246B, Notes 1: Zeroes, poles, and factorisation of meromorphic functions

Previous set of notes: 246A Notes 5. Next set of notes: Notes 2.

— 1. Jensen’s formula —

Suppose is a non-zero rational function , then by the fundamental theorem of algebra one can write

for some non-zero constant , where ranges over the zeroes of (counting multiplicity) and ranges over the zeroes of (counting multiplicity), and assuming avoids the zeroes of . Taking absolute values and then logarithms, we arrive at the formula

as long as avoids the zeroes of both and . (In this set of notes we use for the natural logarithm when applied to a positive real number, and for the standard branch of the complex logarithm (which extends ); the multi-valued complex logarithm will only be used in passing.) Alternatively, taking logarithmic derivatives, we arrive at the closely related formula

again for avoiding the zeroes of both and . Thus we see that the zeroes and poles of a rational function describe the behaviour of that rational function, as well as close relatives of that function such as the log-magnitude and log-derivative . We have already seen these sorts of formulae arise in our treatment of the argument principle in 246A Notes 4.

Exercise 1 Let be a complex polynomial of degree .

• (i) (Gauss-Lucas theorem) Show that the complex roots of are contained in the closed convex hull of the complex roots of .
• (ii) (Laguerre separation theorem) If all the complex roots of are contained in a disk , and , then all the complex roots of are also contained in . (Hint: apply a suitable Möbius transformation to move to infinity, and then apply part (i) to a polynomial that emerges after applying this transformation.)

There are a number of useful ways to extend these formulae to more general meromorphic functions than rational functions. Firstly there is a very handy “local” variant of (1) known as Jensen’s formula:

Theorem 2 (Jensen’s formula) Let be a meromorphic function on an open neighbourhood of a disk , with all removable singularities removed. Then, if is neither a zero nor a pole of , we have

where and range over the zeroes and poles of respectively (counting multiplicity) in the disk .

One can view (3) as a truncated (or localised) variant of (1). Note also that the summands are always non-positive.

Proof: By perturbing slightly if necessary, we may assume that none of the zeroes or poles of (which form a discrete set) lie on the boundary circle . By translating and rescaling, we may then normalise and , thus our task is now to show that

We may remove the poles and zeroes inside the disk by the useful device of Blaschke products. Suppose for instance that has a zero inside the disk . Observe that the function

has magnitude on the unit circle , equals at the origin, has a simple zero at , but has no other zeroes or poles inside the disk. Thus Jensen’s formula (4) already holds if is replaced by . To prove (4) for , it thus suffices to prove it for , which effectively deletes a zero inside the disk from (and replaces it instead with its inversion ). Similarly we may remove all the poles inside the disk. As a meromorphic function only has finitely many poles and zeroes inside a compact set, we may thus reduce to the case when has no poles or zeroes on or inside the disk , at which point our goal is simply to show that

Since has no zeroes or poles inside the disk, it has a holomorphic logarithm (Exercise 46 of 246A Notes 4). In particular, is the real part of . The claim now follows by applying the mean value property (Exercise 17 of 246A Notes 3) to .

An important special case of Jensen’s formula arises when is holomorphic in a neighborhood of , in which case there are no contributions from poles and one simply has

This is quite a useful formula, mainly because the summands are non-negative; it can be viewed as a more precise assertion of the subharmonicity of (see Exercises 60(ix) and 61 of 246A Notes 5). Here are some quick applications of this formula:

Exercise 3 Use (6) to give another proof of Liouville’s theorem: a bounded holomorphic function on the entire complex plane is necessarily constant.

Exercise 4 Use Jensen’s formula to prove the fundamental theorem of algebra: a complex polynomial of degree has exactly complex zeroes (counting multiplicity), and can thus be factored as for some complex numbers with . (Note that the fundamental theorem was invoked previously in this section, but only for motivational purposes, so the proof here is non-circular.)

Exercise 5 (Shifted Jensen’s formula) Let be a meromorphic function on an open neighbourhood of a disk , with all removable singularities removed. Show that

for all in the open disk that are not zeroes or poles of , where and . (The function appearing in the integrand is sometimes known as the Poisson kernel, particularly if one normalises so that and .)

Exercise 6 (Bounded type)

• (i) If is a holomorphic function on that is not identically zero, show that .
• (ii) If is a meromorphic function on that is the ratio of two bounded holomorphic functions that are not identically zero, show that . (Functions of this form are said to be of bounded type and lie in the Nevanlinna class for the unit disk .)

Exercise 7 (Smoothed out Jensen formula) Let be a meromorphic function on an open set , and let be a smooth compactly supported function. Show that

where range over the zeroes and poles of (respectively) in the support of . Informally argue why this identity is consistent with Jensen’s formula.

When applied to entire functions , Jensen’s formula relates the order of growth of near infinity with the density of zeroes of . Here is a typical result:

Proposition 8 Let be an entire function, not identically zero, that obeys a growth bound for some and all . Then there exists a constant such that has at most zeroes (counting multiplicity) for any .

Entire functions that obey a growth bound of the form for every and (where depends on ) are said to be of order at most . The above theorem shows that for such functions that are not identically zero, the number of zeroes in a disk of radius does not grow much faster than . This is often a useful preliminary upper bound on the zeroes of entire functions, as the order of an entire function tends to be relatively easy to compute in practice.

Proof: First suppose that is non-zero. From (6) applied with and one has

Every zero in contribute at least to a summand on the right-hand side, while all other zeroes contribute a non-negative quantity, thus

where denotes the number of zeroes in . This gives the claim for . When , one can shift by a small amount to make non-zero at the origin (using the fact that zeroes of holomorphic functions not identically zero are isolated), modifying in the process, and then repeating the previous arguments.

Just as (3) and (7) give truncated variants of (1), we can create truncated versions of (2). The following crude truncation is adequate for many applications:

Theorem 9 (Truncated formula for log-derivative) Let be a holomorphic function on an open neighbourhood of a disk that is not identically zero on this disk. Suppose that one has a bound of the form for some and all on the circle . Let be constants. Then one has the approximate formula

for all in the disk other than zeroes of . Furthermore, the number of zeroes in the above sum is .

Proof: To abbreviate notation, we allow all implied constants in this proof to depend on .

We mimic the proof of Jensen’s formula. Firstly, we may translate and rescale so that and , so we have when , and our main task is to show that

for . Note that if then vanishes on the unit circle and hence (by the maximum principle) vanishes identically on the disk, a contradiction, so we may assume . From hypothesis we then have

on the unit circle, and so from Jensen’s formula (3) we see that

In particular we see that the number of zeroes with is , as claimed.

Suppose has a zero with . If we factor , where is the Blaschke product (5), then

Observe from Taylor expansion that the distance between and is , and hence for . Thus we see from (9) that we may use Blaschke products to remove all the zeroes in the annulus while only affecting the left-hand side of (8) by ; also, removing the Blaschke products does not affect on the unit circle, and only affects by thanks to (9). Thus we may assume without loss of generality that there are no zeroes in this annulus.

Similarly, given a zero with , we have , so using Blaschke products to remove all of these zeroes also only affects the left-hand side of (8) by (since the number of zeroes here is ), with also modified by at most . Thus we may assume in fact that has no zeroes whatsoever within the unit disk. We may then also normalise , then for all . By Jensen’s formula again, we have

and thus (by using the identity for any real )

On the other hand, from (7) we have

which implies from (10) that and its first derivatives are on the disk . But recall from the proof of Jensen’s formula that is the derivative of a logarithm of , whose real part is . By the Cauchy-Riemann equations for , we conclude that on the disk , as required.

Exercise 10

• (i) (Borel-Carathéodory theorem) If is analytic on an open neighborhood of a disk , show that

  

  (Hint: one can normalise , , , and . Now maps the unit disk to the half-plane . Use a Möbius transformation to map the half-plane to the unit disk and then use the Schwarz lemma.)
• (ii) Use (i) to give an alternate way to conclude the proof of Theorem 9.

A variant of the above argument allows one to make precise the heuristic that holomorphic functions locally look like polynomials:

Exercise 11 (Local Weierstrass factorisation) Let the notation and hypotheses be as in Theorem 9. Then show that

for all in the disk , where is a polynomial whose zeroes are precisely the zeroes of in (counting multiplicity) and is a holomorphic function on of magnitude and first derivative on this disk. Furthermore, show that the degree of is .

Exercise 12 (Preliminary Beurling factorisation) Let denote the space of bounded analytic functions on the unit disk; this is a normed vector space with norm

• (i) If is not identically zero, and denote the zeroes of in counting multiplicity, show that

  

  and

  

• (ii) Let the notation be as in (i). If we define the Blaschke product

  

  where is the order of vanishing of at zero, show that this product converges absolutely to a holomorphic function on , and that for all . (It may be easier to work with finite Blaschke products first to obtain this bound.)
• (iii) Continuing the notation from (i), establish a factorisation for some holomorphic function with for all .
• (iv) (Theorem of F. and M. Riesz, special case) If extends continuously to the boundary , show that the set has zero measure.

Remark 13 The factorisation (iii) can be refined further, with being the Poisson integral of some finite measure on the unit circle. Using the Lebesgue decomposition of this finite measure into absolutely continuous parts one ends up factorising functions into “outer functions” and “inner functions”, giving the Beurling factorisation of . There are also extensions to larger spaces than (which are to as is to ), known as Hardy spaces. We will not discuss this topic further here, but see for instance this text of Garnett for a treatment.

Exercise 14 (Littlewood’s lemma) Let be holomorphic on an open neighbourhood of a rectangle for some and , with non-vanishing on the boundary of the rectangle. Show that

where ranges over the zeroes of inside (counting multiplicity) and one uses a branch of which is continuous on the upper, lower, and right edges of . (This lemma is a popular tool to explore the zeroes of Dirichlet series such as the Riemann zeta function.)

Course announcement: 246B, complex analysis

Just a short announcement that next quarter I will be continuing the recently concluded 246A complex analysis class as 246B. Topics I plan to cover:

• Schwartz-Christoffel transformations and the uniformisation theorem (using the remainder of the 246A notes);
• Jensen’s formula and factorisation theorems (particularly Weierstrass and Hadamard); the Gamma function;
• Connections with the Fourier transform on the real line;
• Elliptic functions and their relatives;
• (if time permits) the Riemann zeta function and the prime number theorem.

Notes for the later material will appear on this blog in due course.

Sendov’s conjecture for sufficiently high degree polynomials

I’ve just uploaded to the arXiv my paper “Sendov’s conjecture for sufficiently high degree polynomials“. This paper is a contribution to an old conjecture of Sendov on the zeroes of polynomials:

Conjecture 1 (Sendov’s conjecture) Let be a polynomial of degree that has all zeroes in the closed unit disk . If is one of these zeroes, then has at least one zero in .

It is common in the literature on this problem to normalise to be monic, and to rotate the zero to be an element of the unit interval . As it turns out, the location of on this unit interval ends up playing an important role in the arguments.

Many cases of this conjecture are already known, for instance

• When (Brown-Xiang 1999);
• When (Gauss-Lucas theorem);
• When (Bojanov 2011);
• When for a fixed , and is sufficiently large depending on (Dégot 2014);
• When for a sufficiently large absolute constant (Chalebgwa 2020);
• When (Rubinstein 1968; Goodman-Rahman-Ratti 1969; Joyal 1969);
• When , where is sufficiently small depending on (Miller 1993; Vajaitu-Zaharescu 1993);
• When (Chijiwa 2011);
• When (Kasmalkar 2014).

In particular, in high degrees the only cases left uncovered by prior results are when is close (but not too close) to , or when is close (but not too close) to ; see Figure 1 of my paper.

Our main result covers the high degree case uniformly for all values of :

Theorem 2 There exists an absolute constant such that Sendov’s conjecture holds for all .

In principle, this reduces the verification of Sendov’s conjecture to a finite time computation, although our arguments use compactness methods and thus do not easily provide an explicit value of . I believe that the compactness arguments can be replaced with quantitative substitutes that provide an explicit , but the value of produced is likely to be extremely large (certainly much larger than ).

Because of the previous results (particularly those of Chalebgwa and Chijiwa), we will only need to establish the following two subcases of the above theorem:

Theorem 3 (Sendov’s conjecture near the origin) Under the additional hypothesis , Sendov’s conjecture holds for sufficiently large .

Theorem 4 (Sendov’s conjecture near the unit circle) Under the additional hypothesis for a fixed , Sendov’s conjecture holds for sufficiently large .

We approach these theorems using the “compactness and contradiction” strategy, assuming that there is a sequence of counterexamples whose degrees going to infinity, using various compactness theorems to extract various asymptotic objects in the limit , and somehow using these objects to derive a contradiction. There are many ways to effect such a strategy; we will use a formalism that I call “cheap nonstandard analysis” and which is common in the PDE literature, in which one repeatedly passes to subsequences as necessary whenever one invokes a compactness theorem to create a limit object. However, the particular choice of asymptotic formalism one selects is not of essential importance for the arguments.

I also found it useful to use the language of probability theory. Given a putative counterexample to Sendov’s conjecture, let be a zero of (chosen uniformly at random among the zeroes of , counting multiplicity), and let similarly be a uniformly random zero of . We introduce the logarithmic potentials

and the Stieltjes transforms

Standard calculations using the fundamental theorem of algebra yield the basic identities

and

and in particular the random variables are linked to each other by the identity

On the other hand, the hypotheses of Sendov’s conjecture (and the Gauss-Lucas theorem) place inside the unit disk . Applying Prokhorov’s theorem, and passing to a subsequence, one can then assume that the random variables converge in distribution to some limiting random variables (possibly defined on a different probability space than the original variables ), also living almost surely inside the unit disk. Standard potential theory then gives the convergence

and

at least in the local sense. Among other things, we then conclude from the identity (2) and some elementary inequalities that

for all . This turns out to have an appealing interpretation in terms of Brownian motion: if one takes two Brownian motions in the complex plane, one originating from and one originating from , then the location where these Brownian motions first exit the unit disk will have the same distribution. (In our paper we actually replace Brownian motion with the closely related formalism of balayage.) This turns out to connect the random variables , quite closely to each other. In particular, with this observation and some additional arguments involving both the unique continuation property for harmonic functions and Grace’s theorem (discussed in this previous post), with the latter drawn from the prior work of Dégot, we can get very good control on these distributions:

Theorem 5

• (i) If , then almost surely lie in the semicircle and have the same distribution.
• (ii) If , then is uniformly distributed on the circle , and is almost surely zero.

In case (i) (and strengthening the hypothesis to to control some technical contributions of “outlier” zeroes of ), we can use this information about and (4) to ensure that the normalised logarithmic derivative has a non-negative winding number in a certain small (but not too small) circle around the origin, which by the argument principle is inconsistent with the hypothesis that has a zero at and that has no zeroes near . This is how we establish Theorem 3.

Case (ii) turns out to be more delicate. This is because there are a number of “near-counterexamples” to Sendov’s conjecture that are compatible with the hypotheses and conclusion of case (ii). The simplest such example is , where the zeroes of are uniformly distributed amongst the roots of unity (including at ), and the zeroes of are all located at the origin. In my paper I also discuss a variant of this construction, in which has zeroes mostly near the origin, but also acquires a bounded number of zeroes at various locations inside the unit disk. Specifically, we take

where for some constants and

By a perturbative analysis to locate the zeroes of , one eventually would be able to arrive at a true counterexample to Sendov’s conjecture if these locations were in the open lune

and if one had the inequality

for all . However, if one takes the mean of this inequality in , one arrives at the inequality

which is incompatible with the hypotheses and . In order to extend this argument to more general polynomials , we require a stability analysis of the endpoint equation

where we now only assume the closed conditions and . The above discussion then places all the zeros on the arc

and if one also takes the second Fourier coefficient of (6) one also obtains the vanishing second moment

These two conditions are incompatible with each other (except in the degenerate case when all the vanish), because all the non-zero elements of the arc (7) have argument in , so in particular their square will have negative real part. It turns out that one can adapt this argument to the more general potential counterexamples to Sendov’s conjecture (in the form of Theorem 4). The starting point is to use (1), (4), and Theorem 5(ii) to obtain good control on , which one then integrates and exponentiates to get good control on , and then on a second integration one gets enough information about to pin down the location of its zeroes to high accuracy. The constraint that these zeroes lie inside the unit disk then gives an inequality resembling (5), and an adaptation of the above stability analysis is then enough to conclude. The arguments here are inspired by the previous arguments of Miller, which treated the case when was extremely close to via a similar perturbative analysis; the main novelty is to control the error terms not in terms of the magnitude of the largest zero of (which is difficult to manage when gets large), but rather by the variance of those zeroes, which ends up being a more tractable expression to keep track of.

Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle

Laura Cladek and I have just uploaded to the arXiv our paper “Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle“. This paper concerns a continuous version of the notion of additive energy. Given a finite measure on and a scale , define the energy at scale to be the quantity

where is the product measure on formed from four copies of the measure on . We will be interested in Cantor-type measures , supported on a compact set and obeying the Ahlfors-David regularity condition

for all balls and some constants , as well as the matching lower bound

when whenever . One should think of as a -dimensional fractal set, and as some vaguely self-similar measure on this set.

Note that once one fixes , the variable in (1) is constrained to a ball of radius , hence we obtain the trivial upper bound

If the set contains a lot of “additive structure”, one can expect this bound to be basically sharp; for instance, if is an integer, is a -dimensional unit disk, and is Lebesgue measure on this disk, one can verify that (where we allow implied constants to depend on . However we show that if the dimension is non-integer, then one obtains a gain:

Theorem 1 If is not an integer, and are as above, then

for some depending only on .

Informally, this asserts that Ahlfors-David regular fractal sets of non-integer dimension cannot behave as if they are approximately closed under addition. In fact the gain we obtain is quasipolynomial in the regularity constant :

(We also obtain a localised version in which the regularity condition is only required to hold at scales between and .) Such a result was previously obtained (with more explicit values of the implied constants) in the one-dimensional case by Dyatlov and Zahl; but in higher dimensions there does not appear to have been any results for this general class of sets and measures . In the paper of Dyatlov and Zahl it is noted that some dependence on is necessary; in particular, cannot be much better than . This reflects the fact that there are fractal sets that do behave reasonably well with respect to addition (basically because they are built out of long arithmetic progressions at many scales); however, such sets are not very Ahlfors-David regular. Among other things, this result readily implies a dimension expansion result

for any non-degenerate smooth map , including the sum map and (in one dimension) the product map , where the non-degeneracy condition required is that the gradients are invertible for every . We refer to the paper for the formal statement.

Our higher-dimensional argument shares many features in common with that of Dyatlov and Zahl, notably a reliance on the modern tools of additive combinatorics (and specifically the Bogulybov-Ruzsa lemma of Sanders). However, in one dimension we were also able to find a completely elementary argument, avoiding any particularly advanced additive combinatorics and instead primarily exploiting the order-theoretic properties of the real line, that gave a superior value of , namely

One of the main reasons for obtaining such improved energy bounds is that they imply a fractal uncertainty principle in some regimes. We focus attention on the model case of obtaining such an uncertainty principle for the semiclassical Fourier transform

where is a small parameter. If are as above, and denotes the -neighbourhood of , then from the Hausdorff-Young inequality one obtains the trivial bound

(There are also variants involving pairs of sets , but for simplicity we focus on the uncertainty principle for a single set .) The fractal uncertainty principle, when it applies, asserts that one can improve this to

for some ; informally, this asserts that a function and its Fourier transform cannot simultaneously be concentrated in the set when , and that a function cannot be concentrated on and have its Fourier transform be of maximum size on when . A modification of the disk example mentioned previously shows that such a fractal uncertainty principle cannot hold if is an integer. However, in one dimension, the fractal uncertainty principle is known to hold for all . The above-mentioned results of Dyatlov and Zahl were able to establish this for close to , and the remaining cases and were later established by Bourgain-Dyatlov and Dyatlov-Jin respectively. Such uncertainty principles have applications to hyperbolic dynamics, in particular in establishing spectral gaps for certain Selberg zeta functions.

It remains a largely open problem to establish a fractal uncertainty principle in higher dimensions. Our results allow one to establish such a principle when the dimension is close to , and is assumed to be odd (to make a non-integer). There is also work of Han and Schlag that obtains such a principle when one of the copies of is assumed to have a product structure. We hope to obtain further higher-dimensional fractal uncertainty principles in subsequent work.

We now sketch how our main theorem is proved. In both one dimension and higher dimensions, the main point is to get a preliminary improvement

over the trivial bound (2) for any small , provided is sufficiently small depending on ; one can then iterate this bound by a fairly standard “induction on scales” argument (which roughly speaking can be used to show that energies behave somewhat multiplicatively in the scale parameter ) to propagate the bound to a power gain at smaller scales. We found that a particularly clean way to run the induction on scales was via use of the Gowers uniformity norm , and particularly via a clean Fubini-type inequality

(ultimately proven using the Gowers-Cauchy-Schwarz inequality) that allows one to “decouple” coarse and fine scale aspects of the Gowers norms (and hence of additive energies).

It remains to obtain the preliminary improvement. In one dimension this is done by identifying some “left edges” of the set that supports : intervals that intersect , but such that a large interval just to the left of this interval is disjoint from . Here is a large constant and is a scale parameter. It is not difficult to show (using in particular the Archimedean nature of the real line) that if one has the Ahlfors-David regularity condition for some then left edges exist in abundance at every scale; for instance most points of would be expected to lie in quite a few of these left edges (much as most elements of, say, the ternary Cantor set would be expected to contain a lot of s in their base expansion). In particular, most pairs would be expected to lie in a pair of left edges of equal length. The key point is then that if lies in such a pair with , then there are relatively few pairs at distance from for which one has the relation , because will both tend to be to the right of respectively. This causes a decrement in the energy at scale , and by carefully combining all these energy decrements one can eventually cobble together the energy bound (3).

We were not able to make this argument work in higher dimension (though perhaps the cases and might not be completely out of reach from these methods). Instead we return to additive combinatorics methods. If the claim (3) failed, then by applying the Balog-Szemeredi-Gowers theorem we can show that the set has high correlation with an approximate group , and hence (by the aforementioned Bogulybov-Ruzsa type theorem of Sanders, which is the main source of the quasipolynomial bounds in our final exponent) will exhibit an approximate “symmetry” along some non-trivial arithmetic progression of some spacing length and some diameter . The -neighbourhood of will then resemble the union of parallel “cylinders” of dimensions . If we focus on a typical -ball of , the set now resembles a Cartesian product of an interval of length with a subset of a -dimensional hyperplane, which behaves approximately like an Ahlfors-David regular set of dimension (this already lets us conclude a contradiction if ). Note that if the original dimension was non-integer then this new dimension will also be non-integer. It is then possible to contradict the failure of (3) by appealing to a suitable induction hypothesis at one lower dimension.

Holomorphic images of disks

Consider a disk in the complex plane. If one applies an affine-linear map to this disk, one obtains

For maps that are merely holomorphic instead of affine-linear, one has some variants of this assertion, which I am recording here mostly for my own reference:

Theorem 1 (Holomorphic images of disks) Let be a disk in the complex plane, and be a holomorphic function with .

• (i) (Open mapping theorem or inverse function theorem) contains a disk for some . (In fact there is even a holomorphic right inverse of from to .)
• (ii) (Bloch theorem) contains a disk for some absolute constant and some . (In fact there is even a holomorphic right inverse of from to .)
• (iii) (Koebe quarter theorem) If is injective, then contains the disk .
• (iv) If is a polynomial of degree , then contains the disk .
• (v) If one has a bound of the form for all and some , then contains the disk for some absolute constant . (In fact there is holomorphic right inverse of from to .)

Parts (i), (ii), (iii) of this theorem are standard, as indicated by the given links. I found part (iv) as (a consequence of) Theorem 2 of this paper of Degot, who remarks that it “seems not already known in spite of its simplicity”; an equivalent form of this result also appears in Lemma 4 of this paper of Miller. The proof is simple:

Proof: (Proof of (iv)) Let , then we have a lower bound for the log-derivative of at :

(with the convention that the left-hand side is infinite when ). But by the fundamental theorem of algebra we have

where are the roots of the polynomial (counting multiplicity). By the pigeonhole principle, there must therefore exist a root of such that

and hence . Thus contains , and the claim follows.

The constant in (iv) is completely sharp: if and is non-zero then contains the disk

but avoids the origin, thus does not contain any disk of the form . This example also shows that despite parts (ii), (iii) of the theorem, one cannot hope for a general inclusion of the form

for an absolute constant .

Part (v) is implicit in the standard proof of Bloch’s theorem (part (ii)), and is easy to establish:

Proof: (Proof of (v)) From the Cauchy inequalities one has for , hence by Taylor’s theorem with remainder for . By Rouche’s theorem, this implies that the function has a unique zero in for any , if is a sufficiently small absolute constant. The claim follows.

Note that part (v) implies part (i). A standard point picking argument also lets one deduce part (ii) from part (v):

Proof: (Proof of (ii)) By shrinking slightly if necessary we may assume that extends analytically to the closure of the disk . Let be the constant in (v) with ; we will prove (iii) with replaced by . If we have for all then we are done by (v), so we may assume without loss of generality that there is such that . If for all then by (v) we have

and we are again done. Hence we may assume without loss of generality that there is such that . Iterating this procedure in the obvious fashion we either are done, or obtain a Cauchy sequence in such that goes to infinity as , which contradicts the analytic nature of (and hence continuous nature of ) on the closure of . This gives the claim.

Here is another classical result stated by Alexander (and then proven by Kakeya and by Szego, but also implied to a classical theorem of Grace and Heawood) that is broadly compatible with parts (iii), (iv) of the above theorem:

Proposition 2 Let be a disk in the complex plane, and be a polynomial of degree with for all . Then is injective on .

The radius is best possible, for the polynomial has non-vanishing on , but one has , and lie on the boundary of .

If one narrows slightly to then one can quickly prove this proposition as follows. Suppose for contradiction that there exist distinct with , thus if we let be the line segment contour from to then . However, by assumption we may factor where all the lie outside of . Elementary trigonometry then tells us that the argument of only varies by less than as traverses , hence the argument of only varies by less than . Thus takes values in an open half-plane avoiding the origin and so it is not possible for to vanish.

To recover the best constant of requires some effort. By taking contrapositives and applying an affine rescaling and some trigonometry, the proposition can be deduced from the following result, known variously as the Grace-Heawood theorem or the complex Rolle theorem.

Proposition 3 (Grace-Heawood theorem) Let be a polynomial of degree such that . Then contains a zero in the closure of .

This is in turn implied by a remarkable and powerful theorem of Grace (which we shall prove shortly). Given two polynomials of degree at most , define the apolar form by

Theorem 4 (Grace’s theorem) Let be a circle or line in , dividing into two open connected regions . Let be two polynomials of degree at most , with all the zeroes of lying in and all the zeroes of lying in . Then .

(Contrapositively: if , then the zeroes of cannot be separated from the zeroes of by a circle or line.)

Indeed, a brief calculation reveals the identity

where is the degree polynomial

The zeroes of are for , so the Grace-Heawood theorem follows by applying Grace’s theorem with equal to the boundary of .

The same method of proof gives the following nice consequence:

Theorem 5 (Perpendicular bisector theorem) Let be a polynomial such that for some distinct . Then the zeroes of cannot all lie on one side of the perpendicular bisector of . For instance, if , then the zeroes of cannot all lie in the halfplane or the halfplane .

I’d be interested in seeing a proof of this latter theorem that did not proceed via Grace’s theorem.

Now we give a proof of Grace’s theorem. The case can be established by direct computation, so suppose inductively that and that the claim has already been established for . Given the involvement of circles and lines it is natural to suspect that a Möbius transformation symmetry is involved. This is indeed the case and can be made precise as follows. Let denote the vector space of polynomials of degree at most , then the apolar form is a bilinear form . Each translation on the complex plane induces a corresponding map on , mapping each polynomial to its shift . We claim that the apolar form is invariant with respect to these translations:

Taking derivatives in , it suffices to establish the skew-adjointness relation

but this is clear from the alternating form of (1).

Next, we see that the inversion map also induces a corresponding map on , mapping each polynomial to its inversion . From (1) we see that this map also (projectively) preserves the apolar form:

More generally, the group of Möbius transformations on the Riemann sphere acts projectively on , with each Möbius transformation mapping each to , where is the unique (up to constants) rational function that maps this a map from to (its divisor is ). Since the Möbius transformations are generated by translations and inversion, we see that the action of Möbius transformations projectively preserves the apolar form; also, we see this action of on also moves the zeroes of each by (viewing polynomials of degree less than in as having zeroes at infinity). In particular, the hypotheses and conclusions of Grace’s theorem are preserved by this Möbius action. We can then apply such a transformation to move one of the zeroes of to infinity (thus making a polynomial of degree ), so that must now be a circle, with the zeroes of inside the circle and the remaining zeroes of outside the circle. But then

By the Gauss-Lucas theorem, the zeroes of are also inside . The claim now follows from the induction hypothesis.

A correction to “An arithmetic regularity lemma, an associated counting lemma, and applications”

Ben Green and I have updated our paper “An arithmetic regularity lemma, an associated counting lemma, and applications” to account for a somewhat serious issue with the paper that was pointed out to us recently by Daniel Altman. This paper contains two core theorems:

• An “arithmetic regularity lemma” that, roughly speaking, decomposes an arbitrary bounded sequence on an interval as an “irrational nilsequence” of controlled complexity, plus some “negligible” errors (where one uses the Gowers uniformity norm as the main norm to control the neglibility of the error); and
• An “arithmetic counting lemma” that gives an asymptotic formula for counting various averages for various affine-linear forms when the functions are given by irrational nilsequences.

The combination of the two theorems is then used to address various questions in additive combinatorics.

There are no direct issues with the arithmetic regularity lemma. However, it turns out that the arithmetic counting lemma is only true if one imposes an additional property (which we call the “flag property”) on the affine-linear forms . Without this property, there does not appear to be a clean asymptotic formula for these averages if the only hypothesis one places on the underlying nilsequences is irrationality. Thus when trying to understand the asymptotics of averages involving linear forms that do not obey the flag property, the paradigm of understanding these averages via a combination of the regularity lemma and a counting lemma seems to require some significant revision (in particular, one would probably have to replace the existing regularity lemma with some variant, despite the fact that the lemma is still technically true in this setting). Fortunately, for most applications studied to date (including the important subclass of translation-invariant affine forms), the flag property holds; however our claim in the paper to have resolved a conjecture of Gowers and Wolf on the true complexity of systems of affine forms must now be narrowed, as our methods only verify this conjecture under the assumption of the flag property.

In a bit more detail: the asymptotic formula for our counting lemma involved some finite-dimensional vector spaces for various natural numbers , defined as the linear span of the vectors as ranges over the parameter space . Roughly speaking, these spaces encode some constraints one would expect to see amongst the forms . For instance, in the case of length four arithmetic progressions when , , and

for , then is spanned by the vectors and and can thus be described as the two-dimensional linear space

while is spanned by the vectors , , and can be described as the hyperplane

As a special case of the counting lemma, we can check that if takes the form for some irrational , some arbitrary , and some smooth , then the limiting value of the average

as is equal to

which reflects the constraints

and

These constraints follow from the descriptions (1), (2), using the containment to dispense with the lower order term (which then plays no further role in the analysis).

The arguments in our paper turn out to be perfectly correct under the assumption of the “flag property” that for all . The problem is that the flag property turns out to not always hold. A counterexample, provided by Daniel Altman, involves the four linear forms

Here it turns out that

and

and is no longer contained in . The analogue of the asymptotic formula given previously for is then valid when vanishes, but not when is non-zero, because the identity

holds in the former case but not the latter. Thus the output of any purported arithmetic regularity lemma in this case is now sensitive to the lower order terms of the nilsequence and cannot be described in a uniform fashion for all “irrational” sequences. There should still be some sort of formula for the asymptotics from the general equidistribution theory of nilsequences, but it could be considerably more complicated than what is presented in this paper.

Fortunately, the flag property does hold in several key cases, most notably the translation invariant case when contains , as well as “complexity one” cases. Nevertheless non-flag property systems of affine forms do exist, thus limiting the range of applicability of the techniques in this paper. In particular, the conjecture of Gowers and Wolf (Theorem 1.13 in the paper) is now open again in the non-flag property case.

Climbing the cosmic distance ladder (book announcement)

Several years ago, I developed a public lecture on the cosmic distance ladder in astronomy from a historical perspective (and emphasising the role of mathematics in building the ladder). I previously blogged about the lecture here; the most recent version of the slides can be found here. Recently, I have begun working with Tanya Klowden (a long time friend with a background in popular writing on a variety of topics, including astronomy) to expand the lecture into a popular science book, with the tentative format being non-technical chapters interspersed with some more mathematical sections to give some technical details. We are still in the middle of the writing process, but we have produced a sample chapter (which deals with what we call the “fourth rung” of the distance ladder – the distances and orbits of the planets – and how the work of Copernicus, Brahe, Kepler and others led to accurate measurements of these orbits, as well as Kepler’s famous laws of planetary motion). As always, any feedback on the chapter is welcome. (Due to various pandemic-related uncertainties, we do not have a definite target deadline for when the book will be completed, but presumably this will occur sometime in the next year.)

The book is currently under contract with Yale University Press. My coauthor Tanya Klowden can be reached at tklowden@gmail.com.

The structure of translational tilings in Z^d

Rachel Greenfeld and I have just uploaded to the arXiv our paper “The structure of translational tilings in “. This paper studies the tilings of a finite tile in a standard lattice , that is to say sets (which we call tiling sets) such that every element of lies in exactly one of the translates of . We also consider more general tilings of level for a natural number (several of our results consider an even more general setting in which is periodic but allowed to be non-constant).

In many cases the tiling set will be periodic (by which we mean translation invariant with respect to some lattice (a finite index subgroup) of ). For instance one simple example of a tiling is when is the unit square and is the lattice . However one can modify some tilings to make them less periodic. For instance, keeping one also has the tiling set

where is an arbitrary function. This tiling set is periodic in a single direction , but is not doubly periodic. For the slightly modified tile , the set

for arbitrary can be verified to be a tiling set, which in general will not exhibit any periodicity whatsoever; however, it is weakly periodic in the sense that it is the disjoint union of finitely many sets, each of which is periodic in one direction.

The most well known conjecture in this area is the Periodic Tiling Conjecture:

Conjecture 1 (Periodic tiling conjecture) If a finite tile has at least one tiling set, then it has a tiling set which is periodic.

This conjecture was stated explicitly by Lagarias and Wang, and also appears implicitly in this text of Grunbaum and Shepard. In one dimension there is a simple pigeonhole principle argument of Newman that shows that all tiling sets are in fact periodic, which certainly implies the periodic tiling conjecture in this case. The case was settled more recently by Bhattacharya, but the higher dimensional cases remain open in general.

We are able to obtain a new proof of Bhattacharya’s result that also gives some quantitative bounds on the periodic tiling set, which are polynomial in the diameter of the set if the cardinality of the tile is bounded:

Theorem 2 (Quantitative periodic tiling in ) If a finite tile has at least one tiling set, then it has a tiling set which is -periodic for some .

Among other things, this shows that the problem of deciding whether a given subset of of bounded cardinality tiles or not is in the NP complexity class with respect to the diameter . (Even the decidability of this problem was not known until the result of Bhattacharya.)

We also have a closely related structural theorem:

Theorem 3 (Quantitative weakly periodic tiling in ) Every tiling set of a finite tile is weakly periodic. In fact, the tiling set is the union of at most disjoint sets, each of which is periodic in a direction of magnitude .

We also have a new bound for the periodicity of tilings in :

Theorem 4 (Universal period for tilings in ) Let be finite, and normalized so that . Then every tiling set of is -periodic, where is the least common multiple of all primes up to , and is the least common multiple of the magnitudes of all .

We remark that the current best complexity bound of determining whether a subset of tiles or not is , due to Biro. It may be that the results in this paper can improve upon this bound, at least for tiles of bounded cardinality.

On the other hand, we discovered a genuine difference between level one tiling and higher level tiling, by locating a counterexample to the higher level analogue of (the qualitative version of) Theorem 3:

Theorem 5 (Counterexample) There exists an eight-element subset and a level tiling such that is not weakly periodic.

We do not know if there is a corresponding counterexample to the higher level periodic tiling conjecture (that if tiles at level , then there is a periodic tiling at the same level ). Note that it is important to keep the level fixed, since one trivially always has a periodic tiling at level from the identity .

The methods of Bhattacharya used the language of ergodic theory. Our investigations also originally used ergodic-theoretic and Fourier-analytic techniques, but we ultimately found combinatorial methods to be more effective in this problem (and in particular led to quite strong quantitative bounds). The engine powering all of our results is the following remarkable fact, valid in all dimensions:

Lemma 6 (Dilation lemma) Suppose that is a tiling of a finite tile . Then is also a tiling of the dilated tile for any coprime to , where is the least common multiple of all the primes up to .

Versions of this dilation lemma have previously appeared in work of Tijdeman and of Bhattacharya. We sketch a proof here. By the fundamental theorem of arithmetic and iteration it suffices to establish the case where is a prime . We need to show that . It suffices to show the claim , since both sides take values in . The convolution algebra (or group algebra) of finitely supported functions from to is a commutative algebra of characteristic , so we have the Frobenius identity for any . As a consequence we see that . The claim now follows by convolving the identity by further copies of .

In our paper we actually establish a more general version of the dilation lemma that can handle tilings of higher level or of a periodic set, and this stronger version is useful to get the best quantitative results, but for simplicity we focus attention just on the above simple special case of the dilation lemma.

By averaging over all in an arithmetic progression, one already gets a useful structural theorem for tilings in any dimension, which appears to be new despite being an easy consequence of Lemma 6:

Corollary 7 (Structure theorem for tilings) Suppose that is a tiling of a finite tile , where we normalize . Then we have a decomposition

where each is a function that is periodic in the direction , where is the least common multiple of all the primes up to .

Proof: From Lemma 6 we have for any , where is the Kronecker delta at . Now average over (extracting a weak limit or generalised limit as necessary) to obtain the conclusion.

The identity (1) turns out to impose a lot of constraints on the functions , particularly in one and two dimensions. On one hand, one can work modulo to eliminate the and terms to obtain the equation

which in two dimensions in particular puts a lot of structure on each individual (roughly speaking it makes the behave in a polynomial fashion, after collecting commensurable terms). On the other hand we have the inequality

which can be used to exclude “equidistributed” polynomial behavior after a certain amount of combinatorial analysis. Only a small amount of further argument is then needed to conclude Theorem 3 and Theorem 2.

For level tilings the analogue of (2) becomes

which is a significantly weaker inequality and now no longer seems to prohibit “equidistributed” behavior. After some trial and error we were able to come up with a completely explicit example of a tiling that actually utilises equidistributed polynomials; indeed the tiling set we ended up with was a finite boolean combination of Bohr sets.

We are currently studying what this machinery can tell us about tilings in higher dimensions, focusing initially on the three-dimensional case.

Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration

Asgar Jamneshan and I have just uploaded to the arXiv our paper “Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration“. This paper arose from our longer-term project to systematically develop “uncountable” ergodic theory – ergodic theory in which the groups acting are not required to be countable, the probability spaces one acts on are not required to be standard Borel, or Polish, and the compact groups that arise in the structural theory (e.g., the theory of group extensions) are not required to be separable. One of the motivations of doing this is to allow ergodic theory results to be applied to ultraproducts of finite dynamical systems, which can then hopefully be transferred to establish combinatorial results with good uniformity properties. An instance of this is the uncountable Mackey-Zimmer theorem, discussed in this companion blog post.

In the course of this project, we ran into the obstacle that many foundational results, such as the Riesz representation theorem, often require one or more of these countability hypotheses when encountered in textbooks. Other technical issues also arise in the uncountable setting, such as the need to distinguish the Borel -algebra from the (two different types of) Baire -algebra. As such we needed to spend some time reviewing and synthesizing the known literature on some foundational results of “uncountable” measure theory, which led to this paper. As such, most of the results of this paper are already in the literature, either explicitly or implicitly, in one form or another (with perhaps the exception of the canonical disintegration, which we discuss below); we view the main contribution of this paper as presenting the results in a coherent and unified fashion. In particular we found that the language of category theory was invaluable in clarifying and organizing all the different results. In subsequent work we (and some other authors) will use the results in this paper for various applications in uncountable ergodic theory.

The foundational results covered in this paper can be divided into a number of subtopics (Gelfand duality, Baire -algebras and Riesz representation, canonical models, and canonical disintegration), which we discuss further below the fold.

An uncountable Mackey-Zimmer theorem

Asgar Jamneshan and I have just uploaded to the arXiv our paper “An uncountable Mackey-Zimmer theorem“. This paper is part of our longer term project to develop “uncountable” versions of various theorems in ergodic theory; see this previous paper of Asgar and myself for the first paper in this series (and another paper will appear shortly).

In this case the theorem in question is the Mackey-Zimmer theorem, previously discussed in this blog post. This theorem gives an important classification of group and homogeneous extensions of measure-preserving systems. Let us first work in the (classical) setting of concrete measure-preserving systems. Let be a measure-preserving system for some group , thus is a (concrete) probability space and is a group homomorphism from to the automorphism group of the probability space. (Here we are abusing notation by using to refer both to the measure-preserving system and to the underlying set. In the notation of the paper we would instead distinguish these two objects as and respectively, reflecting two of the (many) categories one might wish to view as a member of, but for sake of this informal overview we will not maintain such precise distinctions.) If is a compact group, we define a (concrete) cocycle to be a collection of measurable functions for that obey the cocycle equation

for each and all . (One could weaken this requirement by only demanding the cocycle equation to hold for almost all , rather than all ; we will effectively do so later in the post, when we move to opposite probability algebra systems.) Any such cocycle generates a group skew-product of , which is another measure-preserving system where

• is the Cartesian product of and ;
• is the product measure of and Haar probability measure on ; and
• The action is given by the formula

  

The cocycle equation (1) guarantees that is a homomorphism, and the (left) invariance of Haar measure and Fubini’s theorem guarantees that the remain measure preserving. There is also the more general notion of a homogeneous skew-product in which the group is replaced by the homogeneous space for some closed subgroup of , noting that still comes with a left-action of and a Haar measure. Group skew-products are very “explicit” ways to extend a system , as everything is described by the cocycle which is a relatively tractable object to manipulate. (This is not to say that the cohomology of measure-preserving systems is trivial, but at least there are many tools one can use to study them, such as the Moore-Schmidt theorem discussed in this previous post.)

This group skew-product comes with a factor map and a coordinate map , which by (2) are related to the action via the identities

and

where in (4) we are implicitly working in the group of (concretely) measurable functions from to . Furthermore, the combined map is measure-preserving (using the product measure on ), indeed the way we have constructed things this map is just the identity map.

We can now generalize the notion of group skew-product by just working with the maps , and weakening the requirement that be measure-preserving. Namely, define a group extension of by to be a measure-preserving system equipped with a measure-preserving map obeying (3) and a measurable map obeying (4) for some cocycle , such that the -algebra of is generated by . There is also a more general notion of a homogeneous extension in which takes values in rather than . Then every group skew-product is a group extension of by , but not conversely. Here are some key counterexamples:

• (i) If is a closed subgroup of , and is a cocycle taking values in , then can be viewed as a group extension of by , taking to be the vertical coordinate (viewing now as an element of ). This will not be a skew-product by because pushes forward to the wrong measure on : it pushes forward to rather than .
• (ii) If one takes the same example as (i), but twists the vertical coordinate to another vertical coordinate for some measurable “gauge function” , then is still a group extension by , but now with the cocycle replaced by the cohomologous cocycle

  

  Again, this will not be a skew product by , because pushes forward to a twisted version of that is supported (at least in the case where is compact and the cocycle is continuous) on the -bundle .
• (iii) With the situation as in (i), take to be the union for some outside of , where we continue to use the action (2) and the standard vertical coordinate but now use the measure .

As it turns out, group extensions and homogeneous extensions arise naturally in the Furstenberg-Zimmer structural theory of measure-preserving systems; roughly speaking, every compact extension of is an inverse limit of group extensions. It is then of interest to classify such extensions.

Examples such as (iii) are annoying, but they can be excluded by imposing the additional condition that the system is ergodic – all invariant (or essentially invariant) sets are of measure zero or measure one. (An essentially invariant set is a measurable subset of such that is equal modulo null sets to for all .) For instance, the system in (iii) is non-ergodic because the set (or ) is invariant but has measure . We then have the following fundamental result of Mackey and Zimmer:

Theorem 1 (Countable Mackey Zimmer theorem) Let be a group, be a concrete measure-preserving system, and be a compact Hausdorff group. Assume that is at most countable, is a standard Borel space, and is metrizable. Then every (concrete) ergodic group extension of is abstractly isomorphic to a group skew-product (by some closed subgroup of ), and every (concrete) ergodic homogeneous extension of is similarly abstractly isomorphic to a homogeneous skew-product.

We will not define precisely what “abstractly isomorphic” means here, but it roughly speaking means “isomorphic after quotienting out the null sets”. A proof of this theorem can be found for instance in .

The main result of this paper is to remove the “countability” hypotheses from the above theorem, at the cost of working with opposite probability algebra systems rather than concrete systems. (We will discuss opposite probability algebras in a subsequent blog post relating to another paper in this series.)

Theorem 2 (Uncountable Mackey Zimmer theorem) Let be a group, be an opposite probability algebra measure-preserving system, and be a compact Hausdorff group. Then every (abstract) ergodic group extension of is abstractly isomorphic to a group skew-product (by some closed subgroup of ), and every (abstract) ergodic homogeneous extension of is similarly abstractly isomorphic to a homogeneous skew-product.

We plan to use this result in future work to obtain uncountable versions of the Furstenberg-Zimmer and Host-Kra structure theorems.

As one might expect, one locates a proof of Theorem 2 by finding a proof of Theorem 1 that does not rely too strongly on “countable” tools, such as disintegration or measurable selection, so that all of those tools can be replaced by “uncountable” counterparts. The proof we use is based on the one given in this previous post, and begins by comparing the system with the group extension . As the examples (i), (ii) show, these two systems need not be isomorphic even in the ergodic case, due to the different probability measures employed. However one can relate the two after performing an additional averaging in . More precisely, there is a canonical factor map given by the formula

This is a factor map not only of -systems, but actually of -systems, where the opposite group to acts (on the left) by right-multiplication of the second coordinate (this reversal of order is why we need to use the opposite group here). The key point is that the ergodicity properties of the system are closely tied the group that is “secretly” controlling the group extension. Indeed, in example (i), the invariant functions on take the form for some measurable , while in example (ii), the invariant functions on take the form . In either case, the invariant factor is isomorphic to , and can be viewed as a factor of the invariant factor of , which is isomorphic to . Pursuing this reasoning (using an abstract ergodic theorem of Alaoglu and Birkhoff, as discussed in the previous post) one obtains the Mackey range , and also obtains the quotient of to in this process. The main remaining task is to lift the quotient back up to a map that stays measurable, in order to “untwist” a system that looks like (ii) to make it into one that looks like (i). In countable settings this is where a “measurable selection theorem” would ordinarily be invoked, but in the uncountable setting such theorems are not available for concrete maps. However it turns out that they still remain available for abstract maps: any abstractly measurable map from to has an abstractly measurable lift from to . To prove this we first use a canonical model for opposite probability algebras (which we will discuss in a companion post to this one, to appear shortly) to work with continuous maps (on a Stone space) rather than abstractly measurable maps. The measurable map then induces a probability measure on , formed by pushing forward by the graphing map . This measure in turn has several lifts up to a probability measure on ; for instance, one can construct such a measure via the Riesz representation theorem by demanding

for all continuous functions . This measure does not come from a graph of any single lift , but is in some sense an “average” of the entire ensemble of these lifts. But it turns out one can invoke the Krein-Milman theorem to pass to an extremal lifting measure which does come from an (abstract) lift , and this can be used as a substitute for a measurable selection theorem. A variant of this Krein-Milman argument can also be used to express any homogeneous extension as a quotient of a group extension, giving the second part of the Mackey-Zimmer theorem.