis to help ensure that one has written down a key identity involving such functions correctly, much as is done in physics.

In addition to the dimensional analysis arising from the order of derivation, there is another dimensional analysis coming from the value of multiplicative functions at primes (which is more or less equivalent to the order of pole of the Dirichlet series at ). Let us say that a multiplicative function has a *pole of order * if one has on the average for primes , where we will be a bit vague as to what “on the average” means as it usually does not matter in applications. Thus for instance, or has a pole of order (a simple pole), or has a pole of order (i.e. neither a zero or a pole), Dirichlet characters also have a pole of order (although this is slightly nontrivial, requiring Dirichlet’s theorem), has a pole of order (a simple zero), has a pole of order , and so forth. Note that the convolution of a multiplicative function with a pole of order with a multiplicative function with a pole of order will be a multiplicative function with a pole of order . If there is no oscillation in the primes (e.g. if for *all* primes , rather than on the average), it is also true that the product of a multiplicative function with a pole of order with a multiplicative function with a pole of order will be a multiplicative function with a pole of order . The situation is significantly different though in the presence of oscillation; for instance, if is a quadratic character then has a pole of order even though has a pole of order .

A -derived multiplicative function will then be said to have an *underived pole of order * if it is the top order coefficient of a multiplicative function with a pole of order ; in terms of Dirichlet series, this roughly means that the Dirichlet series has a pole of order at . For instance, the singly derived multiplicative function has an underived pole of order , because it is the top order coefficient of , which has a pole of order ; similarly has an underived pole of order , being the top order coefficient of . More generally, and have underived poles of order and respectively for any .

By taking top order coefficients, we then see that the convolution of a -derived multiplicative function with underived pole of order and a -derived multiplicative function with underived pole of order is a -derived multiplicative function with underived pole of order . If there is no oscillation in the primes, the product of these functions will similarly have an underived pole of order , for instance has an underived pole of order . We then have the dimensional consistency property that in any of the standard identities involving derived multiplicative functions, all terms not only have the same derived order, but also the same underived pole order. For instance, in (3), (4), (5) all terms have underived pole order (with any Mobius function terms being counterbalanced by a matching term of or ). This gives a second way to use dimensional analysis as a consistency check. For instance, any identity that involves a linear combination of and is suspect because the underived pole orders do not match (being and respectively), even though the derived orders match (both are ).

One caveat, though: this latter dimensional consistency breaks down for identities that involve infinitely many terms, such as Linnik’s identity

In this case, one can still rewrite things in terms of multiplicative functions as

so the former dimensional consistency is still maintained.

I thank Andrew Granville, Kannan Soundararajan, and Emmanuel Kowalski for helpful conversations on these topics.

Filed under: expository, math.AC, math.NT Tagged: Dirichlet convolution, Heath-Brown identity, infinitesimals, multiplicative number theory, Vaughan identity ]]>

About a decade ago, Ben Green and I showed that the primes contained arbitrarily long arithmetic progressions: given any , one could find a progression with consisting entirely of primes. In fact we showed the same statement was true if the primes were replaced by any subset of the primes of positive relative density.

A little while later, Tamar Ziegler and I obtained the following generalisation: given any and any polynomials with , one could find a “polynomial progression” with consisting entirely of primes. Furthermore, we could make this progression somewhat “narrow” by taking (where denotes a quantity that goes to zero as goes to infinity). Again, the same statement also applies if the primes were replaced by a subset of positive relative density. My previous result with Ben corresponds to the linear case .

In this paper we were able to make the progressions a bit narrower still: given any and any polynomials with , one could find a “polynomial progression” with consisting entirely of primes, and such that , where depends only on and (in fact it depends only on and the degrees of ). The result is still true if the primes are replaced by a subset of positive density , but unfortunately in our arguments we must then let depend on . However, in the linear case , we were able to make independent of (although it is still somewhat large, of the order of ).

The polylogarithmic factor is somewhat necessary: using an upper bound sieve, one can easily construct a subset of the primes of density, say, , whose arithmetic progressions of length all obey the lower bound . On the other hand, the prime tuples conjecture predicts that if one works with the actual primes rather than dense subsets of the primes, then one should have infinitely many length arithmetic progressions of bounded width for any fixed . The case of this is precisely the celebrated theorem of Yitang Zhang that was the focus of the recently concluded Polymath8 project here. The higher case is conjecturally true, but appears to be out of reach of known methods. (Using the multidimensional Selberg sieve of Maynard, one can get primes inside an interval of length , but this is such a sparse set of primes that one would not expect to find even a progression of length three within such an interval.)

The argument in the previous paper was unable to obtain a polylogarithmic bound on the width of the progressions, due to the reliance on a certain technical “correlation condition” on a certain Selberg sieve weight . This correlation condition required one to control arbitrarily long correlations of , which was not compatible with a bounded value of (particularly if one wanted to keep independent of ).

However, thanks to recent advances in this area by Conlon, Fox, and Zhao (who introduced a very nice “densification” technique), it is now possible (in principle, at least) to delete this correlation condition from the arguments. Conlon-Fox-Zhao did this for my original theorem with Ben; and in the current paper we apply the densification method to our previous argument to similarly remove the correlation condition. This method does not fully eliminate the need to control arbitrarily long correlations, but allows most of the factors in such a long correlation to be *bounded*, rather than merely controlled by an unbounded weight such as . This turns out to be significantly easier to control, although in the non-linear case we still unfortunately had to make large compared to due to a certain “clearing denominators” step arising from the complicated nature of the Gowers-type uniformity norms that we were using to control polynomial averages. We believe though that this an artefact of our method, and one should be able to prove our theorem with an that is uniform in .

Here is a simple instance of the densification trick in action. Suppose that one wishes to establish an estimate of the form

for some real-valued functions which are bounded in magnitude by a weight function , but which are not expected to be bounded; this average will naturally arise when trying to locate the pattern in a set such as the primes. Here I will be vague as to exactly what range the parameters are being averaged over. Suppose that the factor (say) has enough uniformity that one can already show a smallness bound

whenever are bounded functions. (One should think of as being like the indicator functions of “dense” sets, in contrast to which are like the normalised indicator functions of “sparse” sets). The bound (2) cannot be directly applied to control (1) because of the unbounded (or “sparse”) nature of and . However one can “densify” and as follows. Since is bounded in magnitude by , we can bound the left-hand side of (1) as

The weight function will be normalised so that , so by the Cauchy-Schwarz inequality it suffices to show that

The left-hand side expands as

Now, it turns out that after an enormous (but finite) number of applications of the Cauchy-Schwarz inequality to steadily eliminate the factors, as well as a certain “polynomial forms condition” hypothesis on , one can show that

(Because of the polynomial shifts, this requires a method known as “PET induction”, but let me skip over this point here.) In view of this estimate, we now just need to show that

Now we can reverse the previous steps. First, we collapse back to

One can bound by , which can be shown to be “bounded on average” in a suitable sense (e.g. bounded norm) via the aforementioned polynomial forms condition. Because of this and the Hölder inequality, the above estimate is equivalent to

By setting to be the signum of , this is equivalent to

This is halfway between (1) and (2); the sparsely supported function has been replaced by its “densification” , but we have not yet densified to . However, one can shift by and repeat the above arguments to achieve a similar densificiation of , at which point one has reduced (1) to (2).

Filed under: math.NT, paper Tagged: densification, Tamar Ziegler ]]>

Finding lower bounds on is more or less equivalent to locating long strings of consecutive composite numbers that are not too large compared to the length of the string. A classic (and quite well known) construction here starts with the observation that for any natural number , the consecutive numbers are all composite, because each , is divisible by some prime , while being strictly larger than that prime . From this and Stirling’s formula, it is not difficult to obtain the bound

A more efficient bound comes from the prime number theorem: there are only primes up to , so just from the pigeonhole principle one can locate a string of consecutive composite numbers up to of length at least , thus

where we use or as shorthand for or .

What about upper bounds? The *Cramér random model* predicts that the primes up to are distributed like a random subset of density . Using this model, Cramér arrived at the conjecture

In fact, if one makes the extremely optimistic assumption that the random model perfectly describes the behaviour of the primes, one would arrive at the even more precise prediction

However, it is no longer widely believed that this optimistic version of the conjecture is true, due to some additional irregularities in the primes coming from the basic fact that large primes cannot be divisible by very small primes. Using the Maier matrix method to capture some of this irregularity, Granville was led to the conjecture that

(note that is slightly larger than ). For comparison, the known upper bounds on are quite weak; unconditionally one has by the work of Baker, Harman, and Pintz, and even on the Riemann hypothesis one only gets down to , as shown by Cramér (a slight improvement is also possible if one additionally assumes the pair correlation conjecture; see this article of Heath-Brown and the references therein).

This conjecture remains out of reach of current methods. In 1931, Westzynthius managed to improve the bound (2) slightly to

which Erdös in 1935 improved to

and Rankin in 1938 improved slightly further to

with . Remarkably, this rather strange bound then proved extremely difficult to advance further on; until recently, the only improvements were to the constant , which was raised to in 1963 by Schönhage, to in 1963 by Rankin, to by Maier and Pomerance, and finally to in 1997 by Pintz.

Erdös listed the problem of making arbitrarily large one of his favourite open problems, even offering (“somewhat rashly”, in his words) a cash prize for the solution. Our main result answers this question in the affirmative:

Theorem 1The bound (3) holds for arbitrarily large .

In principle, we thus have a bound of the form

for some that grows to infinity. Unfortunately, due to various sources of ineffectivity in our methods, we cannot provide any explicit rate of growth on at all.

We decided to announce this result the old-fashioned way, as part of a research lecture; more precisely, Ben Green announced the result in his ICM lecture this Tuesday. (The ICM staff have very efficiently put up video of his talks (and most of the other plenary and prize talks) online; Ben’s talk is here, with the announcement beginning at about 0:48. Note a slight typo in his slides, in that the exponent of in the denominator is instead of .) Ben’s lecture slides may be found here.

By coincidence, an independent proof of this theorem has also been obtained very recently by James Maynard.

I discuss our proof method below the fold.

** — 1. Sketch of proof — **

Our method is a modification of Rankin’s method, combined with some work of myself and Ben Green (and of Tamar Ziegler) on counting various linear patterns in the primes. To explain this, let us first go back to Rankin’s argument, presented in a fashion that allows for comparison with our own methods. Let’s first go back to the easy bound (1) that came from using the consecutive string of composite numbers. This bound was inferior to the prime number theorem bound (2), however this can be easily remedied by replacing with the somewhat smaller primorial , defined as the product of the primes up to and including . It is still easy to see that are all composite, with each divisible by some prime while being larger than that prime. On the other hand, the prime number theorem tells us that . From this, one can recover an alternate proof of (2) (perhaps not so surprising, since the prime number theorem is a key ingredient in both proofs).

This gives hope that further modification of this construction can be used to go beyond (2). If one looks carefully at the above proof, we see that the key fact used here is that the discrete interval of integers is *completely sieved out* by the residue classes for primes , in the sense that each element in this interval is contained in at least one of these residue classes. More generally (and shifting the interval by for a more convenient normalisation), suppose we can find an interval which is completely sieved out by one residue class for each , for some and . Then the string of consecutive numbers will be composite, whenever is an integer larger than or equal to with for each prime , since each of the will be divisible by some prime while being larger than that prime. From the Chinese remainder theorem, one can find such an that is of size at most . From this and the prime number theorem, one can obtain lower bounds on if one can get lower bounds on in terms of . In particular, if for any large one can completely sieve out with a residue class for each , and

then one can establish the bound (3). (The largest one can take for a given is known as the Jacobsthal function of the primorial .) So the task is basically to find a smarter set of congruence classes than just the zero congruence classes that can sieve out a larger interval than . (Unfortunately, this approach by itself is unlikely to reach the Cramér conjecture; it was shown by Iwaniec using the large sieve that is necessarily of size (which somewhat coincidentally matches the Cramér bound), but Maier and Pomerance conjecture that in fact one must have , which would mean that the limit of this method would be to establish a bound of the form .)

So, how can one do better than just using the “Eratosthenes” sieve ? We will divide the sieving into different stages, depending on the size of . It turns out that a reasonably optimal division of primes up to will be into the following four classes:

- Stage 1 primes: primes that are either tiny (less than ) or medium size (between and ), where is a parameter to be chosen later.
- Stage 2 primes: primes that are small (between and ).
- Stage 3 primes: Primes that are very large (between and ).
- Stage 4 primes: Primes that are fairly large (between and ).

We will take an interval , where is given by (4), and sieve out first by Stage 1 primes, then Stage 2 primes, then Stage 3 primes, then Stage 4 primes, until none of the elements of are left.

Let’s first discuss the final sieving step, which is rather trivial. Suppose that our sieving by the first three sieving stages is so efficient that the number of surviving elements of is less than or equal to the number of Stage 4 primes (by the prime number theorem, this will for instance be the case for sufficiently large if there are fewer than survivors). Then one can finish off the remaining survivors simply by using each of the Stage 4 primes to remove one of the surviving integers in by an appropriate choice of residue class . So we can recast our problem as an approximate sieving problem rather than a perfect sieving problem; we now only need to eliminate *most* of the elements of rather than *all* of them, at the cost of only using primes from the Stages 1-3, rather than 1-4. Note though that for given by (4), the Stage 1-3 sieving has to be reasonably efficient, in that the proportion of survivors cannot be too much larger than (ignoring factors of etc.).

Next, we discuss the Stage 1 sieving process. Here, we will simply copy the classic construction and use the Eratosthenes sieve for these primes. The elements of that survive this process are those elements that are not divisible by any Stage 1 primes, that is to say they are only divisible by small (Stage 2) primes, or else contain at least one prime factor larger than , and no prime factors less than . In the latter case, the survivor has no choice but to be a prime in (since from (4) we have for large enough). In the former case, the survivor is a -smooth number – a number with no prime factors less than or equal to . How many such survivors are there? Here we can use a somewhat crude upper bound of Rankin:

Lemma 2Let be large quantities, and write . Suppose thatThen the number of -smooth numbers in is at most .

*Proof:* We use a Dirichlet series method commonly known as “Rankin’s trick”. Let be a quantity to be optimised in later, and abbreviate “-smooth” as “smooth”. Observe that if is a smooth number less than , then

Thus, the number of smooth numbers in is at most

where we have simply discarded the constraint . The point of doing this is that the above expression factors into a tractable Euler product

so that and . Then the above expression simplifies to

To compute the sum here, we first observe from Mertens’ theorem (discussed in this previous blog post) that

so we may bound the previous expression by

which we rewrite using (6) as

Next, we use the convexity inequality

for and , applied with and , to conclude that

Finally, from the prime number theorem we have . The bound follows.

Remark 1One can basically eliminate the factor here (at the cost of worsening the error slightly to ) by a more refined version of the Rankin trick, based on replacing the crude bound (5) by the more sophisticated inequalitywhere denotes the assertion that divides exactly times. (Thanks to Kevin Ford for pointing out this observation to me.) In fact, the number of -smooth numbers in is known to be asymptotically in the range , a result of de Bruijn.

In view of the error term permitted by the Stage 4 process, we would like to take as large as possible while still leaving only smooth numbers in . A somewhat efficient choice of here is

so that and , and then one can check that the above lemma does indeed show that there are smooth numbers in . (If we use the sharper bound in the remark, we can reduce the here to a , although this makes little difference to the final bound.) If we let denote all the primes in , the remaining task is then to sieve out all but of the primes in by using one congruence class from each of the Stage 2 and Stage 3 primes.

Note that is still quite large compared to the error that the Stage 4 primes can handle – it is of size about , whereas we need to get down to a bit less than . Still, this is some progress (the remaining sparsification needed is of the order of rather than ).

For the Stage 2 sieve, we will just use a random construction, choosing uniformly at random for each Stage 2 prime. This sieve is expected to sparsify the set of survivors by a factor

which by Mertens’ theorem is of size

In particular, if is given by (4), then all the strange logarithmic factors cancel out and

In particular, we expect to be cut down to a random set (which we have called in our paper) of size about . This would already finish the job for very small (e.g. ), and indeed Rankin’s original argument proceeds more or less along these lines. But now we want to take to be large.

Fortunately, we still have the Stage 3 primes to play with. But the number of Stage 3 primes is about , which is a bit smaller than the number of surviving primes , which is about . So to make this work, most of the Stage 3 congruence classes need to sieve out many primes from , rather than just one or two. (Rankin’s original argument is based on sieving out one prime per congruence class; the subsequent work of Maier-Pomerance and Pintz is basically based on sieving out two primes per congruence class.)

Here, one has to take some care because the set is already quite sparse inside (its density is about ). So a randomly chosen would in fact most likely catch none of the primes in at all. So we need to restrict attention to congruence classes which already catch a large number of primes in , so that even after the Stage 2 sieving one can hope to be left with many congruence classes that also catch a large number of primes in .

Here’s where my work with Ben came in. Suppose one has an arithmetic progression of length consisting entirely of primes in , and with a multiple of , then the congruence class is guaranteed to pick up at least primes in . My first theorem with Ben shows that no matter how large is, the set does indeed contain some arithmetic progressions of length . This result is not quite suitable for our applications here, because (a) we need the spacing to also be divisible by a Stage 3 prime (in our paper, we take for concreteness, although other choices are certainly possible), and (b) for technical reasons, it is insufficient to simply have a large number of arithmetic progressions of primes strewn around ; they have to be “evenly distributed” in some sense in order to be able to still cover most of after throwing out any progression that is partly or completely sieved out by the Stage 2 primes. Fortunately, though, these distributional results for linear equations in primes were established by a subsequent paper of Ben and myself, contingent on two conjectures (the Mobius-Nilsequences conjecture and the inverse conjecture for the Gowers norms) which we also proved (the latter with Tamar Ziegler) in some further papers. (Actually, strictly speaking our work does not quite cover the case needed here, because the progressions are a little “narrow”; we need progressions of primes in whose spacing is comparable to instead of , whereas our paper only considered the situation in which the spacing was comparable to the elements of the progression. It turns out though that the arguments can be modified (somewhat tediously) to extend to this case though.)

Filed under: math.NT, paper Tagged: arithmetic progressions, prime gaps ]]>

Mathematicians (and likely other academics!) with small children face some unique challenges when traveling to conferences and workshops. The goal of this post is to reflect on these, and to start a constructive discussion what institutions and event organizers could do to improve the experiences of such participants.

The first necessary step is to recognize that different families have different needs. While it is hard to completely address everybody’s needs, there are some general measures that have a good chance to help most of the people traveling with young children. In this post, I will mostly focus on nursing mothers with infants ( months old) because that is my personal experience. Many of the suggestions will apply to other cases such as non-nursing babies, children of single parents, children of couples of mathematicians who are interested in attending the same conference, etc..

The mother of a nursing infant that wishes to attend a conference has three options:

**Bring the infant and a relative/friend to help caring for the infant.**The main challenge in this case is to fund the trip expenses of the relative. This involves trip costs, lodging, and food. The family may need a hotel room with some special amenities such as crib, fridge, microwave, etc. Location is also important, with easy access to facilities such as a grocery store, pharmacy, etc. The mother will need to take regular breaks from the conference in order to nurse the baby (this could be as often as every three hours or so). Depending on personal preferences, she may need to nurse privately. It is convenient, thus, to make a private room available, located as close to the conference venue as possible. The relative may need to have a place to stay with the baby near the conference such as a playground or a room with toys, particularly if the hotel room is far.**Bring the infant and hire someone local (a nanny) to help caring for the infant**. The main challenges in this case are two: finding the caregiver and paying for such services. Finding a caregiver in a place where one does not live is hard, as it is difficult to conduct interviews or get references. There are agencies that can do this for a (quite expensive) fee: they will find a professional caregiver with background checks, CPR certification, many references, etc. It may be worth it, though, as professional caregivers tend to provide high-quality services and peace of mind is priceless for the mother mathematician attending a conference. As in the previous case, the mother may have particular needs regarding the hotel room, location, and all the other facilities mentioned for Option 1.**Travel without the infant and pump milk regularly.**This can be very challenging for the mother, the baby, and the person that stays behind taking care of the baby, but the costs of this arrangement are much lower than in Option 1 or 2 (I am ignoring the possibility that the family needs to hire help at home, which is necessary in some cases). A nursing mother away from her baby has no option but to pump her milk to prevent her from pain and serious health complications. This mother may have to pump milk very often. Pumping is less efficient than nursing, so she will be gone for longer in each break or she will have more breaks compared to a mother that travels with her baby. For pumping, people need a room which should ideally be private, with a sink, and located as close to the conference venue as possible. It is often impossible for these three conditions to be met at the same time, so different mothers give priority to different features. Some people pump milk in washrooms, to have easy access to water. Other people might prefer to pump in a more comfortable setting, such as an office, and go to the washroom to wash the breast pump accessories after. If the mother expects that the baby will drink breastmilk while she is away, then she will also have to pump milk in advance of her trip. This requires some careful planning.

Many pumping mothers try to store the pumped milk and bring it back home. In this case the mother needs a hotel room with a fridge which (ideally, but hard to find) has a freezer. In a perfect world there would also be a fridge in the place where she pumps/where the conference is held.

It is important to keep in mind that each option has its own set of challenges (even when expenses and facilities are all covered) and that different families may be restricted in their choice of options for a variety of reasons. It is therefore important that all these three options be facilitated.

As for the effect these choices have on the conference experience for the mother, Option 1 means that she has to balance her time between the conference and spending time with her relative/friend. This pressure disappears when we consider Option 2, so this option may lead to more participation in the conferences activities. In Option 3, the mother is in principle free to participate in all the conference activities, but the frequent breaks may limit the type of activity. A mother may choose different options depending on the nature of the conference.

I want to stress, for the three options, that having to make choices about what to miss in the conference is very hard. While talks are important, so are the opportunities to meet people and discuss mathematics that happen during breaks and social events. It is very difficult to balance all of this. This is particularly difficult for the pumping mother in Option 3: because she travels without her baby, she is not perceived to be a in special situation or in need of accommodation. However, this mother is probably choosing between going to the last lecture in the morning or having lunch alone, because if she goes to pump right after the last lecture, by the time she is back, everybody has left for lunch.

Here is the Hall of Fame for those organizations that are already supporting nursing mothers’ travels in mathematics:

- The Natural Sciences and Engineering Research Council of Canada (NSERC) (search for “child care”) allows to reimburse the costs of child care with Option 2 out of the mother’s grants. They will also reimburse the travel expenses of a relative with Option 1 up to the amount that would cost to hire a local caregiver.
- The ENFANT/ELEFANT conference (co-organized by Lillian Pierce and Damaris Schindler) provided a good model to follow regarding accommodation for parents with children during conferences that included funding for covering the travel costs of accompanying caretakers (the funding was provided by the Deutsche Forschungsgemeinschaft, and lactation rooms and play rooms near the conference venue (the facilities were provided by the Hausdorff Center for Mathematics).

Additional information (where to go with kids, etc) was provided on site by the organizers and was made available to all participants all the time, by means of a display board that was left standing during the whole week of the conference. - The American Institute of Mathematics (AIM) reimburses up to 500 dollars on childcare for visitors and they have some online resources that assist in finding childcare and nannies.

[UPDATED] Added a few more things to the Hall of Fame

- The Joint Mathematics Meetings have been providing onsite childcare in the last few years.
- The Institute for Applied Mathematics (IPAM) provides childcare resources and funding to workshop participants with small children (comment from juliawolf).
- The Simons Institute for the Theory of Computing at Berkeley has a separate lactation room as part of some female washrooms (comment from juliawolf).
- The London Mathematical Society (LMS) offers Childcare Supplementary Grants of up to £200 to all mathematicians based in the UK travelling to conferences and other research schools, meetings or visits (comments from Peter).

In closing, here is a (possibly incomplete) list of resources that institutes, funding agencies, and conferences could consider providing for nursing mother mathematicians:

- Funding (for cost associated to child care either professional or by an accompanying relative).
- List of childcare resources (nannies, nanny agencies, drop-in childcare centre, etc).
- Nursing rooms and playrooms near the conference venue. Nearby fridge.
- Breaks of at least 20 minutes every 2-3 hours.
- Information about transportation with infants. More specific, taxi and/or shuttle companies that provide infant car seats. Information regarding the law on infant seats in taxis and other public transportation.
- Accessibility for strollers.
- [UPDATED] A nearby playground location. (comment from Peter).

I also find it important that these resources be listed publicly in the institute/conference website. This serves a double purpose: first, it helps those in need of the resources to access them easily, and second, it contributes to make these accommodations normal, setting a good model for future events, and inspiring organizers of future events.

Finally, I am pretty sure that the options and solutions I described do not cover all cases. I would like to finish this note by inviting readers to make suggestions, share experiences, and/or pose questions about this topic.

Filed under: guest blog, non-technical, Uncategorized Tagged: Matilde Lalin ]]>

Subhash Khot is best known for his Unique Games Conjecture, a problem in complexity theory that is perhaps second in importance only to the problem for the purposes of demarcating the mysterious line between “easy” and “hard” problems (if one follows standard practice and uses “polynomial time” as the definition of “easy”). The problem can be viewed as an assertion that it is difficult to find exact solutions to certain standard theoretical computer science problems (such as -SAT); thanks to the NP-completeness phenomenon, it turns out that the precise problem posed here is not of critical importance, and -SAT may be substituted with one of the many other problems known to be NP-complete. The unique games conjecture is similarly an assertion about the difficulty of finding even *approximate* solutions to certain standard problems, in particular “unique games” problems in which one needs to colour the vertices of a graph in such a way that the colour of one vertex of an edge is determined uniquely (via a specified matching) by the colour of the other vertex. This is an easy problem to solve if one insists on exact solutions (in which 100% of the edges have a colouring compatible with the specified matching), but becomes extremely difficult if one permits approximate solutions, with no exact solution available. In analogy with the NP-completeness phenomenon, the threshold for approximate satisfiability of many other problems (such as the MAX-CUT problem) is closely connected with the truth of the unique games conjecture; remarkably, the truth of the unique games conjecture would imply asymptotically sharp thresholds for many of these problems. This has implications for many theoretical computer science constructions which rely on hardness of approximation, such as probabilistically checkable proofs. For a more detailed survey of the unique games conjecture and its implications, see this Bulletin article of Trevisan.

My colleague Stan Osher has worked in many areas of applied mathematics, ranging from image processing to modeling fluids for major animation studies such as Pixar or Dreamworks, but today I would like to talk about one of his contributions that is close to an area of my own expertise, namely compressed sensing. One of the basic reconstruction problem in compressed sensing is the basis pursuit problem of finding the vector in an affine space (where and are given, and is typically somewhat smaller than ) which minimises the -norm of the vector . This is a convex optimisation problem, and thus solvable in principle (it is a polynomial time problem, and thus “easy” in the above theoretical computer science sense). However, once and get moderately large (e.g. of the order of ), standard linear optimisation routines begin to become computationally expensive; also, it is difficult for off-the-shelf methods to exploit any additional structure (e.g. sparsity) in the measurement matrix . Much of the problem comes from the fact that the functional is only barely convex. One way to speed up the optimisation problem is to relax it by replacing the constraint with a convex penalty term , thus one is now trying to minimise the unconstrained functional

This functional is more convex, and is over a computationally simpler domain than the affine space , so is easier (though still not entirely trivial) to optimise over. However, the minimiser to this problem need not match the minimiser to the original problem, particularly if the (sub-)gradient of the original functional is large at , and if is not set to be small. (And setting *too* small will cause other difficulties with numerically solving the optimisation problem, due to the need to divide by very small denominators.) However, if one modifies the objective function by an additional linear term

then some simple convexity considerations reveal that the minimiser to this new problem *will* match the minimiser to the original problem, provided that is (or more precisely, lies in) the (sub-)gradient of at – even if is not small. But, one does not know in advance what the correct value of should be, because one does not know what the minimiser is.

With Yin, Goldfarb and Darbon, Osher introduced a Bregman iteration method in which one solves for and simultaneously; given an initial guess for both and , one first updates to the minimiser of the convex functional

and then updates to the natural value of the subgradient at , namely

(note upon taking the first variation of (1) that is indeed in the subgradient). This procedure converges remarkably quickly (both in theory and in practice) to the true minimiser even for non-small values of , and also has some ability to be parallelised, and has led to actual performance improvements of an order of magnitude or more in certain compressed sensing problems (such as reconstructing an MRI image).

Phillip Griffiths has made many contributions to complex, algebraic and differential geometry, and I am not qualified to describe most of these; my primary exposure to his work is through his text on algebraic geometry with Harris, but as excellent though that text is, it is not really representative of his research. But I thought I would mention one cute result of his related to the famous Nash embedding theorem. Suppose that one has a smooth -dimensional Riemannian manifold that one wants to embed locally into a Euclidean space . The Nash embedding theorem guarantees that one can do this if is large enough depending on , but what is the minimal value of one can get away with? Many years ago, my colleague Robert Greene showed that sufficed (a simplified proof was subsequently given by Gunther). However, this is not believed to be sharp; if one replaces “smooth” with “real analytic” then a standard Cauchy-Kovalevski argument shows that is possible, and no better. So this suggests that is the threshold for the smooth problem also, but this remains open in general. The cases is trivial, and the case is not too difficult (if the curvature is non-zero) as the codimension is one in this case, and the problem reduces to that of solving a Monge-Ampere equation. With Bryant and Yang, Griffiths settled the case, under a non-degeneracy condition on the Einstein tensor. This is quite a serious paper – over 100 pages combining differential geometry, PDE methods (e.g. Nash-Moser iteration), and even some harmonic analysis (e.g. they rely at one key juncture on an extension theorem of my advisor, Elias Stein). The main difficulty is that that the relevant PDE degenerates along a certain characteristic submanifold of the cotangent bundle, which then requires an extremely delicate analysis to handle.

Filed under: math.CO, math.DG, math.NA Tagged: Phillip Griffiths, Stan Osher, Subhash Khot ]]>