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Previous set of notes: Notes 1. Next set of notes: Notes 3.

In Exercise 5 (and Lemma 1) of 246A Notes 4 we already observed some links between complex analysis on the disk (or annulus) and Fourier series on the unit circle:

  • (i) Functions {f} that are holomorphic on a disk {\{ |z| < R \}} are expressed by a convergent Fourier series (and also Taylor series) {f(re^{i\theta}) = \sum_{n=0}^\infty r^n a_n e^{in\theta}} for {0 \leq r < R} (so in particular {a_n = \frac{1}{n!} f^{(n)}(0)}), where

    \displaystyle  \limsup_{n \rightarrow +\infty} |a_n|^{1/n} \leq \frac{1}{R}; \ \ \ \ \ (1)

    conversely, every infinite sequence {(a_n)_{n=0}^\infty} of coefficients obeying (1) arises from such a function {f}.
  • (ii) Functions {f} that are holomorphic on an annulus {\{ r_- < |z| < r_+ \}} are expressed by a convergent Fourier series (and also Laurent series) {f(re^{i\theta}) = \sum_{n=-\infty}^\infty r^n a_n e^{in\theta}}, where

    \displaystyle  \limsup_{n \rightarrow +\infty} |a_n|^{1/n} \leq \frac{1}{r_+}; \limsup_{n \rightarrow -\infty} |a_n|^{1/|n|} \leq \frac{1}{r_-}; \ \ \ \ \ (2)

    conversely, every doubly infinite sequence {(a_n)_{n=-\infty}^\infty} of coefficients obeying (2) arises from such a function {f}.
  • (iii) In the situation of (ii), there is a unique decomposition {f = f_1 + f_2} where {f_1} extends holomorphically to {\{ z: |z| < r_+\}}, and {f_2} extends holomorphically to {\{ z: |z| > r_-\}} and goes to zero at infinity, and are given by the formulae

    \displaystyle  f_1(z) = \sum_{n=0}^\infty a_n z^n = \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z}\ dw

    where {\gamma} is any anticlockwise contour in {\{ z: |z| < r_+\}} enclosing {z}, and and

    \displaystyle  f_2(z) = \sum_{n=-\infty}^{-1} a_n z^n = - \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z}\ dw

    where {\gamma} is any anticlockwise contour in {\{ z: |z| > r_-\}} enclosing {0} but not {z}.

This connection lets us interpret various facts about Fourier series through the lens of complex analysis, at least for some special classes of Fourier series. For instance, the Fourier inversion formula {a_n = \frac{1}{2\pi} \int_0^{2\pi} f(e^{i\theta}) e^{-in\theta}\ d\theta} becomes the Cauchy-type formula for the Laurent or Taylor coefficients of {f}, in the event that the coefficients are doubly infinite and obey (2) for some {r_- < 1 < r_+}, or singly infinite and obey (1) for some {R > 1}.

It turns out that there are similar links between complex analysis on a half-plane (or strip) and Fourier integrals on the real line, which we will explore in these notes.

We first fix a normalisation for the Fourier transform. If {f \in L^1({\bf R})} is an absolutely integrable function on the real line, we define its Fourier transform {\hat f: {\bf R} \rightarrow {\bf C}} by the formula

\displaystyle  \hat f(\xi) := \int_{\bf R} f(x) e^{-2\pi i x \xi}\ dx. \ \ \ \ \ (3)

From the dominated convergence theorem {\hat f} will be a bounded continuous function; from the Riemann-Lebesgue lemma it also decays to zero as {\xi \rightarrow \pm \infty}. My choice to place the {2\pi} in the exponent is a personal preference (it is slightly more convenient for some harmonic analysis formulae such as the identities (4), (5), (6) below), though in the complex analysis and PDE literature there are also some slight advantages in omitting this factor. In any event it is not difficult to adapt the discussion in this notes for other choices of normalisation. It is of interest to extend the Fourier transform beyond the {L^1({\bf R})} class into other function spaces, such as {L^2({\bf R})} or the space of tempered distributions, but we will not pursue this direction here; see for instance these lecture notes of mine for a treatment.

Exercise 1 (Fourier transform of Gaussian) If {a} is a coplex number with {\mathrm{Re} a>0} and {f} is the Gaussian function {f(x) := e^{-\pi a x^2}}, show that the Fourier transform {\hat f} is given by the Gaussian {\hat f(\xi) = a^{-1/2} e^{-\pi \xi^2/a}}, where we use the standard branch for {a^{-1/2}}.

The Fourier transform has many remarkable properties. On the one hand, as long as the function {f} is sufficiently “reasonable”, the Fourier transform enjoys a number of very useful identities, such as the Fourier inversion formula

\displaystyle  f(x) = \int_{\bf R} \hat f(\xi) e^{2\pi i x \xi} d\xi, \ \ \ \ \ (4)

the Plancherel identity

\displaystyle  \int_{\bf R} |f(x)|^2\ dx = \int_{\bf R} |\hat f(\xi)|^2\ d\xi, \ \ \ \ \ (5)

and the Poisson summation formula

\displaystyle  \sum_{n \in {\bf Z}} f(n) = \sum_{k \in {\bf Z}} \hat f(k). \ \ \ \ \ (6)

On the other hand, the Fourier transform also intertwines various qualitative properties of a function {f} with “dual” qualitative properties of its Fourier transform {\hat f}; in particular, “decay” properties of {f} tend to be associated with “regularity” properties of {\hat f}, and vice versa. For instance, the Fourier transform of rapidly decreasing functions tend to be smooth. There are complex analysis counterparts of this Fourier dictionary, in which “decay” properties are described in terms of exponentially decaying pointwise bounds, and “regularity” properties are expressed using holomorphicity on various strips, half-planes, or the entire complex plane. The following exercise gives some examples of this:

Exercise 2 (Decay of {f} implies regularity of {\hat f}) Let {f \in L^1({\bf R})} be an absolutely integrable function.
  • (i) If {f} has super-exponential decay in the sense that {f(x) \lesssim_{f,M} e^{-M|x|}} for all {x \in {\bf R}} and {M>0} (that is to say one has {|f(x)| \leq C_{f,M} e^{-M|x|}} for some finite quantity {C_{f,M}} depending only on {f,M}), then {\hat f} extends uniquely to an entire function {\hat f : {\bf C} \rightarrow {\bf C}}. Furthermore, this function continues to be defined by (3).
  • (ii) If {f} is supported on a compact interval {[a,b]} then the entire function {\hat f} from (i) obeys the bounds {\hat f(\xi) \lesssim_f \max( e^{2\pi a \mathrm{Im} \xi}, e^{2\pi b \mathrm{Im} \xi} )} for {\xi \in {\bf C}}. In particular, if {f} is supported in {[-M,M]} then {\hat f(\xi) \lesssim_f e^{2\pi M |\mathrm{Im}(\xi)|}}.
  • (iii) If {f} obeys the bound {f(x) \lesssim_{f,a} e^{-2\pi a|x|}} for all {x \in {\bf R}} and some {a>0}, then {\hat f} extends uniquely to a holomorphic function {\hat f} on the horizontal strip {\{ \xi: |\mathrm{Im} \xi| < a \}}, and obeys the bound {\hat f(\xi) \lesssim_{f,a} \frac{1}{a - |\mathrm{Im}(\xi)|}} in this strip. Furthermore, this function continues to be defined by (3).
  • (iv) If {f} is supported on {[0,+\infty)} (resp. {(-\infty,0]}), then there is a unique continuous extension of {\hat f} to the lower half-plane {\{ \xi: \mathrm{Im} \xi \leq 0\}} (resp. the upper half-plane {\{ \xi: \mathrm{Im} \xi \geq 0 \}} which is holomorphic in the interior of this half-plane, and such that {\hat f(\xi) \rightarrow 0} uniformly as {\mathrm{Im} \xi \rightarrow -\infty} (resp. {\mathrm{Im} \xi \rightarrow +\infty}). Furthermore, this function continues to be defined by (3).
Hint: to establish holomorphicity in each of these cases, use Morera’s theorem and the Fubini-Tonelli theorem. For uniqueness, use analytic continuation, or (for part (iv)) the Cauchy integral formula.

Later in these notes we will give a partial converse to part (ii) of this exercise, known as the Paley-Wiener theorem; there are also partial converses to the other parts of this exercise.

From (3) we observe the following intertwining property between multiplication by an exponential and complex translation: if {\xi_0} is a complex number and {f: {\bf R} \rightarrow {\bf C}} is an absolutely integrable function such that the modulated function {f_{\xi_0}(x) := e^{2\pi i \xi_0 x} f(x)} is also absolutely integrable, then we have the identity

\displaystyle  \widehat{f_{\xi_0}}(\xi) = \hat f(\xi - \xi_0) \ \ \ \ \ (7)

whenever {\xi} is a complex number such that at least one of the two sides of the equation in (7) is well defined. Thus, multiplication of a function by an exponential weight corresponds (formally, at least) to translation of its Fourier transform. By using contour shifting, we will also obtain a dual relationship: under suitable holomorphicity and decay conditions on {f}, translation by a complex shift will correspond to multiplication of the Fourier transform by an exponential weight. It turns out to be possible to exploit this property to derive many Fourier-analytic identities, such as the inversion formula (4) and the Poisson summation formula (6), which we do later in these notes. (The Plancherel theorem can also be established by complex analytic methods, but this requires a little more effort; see Exercise 8.)

The material in these notes is loosely adapted from Chapter 4 of Stein-Shakarchi’s “Complex Analysis”.

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Marcel Filoche, Svitlana Mayboroda, and I have just uploaded to the arXiv our preprint “The effective potential of an {M}-matrix“. This paper explores the analogue of the effective potential of Schrödinger operators {-\Delta + V} provided by the “landscape function” {u}, when one works with a certain type of self-adjoint matrix known as an {M}-matrix instead of a Schrödinger operator.

Suppose one has an eigenfunction

\displaystyle  (-\Delta + V) \phi = E \phi

of a Schrödinger operator {-\Delta+V}, where {\Delta} is the Laplacian on {{\bf R}^d}, {V: {\bf R}^d \rightarrow {\bf R}} is a potential, and {E} is an energy. Where would one expect the eigenfunction {\phi} to be concentrated? If the potential {V} is smooth and slowly varying, the correspondence principle suggests that the eigenfunction {\phi} should be mostly concentrated in the potential energy wells {\{ x: V(x) \leq E \}}, with an exponentially decaying amount of tunnelling between the wells. One way to rigorously establish such an exponential decay is through an argument of Agmon, which we will sketch later in this post, which gives an exponentially decaying upper bound (in an {L^2} sense) of eigenfunctions {\phi} in terms of the distance to the wells {\{ V \leq E \}} in terms of a certain “Agmon metric” on {{\bf R}^d} determined by the potential {V} and energy level {E} (or any upper bound {\overline{E}} on this energy). Similar exponential decay results can also be obtained for discrete Schrödinger matrix models, in which the domain {{\bf R}^d} is replaced with a discrete set such as the lattice {{\bf Z}^d}, and the Laplacian {\Delta} is replaced by a discrete analogue such as a graph Laplacian.

When the potential {V} is very “rough”, as occurs for instance in the random potentials arising in the theory of Anderson localisation, the Agmon bounds, while still true, become very weak because the wells {\{ V \leq E \}} are dispersed in a fairly dense fashion throughout the domain {{\bf R}^d}, and the eigenfunction can tunnel relatively easily between different wells. However, as was first discovered in 2012 by my two coauthors, in these situations one can replace the rough potential {V} by a smoother effective potential {1/u}, with the eigenfunctions typically localised to a single connected component of the effective wells {\{ 1/u \leq E \}}. In fact, a good choice of effective potential comes from locating the landscape function {u}, which is the solution to the equation {(-\Delta + V) u = 1} with reasonable behavior at infinity, and which is non-negative from the maximum principle, and then the reciprocal {1/u} of this landscape function serves as an effective potential.

There are now several explanations for why this particular choice {1/u} is a good effective potential. Perhaps the simplest (as found for instance in this recent paper of Arnold, David, Jerison, and my two coauthors) is the following observation: if {\phi} is an eigenvector for {-\Delta+V} with energy {E}, then {\phi/u} is an eigenvector for {-\frac{1}{u^2} \mathrm{div}(u^2 \nabla \cdot) + \frac{1}{u}} with the same energy {E}, thus the original Schrödinger operator {-\Delta+V} is conjugate to a (variable coefficient, but still in divergence form) Schrödinger operator with potential {1/u} instead of {V}. Closely related to this, we have the integration by parts identity

\displaystyle  \int_{{\bf R}^d} |\nabla f|^2 + V |f|^2\ dx = \int_{{\bf R}^d} u^2 |\nabla(f/u)|^2 + \frac{1}{u} |f|^2\ dx \ \ \ \ \ (1)

for any reasonable function {f}, thus again highlighting the emergence of the effective potential {1/u}.

These particular explanations seem rather specific to the Schrödinger equation (continuous or discrete); we have for instance not been able to find similar identities to explain an effective potential for the bi-Schrödinger operator {\Delta^2 + V}.

In this paper, we demonstrate the (perhaps surprising) fact that effective potentials continue to exist for operators that bear very little resemblance to Schrödinger operators. Our chosen model is that of an {M}-matrix: self-adjoint positive definite matrices {A} whose off-diagonal entries are negative. This model includes discrete Schrödinger operators (with non-negative potentials) but can allow for significantly more non-local interactions. The analogue of the landscape function would then be the vector {u := A^{-1} 1}, where {1} denotes the vector with all entries {1}. Our main result, roughly speaking, asserts that an eigenvector {A \phi = E \phi} of {A} will then be exponentially localised to the “potential wells” {K := \{ j: \frac{1}{u_j} \leq E \}}, where {u_j} denotes the coordinates of the landscape function {u}. In particular, we establish the inequality

\displaystyle  \sum_k \phi_k^2 e^{2 \rho(k,K) / \sqrt{W}} ( \frac{1}{u_k} - E )_+ \leq W \max_{i,j} |a_{ij}|

if {\phi} is normalised in {\ell^2}, where the connectivity {W} is the maximum number of non-zero entries of {A} in any row or column, {a_{ij}} are the coefficients of {A}, and {\rho} is a certain moderately complicated but explicit metric function on the spatial domain. Informally, this inequality asserts that the eigenfunction {\phi_k} should decay like {e^{-\rho(k,K) / \sqrt{W}}} or faster. Indeed, our numerics show a very strong log-linear relationship between {\phi_k} and {\rho(k,K)}, although it appears that our exponent {1/\sqrt{W}} is not quite optimal. We also provide an associated localisation result which is technical to state but very roughly asserts that a given eigenvector will in fact be localised to a single connected component of {K} unless there is a resonance between two wells (by which we mean that an eigenvalue for a localisation of {A} associated to one well is extremely close to an eigenvalue for a localisation of {A} associated to another well); such localisation is also strongly supported by numerics. (Analogous results for Schrödinger operators had been previously obtained by the previously mentioned paper of Arnold, David, Jerison, and my two coauthors, and to quantum graphs in a very recent paper of Harrell and Maltsev.)

Our approach is based on Agmon’s methods, which we interpret as a double commutator method, and in particular relying on exploiting the negative definiteness of certain double commutator operators. In the case of Schrödinger operators {-\Delta+V}, this negative definiteness is provided by the identity

\displaystyle  \langle [[-\Delta+V,g],g] u, u \rangle = -2\int_{{\bf R}^d} |\nabla g|^2 |u|^2\ dx \leq 0 \ \ \ \ \ (2)

for any sufficiently reasonable functions {u, g: {\bf R}^d \rightarrow {\bf R}}, where we view {g} (like {V}) as a multiplier operator. To exploit this, we use the commutator identity

\displaystyle  \langle g [\psi, -\Delta+V] u, g \psi u \rangle = \frac{1}{2} \langle [[-\Delta+V, g \psi],g\psi] u, u \rangle

\displaystyle -\frac{1}{2} \langle [[-\Delta+V, g],g] \psi u, \psi u \rangle

valid for any {g,\psi,u: {\bf R}^d \rightarrow {\bf R}} after a brief calculation. The double commutator identity then tells us that

\displaystyle  \langle g [\psi, -\Delta+V] u, g \psi u \rangle \leq \int_{{\bf R}^d} |\nabla g|^2 |\psi u|^2\ dx.

If we choose {u} to be a non-negative weight and let {\psi := \phi/u} for an eigenfunction {\phi}, then we can write

\displaystyle  [\psi, -\Delta+V] u = [\psi, -\Delta+V - E] u = \psi (-\Delta+V - E) u

and we conclude that

\displaystyle  \int_{{\bf R}^d} \frac{(-\Delta+V-E)u}{u} |g|^2 |\phi|^2\ dx \leq \int_{{\bf R}^d} |\nabla g|^2 |\phi|^2\ dx. \ \ \ \ \ (3)

We have considerable freedom in this inequality to select the functions {u,g}. If we select {u=1}, we obtain the clean inequality

\displaystyle  \int_{{\bf R}^d} (V-E) |g|^2 |\phi|^2\ dx \leq \int_{{\bf R}^d} |\nabla g|^2 |\phi|^2\ dx.

If we take {g} to be a function which equals {1} on the wells {\{ V \leq E \}} but increases exponentially away from these wells, in such a way that

\displaystyle  |\nabla g|^2 \leq \frac{1}{2} (V-E) |g|^2

outside of the wells, we can obtain the estimate

\displaystyle  \int_{V > E} (V-E) |g|^2 |\phi|^2\ dx \leq 2 \int_{V < E} (E-V) |\phi|^2\ dx,

which then gives an exponential type decay of {\phi} away from the wells. This is basically the classic exponential decay estimate of Agmon; one can basically take {g} to be the distance to the wells {\{ V \leq E \}} with respect to the Euclidean metric conformally weighted by a suitably normalised version of {V-E}. If we instead select {u} to be the landscape function {u = (-\Delta+V)^{-1} 1}, (3) then gives

\displaystyle  \int_{{\bf R}^d} (\frac{1}{u} - E) |g|^2 |\phi|^2\ dx \leq \int_{{\bf R}^d} |\nabla g|^2 |\phi|^2\ dx,

and by selecting {g} appropriately this gives an exponential decay estimate away from the effective wells {\{ \frac{1}{u} \leq E \}}, using a metric weighted by {\frac{1}{u}-E}.

It turns out that this argument extends without much difficulty to the {M}-matrix setting. The analogue of the crucial double commutator identity (2) is

\displaystyle  \langle [[A,D],D] u, u \rangle	= \sum_{i \neq j} a_{ij} u_i u_j (d_{ii} - d_{jj})^2 \leq 0

for any diagonal matrix {D = \mathrm{diag}(d_{11},\dots,d_{NN})}. The remainder of the Agmon type arguments go through after making the natural modifications.

Numerically we have also found some aspects of the landscape theory to persist beyond the {M}-matrix setting, even though the double commutators cease being negative definite, so this may not yet be the end of the story, but it does at least demonstrate that utility the landscape does not purely rely on identities such as (1).

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

— 1. Jensen’s formula —

Suppose {f} is a non-zero rational function {f =P/Q}, then by the fundamental theorem of algebra one can write

\displaystyle  f(z) = c \frac{\prod_\rho (z-\rho)}{\prod_\zeta (z-\zeta)}

for some non-zero constant {c}, where {\rho} ranges over the zeroes of {P} (counting multiplicity) and {\zeta} ranges over the zeroes of {Q} (counting multiplicity), and assuming {z} avoids the zeroes of {Q}. Taking absolute values and then logarithms, we arrive at the formula

\displaystyle  \log |f(z)| = \log |c| + \sum_\rho \log|z-\rho| - \sum_\zeta \log |z-\zeta|, \ \ \ \ \ (1)

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

\displaystyle  \frac{f'(z)}{f(z)} = \sum_\rho \frac{1}{z-\rho} - \sum_\zeta \frac{1}{z-\zeta}, \ \ \ \ \ (2)

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

Exercise 1 Let {P(z)} be a complex polynomial of degree {n \geq 1}.
  • (i) (Gauss-Lucas theorem) Show that the complex roots of {P'(z)} are contained in the closed convex hull of the complex roots of {P(z)}.
  • (ii) (Laguerre separation theorem) If all the complex roots of {P(z)} are contained in a disk {D(z_0,r)}, and {\zeta \not \in D(z_0,r)}, then all the complex roots of {nP(z) + (\zeta - z) P'(z)} are also contained in {D(z_0,r)}. (Hint: apply a suitable Möbius transformation to move {\zeta} 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 {f} be a meromorphic function on an open neighbourhood of a disk {\overline{D(z_0,r)} = \{ z: |z-z_0| \leq r \}}, with all removable singularities removed. Then, if {z_0} is neither a zero nor a pole of {f}, we have

\displaystyle  \log |f(z_0)| = \int_0^1 \log |f(z_0+re^{2\pi i t})|\ dt + \sum_{\rho: |\rho-z_0| \leq r} \log \frac{|\rho-z_0|}{r} \ \ \ \ \ (3)

\displaystyle  - \sum_{\zeta: |\zeta-z_0| \leq r} \log \frac{|\zeta-z_0|}{r}

where {\rho} and {\zeta} range over the zeroes and poles of {f} respectively (counting multiplicity) in the disk {\overline{D(z_0,r)}}.

One can view (3) as a truncated (or localised) variant of (1). Note also that the summands {\log \frac{|\rho-z_0|}{r}, \log \frac{|\zeta-z_0|}{r}} are always non-positive.

Proof: By perturbing {r} slightly if necessary, we may assume that none of the zeroes or poles of {f} (which form a discrete set) lie on the boundary circle {\{ z: |z-z_0| = r \}}. By translating and rescaling, we may then normalise {z_0=0} and {r=1}, thus our task is now to show that

\displaystyle  \log |f(0)| = \int_0^1 \log |f(e^{2\pi i t})|\ dt + \sum_{\rho: |\rho| < 1} \log |\rho| - \sum_{\zeta: |\zeta| < 1} \log |\zeta|. \ \ \ \ \ (4)

We may remove the poles and zeroes inside the disk {D(0,1)} by the useful device of Blaschke products. Suppose for instance that {f} has a zero {\rho} inside the disk {D(0,1)}. Observe that the function

\displaystyle  B_\rho(z) := \frac{\rho - z}{1 - \overline{\rho} z} \ \ \ \ \ (5)

has magnitude {1} on the unit circle {\{ z: |z| = 1\}}, equals {\rho} at the origin, has a simple zero at {\rho}, but has no other zeroes or poles inside the disk. Thus Jensen’s formula (4) already holds if {f} is replaced by {B_\rho}. To prove (4) for {f}, it thus suffices to prove it for {f/B_\rho}, which effectively deletes a zero {\rho} inside the disk {D(0,1)} from {f} (and replaces it instead with its inversion {1/\overline{\rho}}). 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 {f} has no poles or zeroes on or inside the disk {D(0,1)}, at which point our goal is simply to show that

\displaystyle  \log |f(0)| = \int_0^1 \log |f(e^{2\pi i t})|\ dt.

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

An important special case of Jensen’s formula arises when {f} is holomorphic in a neighborhood of {\overline{D(z_0,r)}}, in which case there are no contributions from poles and one simply has

\displaystyle  \int_0^1 \log |f(z_0+re^{2\pi i t})|\ dt = \log |f(z_0)| + \sum_{\rho: |\rho-z_0| \leq r} \log \frac{r}{|\rho-z_0|}. \ \ \ \ \ (6)

This is quite a useful formula, mainly because the summands {\log \frac{r}{|\rho-z_0|}} are non-negative; it can be viewed as a more precise assertion of the subharmonicity of {\log |f|} (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 {f} on the entire complex plane is necessarily constant.

Exercise 4 Use Jensen’s formula to prove the fundamental theorem of algebra: a complex polynomial {P(z)} of degree {n} has exactly {n} complex zeroes (counting multiplicity), and can thus be factored as {P(z) = c (z-z_1) \dots (z-z_n)} for some complex numbers {c,z_1,\dots,z_n} with {c \neq 0}. (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 {f} be a meromorphic function on an open neighbourhood of a disk {\{ z: |z-z_0| \leq r \}}, with all removable singularities removed. Show that

\displaystyle  \log |f(z)| = \int_0^1 \log |f(z_0+re^{2\pi i t})| \mathrm{Re} \frac{r e^{2\pi i t} + (z-z_0)}{r e^{2\pi i t} - (z-z_0)}\ dt \ \ \ \ \ (7)

\displaystyle  + \sum_{\rho: |\rho-z_0| \leq r} \log \frac{|\rho-z|}{|r - \rho^* (z-z_0)|}

\displaystyle - \sum_{\zeta: |\zeta-z_0| \leq r} \log \frac{|\zeta-z|}{|r - \zeta^* (z-z_0)|}

for all {z} in the open disk {\{ z: |z-z_0| < r\}} that are not zeroes or poles of {f}, where {\rho^* = \frac{\overline{\rho-z_0}}{r}} and {\zeta^* = \frac{\overline{\zeta-z_0}}{r}}. (The function {\Re \frac{r e^{2\pi i t} + (z-z_0)}{r e^{2\pi i t} - (z-z_0)}} appearing in the integrand is sometimes known as the Poisson kernel, particularly if one normalises so that {z_0=0} and {r=1}.)

Exercise 6 (Bounded type)
  • (i) If {f} is a holomorphic function on {D(0,1)} that is not identically zero, show that {\liminf_{r \rightarrow 1^-} \int_0^{2\pi} \log |f(re^{i\theta})|\ d\theta > -\infty}.
  • (ii) If {f} is a meromorphic function on {D(0,1)} that is the ratio of two bounded holomorphic functions that are not identically zero, show that {\limsup_{r \rightarrow 1^-} \int_0^{2\pi} |\log |f(re^{i\theta})||\ d\theta < \infty}. (Functions {f} of this form are said to be of bounded type and lie in the Nevanlinna class for the unit disk {D(0,1)}.)

Exercise 7 (Smoothed out Jensen formula) Let {f} be a meromorphic function on an open set {U}, and let {\phi: U \rightarrow {\bf C}} be a smooth compactly supported function. Show that

\displaystyle \sum_\rho \phi(\rho) - \sum_\zeta \phi(\zeta)

\displaystyle  = \frac{-1}{2\pi} \int\int_U ((\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}) \phi(x+iy)) \frac{f'}{f}(x+iy)\ dx dy

\displaystyle  = \frac{1}{2\pi} \int\int_U ((\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y}^2) \phi(x+iy)) \log |f(x+iy)|\ dx dy

where {\rho, \zeta} range over the zeroes and poles of {f} (respectively) in the support of {\phi}. Informally argue why this identity is consistent with Jensen’s formula.

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

Proposition 8 Let {f: {\bf C} \rightarrow {\bf C}} be an entire function, not identically zero, that obeys a growth bound {|f(z)| \leq C \exp( C|z|^\alpha)} for some {C, \alpha > 0} and all {z}. Then there exists a constant {C'>0} such that {D(0,R)} has at most {C' R^\alpha} zeroes (counting multiplicity) for any {R \geq 1}.

Entire functions that obey a growth bound of the form {|f(z)| \leq C_\varepsilon \exp( C_\varepsilon |z|^{\rho+\varepsilon})} for every {\varepsilon>0} and {z} (where {C_\varepsilon} depends on {\varepsilon}) are said to be of order at most {\rho}. The above theorem shows that for such functions that are not identically zero, the number of zeroes in a disk of radius {R} does not grow much faster than {R^\rho}. 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 {f(0)} is non-zero. From (6) applied with {r=2R} and {z_0=0} one has

\displaystyle  \int_0^1 \log(C \exp( C (2R)^\alpha ) )\ dt \geq \log |f(0)| + \sum_{\rho: |\rho| \leq 2R} \log \frac{2R}{|\rho|}.

Every zero in {D(0,R)} contribute at least {\log 2} to a summand on the right-hand side, while all other zeroes contribute a non-negative quantity, thus

\displaystyle  \log C + C (2R)^\alpha \geq \log |f(0)| + N_R \log 2

where {N_R} denotes the number of zeroes in {D(0,R)}. This gives the claim for {f(0) \neq 0}. When {f(0)=0}, one can shift {f} by a small amount to make {f} non-zero at the origin (using the fact that zeroes of holomorphic functions not identically zero are isolated), modifying {C} in the process, and then repeating the previous arguments. \Box

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 {f} be a holomorphic function on an open neighbourhood of a disk {\{ z: |z-z_0| \leq r \}} that is not identically zero on this disk. Suppose that one has a bound of the form {|f(z)| \leq M^{O_{c_1,c_2}(1)} |f(z_0)|} for some {M \geq 1} and all {z} on the circle {\{ z: |z-z_0| = r\}}. Let {0 < c_2 < c_1 < 1} be constants. Then one has the approximate formula

\displaystyle  \frac{f'(z)}{f(z)} = \sum_{\rho: |\rho - z_0| \leq c_1 r} \frac{1}{z-\rho} + O_{c_1,c_2}( \frac{\log M}{r} )

for all {z} in the disk {\{ z: |z-z_0| < c_2 r \}} other than zeroes of {f}. Furthermore, the number of zeroes {\rho} in the above sum is {O_{c_1,c_2}(\log M)}.

Proof: To abbreviate notation, we allow all implied constants in this proof to depend on {c_1,c_2}.

We mimic the proof of Jensen’s formula. Firstly, we may translate and rescale so that {z_0=0} and {r=1}, so we have {|f(z)| \leq M^{O(1)} |f(0)|} when {|z|=1}, and our main task is to show that

\displaystyle  \frac{f'(z)}{f(z)} - \sum_{\rho: |\rho| \leq c_1} \frac{1}{z-\rho} = O( \log M ) \ \ \ \ \ (8)

for {|z| \leq c_2}. Note that if {f(0)=0} then {f} vanishes on the unit circle and hence (by the maximum principle) vanishes identically on the disk, a contradiction, so we may assume {f(0) \neq 0}. From hypothesis we then have

\displaystyle  \log |f(z)| \leq \log |f(0)| + O(\log M)

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

\displaystyle  \sum_{\rho: |\rho| \leq 1} \log \frac{1}{|\rho|} = O(\log M). \ \ \ \ \ (9)

In particular we see that the number of zeroes with {|\rho| \leq c_1} is {O(\log M)}, as claimed.

Suppose {f} has a zero {\rho} with {c_1 < |\rho| \leq 1}. If we factor {f = B_\rho g}, where {B_\rho} is the Blaschke product (5), then

\displaystyle  \frac{f'}{f} = \frac{B'_\rho}{B_\rho} + \frac{g'}{g}

\displaystyle  = \frac{g'}{g} + \frac{1}{z-\rho} - \frac{1}{z-1/\overline{\rho}}.

Observe from Taylor expansion that the distance between {\rho} and {1/\overline{\rho}} is {O( \log \frac{1}{|\rho|} )}, and hence {\frac{1}{z-\rho} - \frac{1}{z-1/\overline{\rho}} = O( \log \frac{1}{|\rho|} )} for {|z| \leq c_2}. Thus we see from (9) that we may use Blaschke products to remove all the zeroes in the annulus {c_1 < |\rho| \leq 1} while only affecting the left-hand side of (8) by {O( \log M)}; also, removing the Blaschke products does not affect {|f(z)|} on the unit circle, and only affects {\log |f(0)|} by {O(\log M)} thanks to (9). Thus we may assume without loss of generality that there are no zeroes in this annulus.

Similarly, given a zero {\rho} with {|\rho| \leq c_1}, we have {\frac{1}{z-1/\overline{\rho}} = O(1)}, so using Blaschke products to remove all of these zeroes also only affects the left-hand side of (8) by {O(\log M)} (since the number of zeroes here is {O(\log M)}), with {\log |f(0)|} also modified by at most {O(\log M)}. Thus we may assume in fact that {f} has no zeroes whatsoever within the unit disk. We may then also normalise {f(0) = 1}, then {\log |f(e^{2\pi i t})| \leq O(\log M)} for all {t \in [0,1]}. By Jensen’s formula again, we have

\displaystyle  \int_0^1 \log |f(e^{2\pi i t})|\ dt = 0

and thus (by using the identity {|x| = 2 \max(x,0) - x} for any real {x})

\displaystyle  \int_0^1 \log |f(e^{2\pi i t})|\ dt \ll \log M. \ \ \ \ \ (10)

On the other hand, from (7) we have

\displaystyle  \log |f(z)| = \int_0^1 \log |f(e^{2\pi i t})| \mathrm{Re} \frac{e^{2\pi i t} + z}{e^{2\pi i t} - z}\ dt

which implies from (10) that {\log |f(z)|} and its first derivatives are {O( \log M )} on the disk {\{ z: |z| \leq c_2 \}}. But recall from the proof of Jensen’s formula that {\frac{f'}{f}} is the derivative of a logarithm {\log f} of {f}, whose real part is {\log |f|}. By the Cauchy-Riemann equations for {\log f}, we conclude that {\frac{f'}{f} = O(\log M)} on the disk {\{ z: |z| \leq c_2 \}}, as required. \Box

Exercise 10
  • (i) (Borel-Carathéodory theorem) If {f: U \rightarrow {\bf C}} is analytic on an open neighborhood of a disk {\overline{D(z_0,R)}}, show that

    \displaystyle  \sup_{z \in D(z_0,r)} |f(z)| \leq \frac{2r}{R-r} \sup_{z \in \overline{D(z_0,R)}} \mathrm{Re} f(z) + \frac{R+r}{R-r} |f(z_0)|.

    (Hint: one can normalise {z_0=0}, {R=1}, {f(0)=0}, and {\sup_{|z-z_0| \leq R} \mathrm{Re} f(z)=1}. Now {f} maps the unit disk to the half-plane {\{ \mathrm{Re} z \leq 1 \}}. 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

\displaystyle  f(z) = P(z) \exp( g(z) )

for all {z} in the disk {\{ z: |z-z_0| < c_2 r \}}, where {P} is a polynomial whose zeroes are precisely the zeroes of {f} in {\{ z: |z-z_0| \leq c_1r \}} (counting multiplicity) and {g} is a holomorphic function on {\{ z: |z-z_0| < c_2 r \}} of magnitude {O_{c_1,c_2}( \log M )} and first derivative {O_{c_1,c_2}( \log M / r )} on this disk. Furthermore, show that the degree of {P} is {O_{c_1,c_2}(\log M)}.

Exercise 12 (Preliminary Beurling factorisation) Let {H^\infty(D(0,1))} denote the space of bounded analytic functions {f: D(0,1) \rightarrow {\bf C}} on the unit disk; this is a normed vector space with norm

\displaystyle  \|f\|_{H^\infty(D(0,1))} := \sup_{z \in D(0,1)} |f(z)|.

  • (i) If {f \in H^\infty(D(0,1))} is not identically zero, and {z_n} denote the zeroes of {f} in {D(0,1)} counting multiplicity, show that

    \displaystyle  \sum_n (1-|z_n|) < \infty

    and

    \displaystyle  \sup_{1/2 < r < 1} \int_0^{2\pi} | \log |f(re^{i\theta})| |\ d\theta < \infty.

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

    \displaystyle  B(z) := z^m \prod_{|z_n| \neq 0} \frac{|z_n|}{z_n} \frac{z_n-z}{1-\overline{z_n} z}

    where {m} is the order of vanishing of {f} at zero, show that this product converges absolutely to a holomorphic function on {D(0,1)}, and that {|f(z)| \leq \|f\|_{H^\infty(D(0,1)} |B(z)|} for all {z \in D(0,1)}. (It may be easier to work with finite Blaschke products first to obtain this bound.)
  • (iii) Continuing the notation from (i), establish a factorisation {f(z) = B(z) \exp(g(z))} for some holomorphic function {g: D(0,1) \rightarrow {\bf C}} with {\mathrm{Re}(g(z)) \leq \log \|f\|_{H^\infty(D(0,1)}} for all {z\in D(0,1)}.
  • (iv) (Theorem of F. and M. Riesz, special case) If {f \in H^\infty(D(0,1))} extends continuously to the boundary {\{e^{i\theta}: 0 \leq \theta < 2\pi\}}, show that the set {\{ 0 \leq \theta < 2\pi: f(e^{i\theta})=0 \}} has zero measure.

Remark 13 The factorisation (iii) can be refined further, with {g} 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 {H^\infty(D(0,1))} functions into “outer functions” and “inner functions”, giving the Beurling factorisation of {H^\infty}. There are also extensions to larger spaces {H^p(D(0,1))} than {H^\infty(D(0,1))} (which are to {H^\infty} as {L^p} is to {L^\infty}), 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 {f} be holomorphic on an open neighbourhood of a rectangle {R = \{ \sigma+it: \sigma_0 \leq \sigma \leq \sigma_1; 0 \leq t \leq T \}} for some {\sigma_0 < \sigma_1} and {T>0}, with {f} non-vanishing on the boundary of the rectangle. Show that

\displaystyle  2\pi \sum_\rho (\mathrm{Re}(\rho)-\sigma_0) = \int_0^T \log |f(\sigma_0+it)|\ dt - \int_0^T \log |f(\sigma_1+it)|\ dt

\displaystyle  + \int_{\sigma_0}^{\sigma_1} \mathrm{arg} f(\sigma+iT)\ d\sigma - \int_{\sigma_0}^{\sigma_1} \mathrm{arg} f(\sigma)\ d\sigma

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

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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:

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

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 {f: {\bf C} \rightarrow {\bf C}} be a polynomial of degree {n \geq 2} that has all zeroes in the closed unit disk {\{ z: |z| \leq 1 \}}. If {\lambda_0} is one of these zeroes, then {f'} has at least one zero in {\{z: |z-\lambda_0| \leq 1\}}.

It is common in the literature on this problem to normalise {f} to be monic, and to rotate the zero {\lambda_0} to be an element {a} of the unit interval {[0,1]}. As it turns out, the location of {a} on this unit interval {[0,1]} ends up playing an important role in the arguments.

Many cases of this conjecture are already known, for instance

In particular, in high degrees the only cases left uncovered by prior results are when {a} is close (but not too close) to {0}, or when {a} is close (but not too close) to {1}; see Figure 1 of my paper.

Our main result covers the high degree case uniformly for all values of {a \in [0,1]}:

Theorem 2 There exists an absolute constant {n_0} such that Sendov’s conjecture holds for all {n \geq n_0}.

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 {n_0}. I believe that the compactness arguments can be replaced with quantitative substitutes that provide an explicit {n_0}, but the value of {n_0} produced is likely to be extremely large (certainly much larger than {9}).

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 {a = o(1/\log n)}, Sendov’s conjecture holds for sufficiently large {n}.

Theorem 4 (Sendov’s conjecture near the unit circle) Under the additional hypothesis {1-o(1) \leq a \leq 1 - \varepsilon_0^n} for a fixed {\varepsilon_0>0}, Sendov’s conjecture holds for sufficiently large {n}.

We approach these theorems using the “compactness and contradiction” strategy, assuming that there is a sequence of counterexamples whose degrees {n} going to infinity, using various compactness theorems to extract various asymptotic objects in the limit {n \rightarrow \infty}, 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 {f} to Sendov’s conjecture, let {\lambda} be a zero of {f} (chosen uniformly at random among the {n} zeroes of {f}, counting multiplicity), and let {\zeta} similarly be a uniformly random zero of {f'}. We introduce the logarithmic potentials

\displaystyle  U_\lambda(z) := {\bf E} \log \frac{1}{|z-\lambda|}; \quad U_\zeta(z) := {\bf E} \log \frac{1}{|z-\zeta|}

and the Stieltjes transforms

\displaystyle  s_\lambda(z) := {\bf E} \frac{1}{z-\lambda}; \quad s_\zeta(z) := {\bf E} \log \frac{1}{z-\zeta}.

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

\displaystyle  U_\lambda(z) = \frac{1}{n} \log \frac{1}{|f(z)|}; \quad U_\zeta(z) = \frac{1}{n-1} \log \frac{n}{|f'(z)|}

and

\displaystyle  s_\lambda(z) = \frac{1}{n} \frac{f'(z)}{f(z)}; \quad s_\zeta(z) = \frac{1}{n-1} \frac{f''(z)}{f'(z)} \ \ \ \ \ (1)

and in particular the random variables {\lambda, \zeta} are linked to each other by the identity

\displaystyle  U_\lambda(z) - \frac{n-1}{n} U_\zeta(z) = \frac{1}{n} \log |s_\lambda(z)|. \ \ \ \ \ (2)

On the other hand, the hypotheses of Sendov’s conjecture (and the Gauss-Lucas theorem) place {\lambda,\zeta} inside the unit disk {\{ z:|z| \leq 1\}}. Applying Prokhorov’s theorem, and passing to a subsequence, one can then assume that the random variables {\lambda,\zeta} converge in distribution to some limiting random variables {\lambda^{(\infty)}, \zeta^{(\infty)}} (possibly defined on a different probability space than the original variables {\lambda,\zeta}), also living almost surely inside the unit disk. Standard potential theory then gives the convergence

\displaystyle  U_\lambda(z) \rightarrow U_{\lambda^{(\infty)}}(z); \quad U_\zeta(z) \rightarrow U_{\zeta^{(\infty)}}(z) \ \ \ \ \ (3)

and

\displaystyle  s_\lambda(z) \rightarrow s_{\lambda^{(\infty)}}(z); \quad s_\zeta(z) \rightarrow s_{\zeta^{(\infty)}}(z) \ \ \ \ \ (4)

at least in the local {L^1} sense. Among other things, we then conclude from the identity (2) and some elementary inequalities that

\displaystyle  U_{\lambda^{(\infty)}}(z) = U_{\zeta^{(\infty)}}(z)

for all {|z|>1}. 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 {\lambda^{(\infty)}} and one originating from {\zeta^{(\infty)}}, then the location where these Brownian motions first exit the unit disk {\{ z: |z| \leq 1 \}} 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 {\lambda^{(\infty)}}, {\zeta^{(\infty)}} 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 {a = o(1)}, then {\lambda^{(\infty)}, \zeta^{(\infty)}} almost surely lie in the semicircle {\{ e^{i\theta}: \pi/2 \leq \theta \leq 3\pi/2\}} and have the same distribution.
  • (ii) If {a = 1-o(1)}, then {\lambda^{(\infty)}} is uniformly distributed on the circle {\{ z: |z|=1\}}, and {\zeta^{(\infty)}} is almost surely zero.

In case (i) (and strengthening the hypothesis {a=o(1)} to {a=o(1/\log n)} to control some technical contributions of “outlier” zeroes of {f}), we can use this information about {\lambda^{(\infty)}} and (4) to ensure that the normalised logarithmic derivative {\frac{1}{n} \frac{f'}{f} = s_\lambda} 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 {f} has a zero at {a = o(1)} and that {f'} has no zeroes near {a}. 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 {f(z) = z^n - 1}, where the zeroes {\lambda} of {f} are uniformly distributed amongst the {n^{th}} roots of unity (including at {a=1}), and the zeroes of {f'} are all located at the origin. In my paper I also discuss a variant of this construction, in which {f'} has zeroes mostly near the origin, but also acquires a bounded number of zeroes at various locations {\lambda_1+o(1),\dots,\lambda_m+o(1)} inside the unit disk. Specifically, we take

\displaystyle  f(z) := \left(z + \frac{c_2}{n}\right)^{n-m} P(z) - \left(a + \frac{c_2}{n}\right)^{n-m} P(a)

where {a = 1 - \frac{c_1}{n}} for some constants {0 < c_1 < c_2} and

\displaystyle  P(z) := (z-\lambda_1) \dots (z-\lambda_m).

By a perturbative analysis to locate the zeroes of {f}, one eventually would be able to arrive at a true counterexample to Sendov’s conjecture if these locations {\lambda_1,\dots,\lambda_m} were in the open lune

\displaystyle  \{ \lambda: |\lambda| < 1 < |\lambda-1| \}

and if one had the inequality

\displaystyle  c_2 - c_1 - c_2 \cos \theta + \sum_{j=1}^m \log \left|\frac{1 - \lambda_j}{e^{i\theta} - \lambda_j}\right| < 0 \ \ \ \ \ (5)

for all {0 \leq \theta \leq 2\pi}. However, if one takes the mean of this inequality in {\theta}, one arrives at the inequality

\displaystyle  c_2 - c_1 + \sum_{j=1}^m \log |1 - \lambda_j| < 0

which is incompatible with the hypotheses {c_2 > c_1} and {|\lambda_j-1| > 1}. In order to extend this argument to more general polynomials {f}, we require a stability analysis of the endpoint equation

\displaystyle  c_2 - c_1 + c_2 \cos \theta + \sum_{j=1}^m \log \left|\frac{1 - \lambda_j}{e^{i\theta} - \lambda_j}\right| = 0 \ \ \ \ \ (6)

where we now only assume the closed conditions {c_2 \geq c_1} and {|\lambda_j-1| \geq 1}. The above discussion then places all the zeros {\lambda_j} on the arc

\displaystyle  \{ \lambda: |\lambda| < 1 = |\lambda-1|\} \ \ \ \ \ (7)

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

\displaystyle  \sum_{j=1}^m \lambda_j^2 = 0.

These two conditions are incompatible with each other (except in the degenerate case when all the {\lambda_j} vanish), because all the non-zero elements {\lambda} of the arc (7) have argument in {\pm [\pi/3,\pi/2]}, so in particular their square {\lambda^2} 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 {f''/f'}, which one then integrates and exponentiates to get good control on {f'}, and then on a second integration one gets enough information about {f} 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 {a} was extremely close to {1} via a similar perturbative analysis; the main novelty is to control the error terms not in terms of the magnitude of the largest zero {\zeta} of {f'} (which is difficult to manage when {n} gets large), but rather by the variance of those zeroes, which ends up being a more tractable expression to keep track of.

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 {\mu} on {{\bf R}^d} and a scale {r>0}, define the energy {\mathrm{E}(\mu,r)} at scale {r} to be the quantity

\displaystyle  \mathrm{E}(\mu,r) := \mu^4\left( \{ (x_1,x_2,x_3,x_4) \in ({\bf R}^d)^4: |x_1+x_2-x_3-x_4| \leq r \}\right) \ \ \ \ \ (1)

where {\mu^4} is the product measure on {({\bf R}^d)^4} formed from four copies of the measure {\mu} on {{\bf R}^d}. We will be interested in Cantor-type measures {\mu}, supported on a compact set {X \subset B(0,1)} and obeying the Ahlfors-David regularity condition

\displaystyle  \mu(B(x,r)) \leq C r^\delta

for all balls {B(x,r)} and some constants {C, \delta > 0}, as well as the matching lower bound

\displaystyle  \mu(B(x,r)) \geq C^{-1} r^\delta

when {x \in X} whenever {0 < r < 1}. One should think of {X} as a {\delta}-dimensional fractal set, and {\mu} as some vaguely self-similar measure on this set.

Note that once one fixes {x_1,x_2,x_3}, the variable {x_4} in (1) is constrained to a ball of radius {r}, hence we obtain the trivial upper bound

\displaystyle  \mathrm{E}(\mu,r) \leq C^4 r^\delta. \ \ \ \ \ (2)

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

Theorem 1 If {0 < \delta < d} is not an integer, and {X, \mu} are as above, then

\displaystyle  \mathrm{E}(\mu,r) \lesssim_{C,\delta,d} r^{\delta+\beta}

for some {\beta>0} depending only on {C,\delta,d}.

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 {\beta} we obtain is quasipolynomial in the regularity constant {C}:

\displaystyle  \beta = \exp\left( - O_{\delta,d}( 1 + \log^{O_{\delta,d}(1)}(C) ) \right).

(We also obtain a localised version in which the regularity condition is only required to hold at scales between {r} and {1}.) Such a result was previously obtained (with more explicit values of the {O_{\delta,d}()} implied constants) in the one-dimensional case {d=1} by Dyatlov and Zahl; but in higher dimensions there does not appear to have been any results for this general class of sets {X} and measures {\mu}. In the paper of Dyatlov and Zahl it is noted that some dependence on {C} is necessary; in particular, {\beta} cannot be much better than {1/\log C}. 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

\displaystyle  \mathrm{dim}( f( X, X) ) \geq \delta + \beta

for any non-degenerate smooth map {f: {\bf R}^d \times {\bf R}^d \rightarrow {\bf R}^d}, including the sum map {f(x,y) := x+y} and (in one dimension) the product map {f(x,y) := x \cdot y}, where the non-degeneracy condition required is that the gradients {D_x f(x,y), D_y f(x,y): {\bf R}^d \rightarrow {\bf R}^d} are invertible for every {x,y}. 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 {\beta}, namely

\displaystyle  \beta := c \min(\delta,1-\delta) C^{-25}.

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

\displaystyle  {\mathcal F}_h f(\xi) := (2\pi h)^{-d/2} \int_{{\bf R}^d} e^{-i x \cdot \xi/h} f(x)\ dx

where {h>0} is a small parameter. If {X, \mu, \delta} are as above, and {X_h} denotes the {h}-neighbourhood of {X}, then from the Hausdorff-Young inequality one obtains the trivial bound

\displaystyle  \| 1_{X_h} {\mathcal F}_h 1_{X_h} \|_{L^2({\bf R}^d) \rightarrow L^2({\bf R}^d)} \lesssim_{C,d} h^{\max\left(\frac{d}{2}-\delta,0\right)}.

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

\displaystyle  \| 1_{X_h} {\mathcal F}_h 1_{X_h} \|_{L^2({\bf R}^d) \rightarrow L^2({\bf R}^d)} \lesssim_{C,d} h^{\max\left(\frac{d}{2}-\delta,0\right) + \beta}

for some {\beta>0}; informally, this asserts that a function and its Fourier transform cannot simultaneously be concentrated in the set {X_h} when {\delta \leq \frac{d}{2}}, and that a function cannot be concentrated on {X_h} and have its Fourier transform be of maximum size on {X_h} when {\delta \geq \frac{d}{2}}. A modification of the disk example mentioned previously shows that such a fractal uncertainty principle cannot hold if {\delta} is an integer. However, in one dimension, the fractal uncertainty principle is known to hold for all {0 < \delta < 1}. The above-mentioned results of Dyatlov and Zahl were able to establish this for {\delta} close to {1/2}, and the remaining cases {1/2 < \delta < 1} and {0 < \delta < 1/2} 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 {\delta} is close to {d/2}, and {d} is assumed to be odd (to make {d/2} a non-integer). There is also work of Han and Schlag that obtains such a principle when one of the copies of {X_h} 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

\displaystyle  \mathrm{E}(\mu,r_0) \leq \varepsilon r_0^\delta \ \ \ \ \ (3)

over the trivial bound (2) for any small {\varepsilon>0}, provided {r_0} is sufficiently small depending on {\varepsilon, \delta, d}; one can then iterate this bound by a fairly standard “induction on scales” argument (which roughly speaking can be used to show that energies {\mathrm{E}(\mu,r)} behave somewhat multiplicatively in the scale parameter {r}) 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 {U^2}, and particularly via a clean Fubini-type inequality

\displaystyle  \| f \|_{U^2(V \times V')} \leq \|f\|_{U^2(V; U^2(V'))}

(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 {X} that supports {\mu}: intervals {[x, x+K^{-n}]} that intersect {X}, but such that a large interval {[x-K^{-n+1},x]} just to the left of this interval is disjoint from {X}. Here {K} is a large constant and {n} 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 {0 < \delta < 1} then left edges exist in abundance at every scale; for instance most points of {X} would be expected to lie in quite a few of these left edges (much as most elements of, say, the ternary Cantor set {\{ \sum_{n=1}^\infty \varepsilon_n 3^{-n} \varepsilon_n \in \{0,1\} \}} would be expected to contain a lot of {0}s in their base {3} expansion). In particular, most pairs {(x_1,x_2) \in X \times X} would be expected to lie in a pair {[x,x+K^{-n}] \times [y,y+K^{-n}]} of left edges of equal length. The key point is then that if {(x_1,x_2) \in X \times X} lies in such a pair with {K^{-n} \geq r}, then there are relatively few pairs {(x_3,x_4) \in X \times X} at distance {O(K^{-n+1})} from {(x_1,x_2)} for which one has the relation {x_1+x_2 = x_3+x_4 + O(r)}, because {x_3,x_4} will both tend to be to the right of {x_1,x_2} respectively. This causes a decrement in the energy at scale {K^{-n+1}}, 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 {0 < \delta < 1} and {d-1 < \delta < d} 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 {X} has high correlation with an approximate group {H}, and hence (by the aforementioned Bogulybov-Ruzsa type theorem of Sanders, which is the main source of the quasipolynomial bounds in our final exponent) {X} will exhibit an approximate “symmetry” along some non-trivial arithmetic progression of some spacing length {r} and some diameter {R \gg r}. The {r}-neighbourhood {X_r} of {X} will then resemble the union of parallel “cylinders” of dimensions {r \times R}. If we focus on a typical {R}-ball of {X_r}, the set now resembles a Cartesian product of an interval of length {R} with a subset of a {d-1}-dimensional hyperplane, which behaves approximately like an Ahlfors-David regular set of dimension {\delta-1} (this already lets us conclude a contradiction if {\delta<1}). Note that if the original dimension {\delta} was non-integer then this new dimension {\delta-1} 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.

Consider a disk {D(z_0,r) := \{ z: |z-z_0| < r \}} in the complex plane. If one applies an affine-linear map {f(z) = az+b} to this disk, one obtains

\displaystyle  f(D(z_0,r)) = D(f(z_0), |f'(z_0)| r).

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 {D(z_0,r)} be a disk in the complex plane, and {f: D(z_0,r) \rightarrow {\bf C}} be a holomorphic function with {f'(z_0) \neq 0}.
  • (i) (Open mapping theorem or inverse function theorem) {f(D(z_0,r))} contains a disk {D(f(z_0),\varepsilon)} for some {\varepsilon>0}. (In fact there is even a holomorphic right inverse of {f} from {D(f(z_0), \varepsilon)} to {D(z_0,r)}.)
  • (ii) (Bloch theorem) {f(D(z_0,r))} contains a disk {D(w, c |f'(z_0)| r)} for some absolute constant {c>0} and some {w \in {\bf C}}. (In fact there is even a holomorphic right inverse of {f} from {D(w, c |f'(z_0)| r)} to {D(z_0,r)}.)
  • (iii) (Koebe quarter theorem) If {f} is injective, then {f(D(z_0,r))} contains the disk {D(f(z_0), \frac{1}{4} |f'(z_0)| r)}.
  • (iv) If {f} is a polynomial of degree {n}, then {f(D(z_0,r))} contains the disk {D(f(z_0), \frac{1}{n} |f'(z_0)| r)}.
  • (v) If one has a bound of the form {|f'(z)| \leq A |f'(z_0)|} for all {z \in D(z_0,r)} and some {A>1}, then {f(D(z_0,r))} contains the disk {D(f(z_0), \frac{c}{A} |f'(z_0)| r)} for some absolute constant {c>0}. (In fact there is holomorphic right inverse of {f} from {D(f(z_0), \frac{c}{A} |f'(z_0)| r)} to {D(z_0,r)}.)

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 {w \in D(f(z_0), \frac{1}{n} |f'(z_0)| r)}, then we have a lower bound for the log-derivative of {f(z)-w} at {z_0}:

\displaystyle  \frac{|f'(z_0)|}{|f(z_0)-w|} > \frac{n}{r}

(with the convention that the left-hand side is infinite when {f(z_0)=w}). But by the fundamental theorem of algebra we have

\displaystyle  \frac{f'(z_0)}{f(z_0)-w} = \sum_{j=1}^n \frac{1}{z_0-\zeta_j}

where {\zeta_1,\dots,\zeta_n} are the roots of the polynomial {f(z)-w} (counting multiplicity). By the pigeonhole principle, there must therefore exist a root {\zeta_j} of {f(z) - w} such that

\displaystyle  \frac{1}{|z_0-\zeta_j|} > \frac{1}{r}

and hence {\zeta_j \in D(z_0,r)}. Thus {f(D(z_0,r))} contains {w}, and the claim follows. \Box

The constant {\frac{1}{n}} in (iv) is completely sharp: if {f(z) = z^n} and {z_0} is non-zero then {f(D(z_0,|z_0|))} contains the disk

\displaystyle D(f(z_0), \frac{1}{n} |f'(z_0)| r) = D( z_0^n, |z_0|^n)

but avoids the origin, thus does not contain any disk of the form {D( z_0^n, |z_0|^n+\varepsilon)}. This example also shows that despite parts (ii), (iii) of the theorem, one cannot hope for a general inclusion of the form

\displaystyle  f(D(z_0,r)) \supset D(f(z_0), c |f'(z_0)| r )

for an absolute constant {c>0}.

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 {f''(z) = O(\frac{A}{r} |f'(z_0)|)} for {z \in D(z_0,r/2)}, hence by Taylor’s theorem with remainder {f(z) = f(z_0) + f'(z_0) (z-z_0) (1 + O( A \frac{|z-z_0|}{r} ) )} for {z \in D(z_0, r/2)}. By Rouche’s theorem, this implies that the function {f(z)-w} has a unique zero in {D(z_0, 2cr/A)} for any {w \in D(f(z_0), cr|f'(z_0)|/A)}, if {c>0} is a sufficiently small absolute constant. The claim follows. \Box

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 {r} slightly if necessary we may assume that {f} extends analytically to the closure of the disk {D(z_0,r)}. Let {c} be the constant in (v) with {A=2}; we will prove (iii) with {c} replaced by {c/2}. If we have {|f'(z)| \leq 2 |f'(z_0)|} for all {z \in D(z_0,r/2)} then we are done by (v), so we may assume without loss of generality that there is {z_1 \in D(z_0,r/2)} such that {|f'(z_1)| > 2 |f'(z_0)|}. If {|f'(z)| \leq 2 |f'(z_1)|} for all {z \in D(z_1,r/4)} then by (v) we have

\displaystyle  f( D(z_0, r) ) \supset f( D(z_1,r/2) ) \supset D( f(z_1), \frac{c}{2} |f'(z_1)| \frac{r}{2} )

\displaystyle \supset D( f(z_1), \frac{c}{2} |f'(z_0)| r )

and we are again done. Hence we may assume without loss of generality that there is {z_2 \in D(z_1,r/4)} such that {|f'(z_2)| > 2 |f'(z_1)|}. Iterating this procedure in the obvious fashion we either are done, or obtain a Cauchy sequence {z_0, z_1, \dots} in {D(z_0,r)} such that {f'(z_j)} goes to infinity as {j \rightarrow \infty}, which contradicts the analytic nature of {f} (and hence continuous nature of {f'}) on the closure of {D(z_0,r)}. This gives the claim. \Box

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 {D(z_0,r)} be a disk in the complex plane, and {f: D(z_0,r) \rightarrow {\bf C}} be a polynomial of degree {n \geq 1} with {f'(z) \neq 0} for all {z \in D(z_0,r)}. Then {f} is injective on {D(z_0, \sin\frac{\pi}{n})}.

The radius {\sin \frac{\pi}{n}} is best possible, for the polynomial {f(z) = z^n} has {f'} non-vanishing on {D(1,1)}, but one has {f(\cos(\pi/n) e^{i \pi/n}) = f(\cos(\pi/n) e^{-i\pi/n})}, and {\cos(\pi/n) e^{i \pi/n}, \cos(\pi/n) e^{-i\pi/n}} lie on the boundary of {D(1,\sin \frac{\pi}{n})}.

If one narrows {\sin \frac{\pi}{n}} slightly to {\sin \frac{\pi}{2n}} then one can quickly prove this proposition as follows. Suppose for contradiction that there exist distinct {z_1, z_2 \in D(z_0, \sin\frac{\pi}{n})} with {f(z_1)=f(z_2)}, thus if we let {\gamma} be the line segment contour from {z_1} to {z_2} then {\int_\gamma f'(z)\ dz}. However, by assumption we may factor {f'(z) = c (z-\zeta_1) \dots (z-\zeta_{n-1})} where all the {\zeta_j} lie outside of {D(z_0,r)}. Elementary trigonometry then tells us that the argument of {z-\zeta_j} only varies by less than {\frac{\pi}{n}} as {z} traverses {\gamma}, hence the argument of {f'(z)} only varies by less than {\pi}. Thus {f'(z)} takes values in an open half-plane avoiding the origin and so it is not possible for {\int_\gamma f'(z)\ dz} to vanish.

To recover the best constant of {\sin \frac{\pi}{n}} 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 {f: {\bf C} \rightarrow {\bf C}} be a polynomial of degree {n \geq 1} such that {f(1)=f(-1)}. Then {f'} contains a zero in the closure of {D( 0, \cot \frac{\pi}{n} )}.

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

\displaystyle  (f,g)_n := \sum_{k=0}^n (-1)^k f^{(k)}(0) g^{(n-k)}(0). \ \ \ \ \ (1)

Theorem 4 (Grace’s theorem) Let {C} be a circle or line in {{\bf C}}, dividing {{\bf C} \backslash C} into two open connected regions {\Omega_1, \Omega_2}. Let {f,g} be two polynomials of degree at most {n \geq 1}, with all the zeroes of {f} lying in {\Omega_1} and all the zeroes of {g} lying in {\Omega_2}. Then {(f,g)_n \neq 0}.

(Contrapositively: if {(f,g)_n=0}, then the zeroes of {f} cannot be separated from the zeroes of {g} by a circle or line.)

Indeed, a brief calculation reveals the identity

\displaystyle  f(1) - f(-1) = (f', g)_{n-1}

where {g} is the degree {n-1} polynomial

\displaystyle  g(z) := \frac{1}{n!} ((z+1)^n - (z-1)^n).

The zeroes of {g} are {i \cot \frac{\pi j}{n}} for {j=1,\dots,n-1}, so the Grace-Heawood theorem follows by applying Grace’s theorem with {C} equal to the boundary of {D(0, \cot \frac{\pi}{n})}.

The same method of proof gives the following nice consequence:

Theorem 5 (Perpendicular bisector theorem) Let {f: {\bf C} \rightarrow C} be a polynomial such that {f(z_1)=f(z_2)} for some distinct {z_1,z_2}. Then the zeroes of {f'} cannot all lie on one side of the perpendicular bisector of {z_1,z_2}. For instance, if {f(1)=f(-1)}, then the zeroes of {f'} cannot all lie in the halfplane {\{ z: \mathrm{Re} z > 0 \}} or the halfplane {\{ z: \mathrm{Re} z < 0 \}}.

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 {n=1} can be established by direct computation, so suppose inductively that {n>1} and that the claim has already been established for {n-1}. 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 {V_n} denote the vector space of polynomials {f} of degree at most {n}, then the apolar form is a bilinear form {(,)_n: V_n \times V_n \rightarrow {\bf C}}. Each translation {z \mapsto z+a} on the complex plane induces a corresponding map on {V_n}, mapping each polynomial {f} to its shift {\tau_a f(z) := f(z-a)}. We claim that the apolar form is invariant with respect to these translations:

\displaystyle  ( \tau_a f, \tau_a g )_n = (f,g)_n.

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

\displaystyle  (f', g)_n + (f,g')_n = 0

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

Next, we see that the inversion map {z \mapsto 1/z} also induces a corresponding map on {V_n}, mapping each polynomial {f \in V_n} to its inversion {\iota f(z) := z^n f(1/z)}. From (1) we see that this map also (projectively) preserves the apolar form:

\displaystyle  (\iota f, \iota g)_n = (-1)^n (f,g)_n.

More generally, the group of Möbius transformations on the Riemann sphere acts projectively on {V_n}, with each Möbius transformation {T: {\bf C} \rightarrow {\bf C}} mapping each {f \in V_n} to {Tf(z) := g_T(z) f(T^{-1} z)}, where {g_T} is the unique (up to constants) rational function that maps this a map from {V_n} to {V_n} (its divisor is {n(T \infty) - n(\infty)}). 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 {T} on {V_n} also moves the zeroes of each {f \in V_n} by {T} (viewing polynomials of degree less than {n} in {V_n} 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 {f} to infinity (thus making {f} a polynomial of degree {n-1}), so that {C} must now be a circle, with the zeroes of {g} inside the circle and the remaining zeroes of {f} outside the circle. But then

\displaystyle  (f,g)_n = (f, g')_{n-1}.

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

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 {f(n)} on an interval {\{1,\dots,N\}} as an “irrational nilsequence” {F(g(n) \Gamma)} 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 {{\mathbb E}_{{\bf n} \in {\bf Z}^d \cap P} f(\psi_1({\bf n})) \dots f(\psi_t({\bf n}))} for various affine-linear forms {\psi_1,\dots,\psi_t} when the functions {f} 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 {\psi_1,\dots,\psi_t}. 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 {\Psi^{[i]}} for various natural numbers {i}, defined as the linear span of the vectors {(\psi^i_1({\bf n}), \dots, \psi^i_t({\bf n}))} as {{\bf n}} ranges over the parameter space {{\bf Z}^d}. Roughly speaking, these spaces encode some constraints one would expect to see amongst the forms {\psi^i_1({\bf n}), \dots, \psi^i_t({\bf n})}. For instance, in the case of length four arithmetic progressions when {d=2}, {{\bf n} = (n,r)}, and

\displaystyle  \psi_i({\bf n}) = n + (i-1)r

for {i=1,2,3,4}, then {\Psi^{[1]}} is spanned by the vectors {(1,1,1,1)} and {(1,2,3,4)} and can thus be described as the two-dimensional linear space

\displaystyle  \Psi^{[1]} = \{ (a,b,c,d): a-2b+c = b-2c+d = 0\} \ \ \ \ \ (1)

while {\Psi^{[2]}} is spanned by the vectors {(1,1,1,1)}, {(1,2,3,4)}, {(1^2,2^2,3^2,4^2)} and can be described as the hyperplane

\displaystyle  \Psi^{[2]} = \{ (a,b,c,d): a-3b+3c-d = 0 \}. \ \ \ \ \ (2)

As a special case of the counting lemma, we can check that if {f} takes the form {f(n) = F( \alpha n, \beta n^2 + \gamma n)} for some irrational {\alpha,\beta \in {\bf R}/{\bf Z}}, some arbitrary {\gamma \in {\bf R}/{\bf Z}}, and some smooth {F: {\bf R}/{\bf Z} \times {\bf R}/{\bf Z} \rightarrow {\bf C}}, then the limiting value of the average

\displaystyle  {\bf E}_{n, r \in [N]} f(n) f(n+r) f(n+2r) f(n+3r)

as {N \rightarrow \infty} is equal to

\displaystyle  \int_{a_1,b_1,c_1,d_1 \in {\bf R}/{\bf Z}: a_1-2b_1+c_1=b_1-2c_1+d_1=0} \int_{a_2,b_2,c_2,d_2 \in {\bf R}/{\bf Z}: a_2-3b_2+3c_2-d_2=0}

\displaystyle  F(a_1,a_2) F(b_1,b_2) F(c_1,c_2) F(d_1,d_2)

which reflects the constraints

\displaystyle  \alpha n - 2 \alpha(n+r) + \alpha(n+2r) = \alpha(n+r) - 2\alpha(n+2r)+\alpha(n+3r)=0

and

\displaystyle  (\beta n^2 + \gamma n) - 3 (\beta(n+r)^2+\gamma(n+r))

\displaystyle + 3 (\beta(n+2r)^2 +\gamma(n+2r)) - (\beta(n+3r)^2+\gamma(n+3r))=0.

These constraints follow from the descriptions (1), (2), using the containment {\Psi^{[1]} \subset \Psi^{[2]}} to dispense with the lower order term {\gamma n} (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 {\Psi^{[i]} \subset \Psi^{[i+1]}} for all {i}. The problem is that the flag property turns out to not always hold. A counterexample, provided by Daniel Altman, involves the four linear forms

\displaystyle  \psi_1(n,r) = r; \psi_2(n,r) = 2n+2r; \psi_3(n,r) = n+3r; \psi_4(n,r) = n.

Here it turns out that

\displaystyle  \Psi^{[1]} = \{ (a,b,c,d): d-c=3a; b-2a=2d\}

and

\displaystyle  \Psi^{[2]} = \{ (a,b,c,d): 24a+3b-4c-8d=0 \}

and {\Psi^{[1]}} is no longer contained in {\Psi^{[2]}}. The analogue of the asymptotic formula given previously for {f(n) = F( \alpha n, \beta n^2 + \gamma n)} is then valid when {\gamma} vanishes, but not when {\gamma} is non-zero, because the identity

\displaystyle  24 (\beta \psi_1(n,r)^2 + \gamma \psi_1(n,r)) + 3 (\beta \psi_2(n,r)^2 + \gamma \psi_2(n,r))

\displaystyle - 4 (\beta \psi_3(n,r)^2 + \gamma \psi_3(n,r)) - 8 (\beta \psi_4(n,r)^2 + \gamma \psi_4(n,r)) = 0

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 {\Psi^{[1]}} contains {(1,\dots,1)}, 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.

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.

Rachel Greenfeld and I have just uploaded to the arXiv our paper “The structure of translational tilings in {{\bf Z}^d}“. This paper studies the tilings {1_F * 1_A = 1} of a finite tile {F} in a standard lattice {{\bf Z}^d}, that is to say sets {A \subset {\bf Z}^d} (which we call tiling sets) such that every element of {{\bf Z}^d} lies in exactly one of the translates {a+F, a \in A} of {F}. We also consider more general tilings of level {k} {1_F * 1_A = k} for a natural number {k} (several of our results consider an even more general setting in which {1_F * 1_A} is periodic but allowed to be non-constant).

In many cases the tiling set {A} will be periodic (by which we mean translation invariant with respect to some lattice (a finite index subgroup) of {{\bf Z}^d}). For instance one simple example of a tiling is when {F \subset {\bf Z}^2} is the unit square {F = \{0,1\}^2} and {A} is the lattice {2{\bf Z}^2 = \{ 2x: x \in {\bf Z}^2\}}. However one can modify some tilings to make them less periodic. For instance, keeping {F = \{0,1\}^2} one also has the tiling set

\displaystyle  A = \{ (2x, 2y+a(x)): x,y \in {\bf Z} \}

where {a: {\bf Z} \rightarrow \{0,1\}} is an arbitrary function. This tiling set is periodic in a single direction {(0,2)}, but is not doubly periodic. For the slightly modified tile {F = \{0,1\} \times \{0,2\}}, the set

\displaystyle  A = \{ (2x, 4y+2a(x)): x,y \in {\bf Z} \} \cup \{ (2x+b(y), 4y+1): x,y \in {\bf Z}\}

for arbitrary {a,b: {\bf Z} \rightarrow \{0,1\}} 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 {F \subset {\bf Z}^d} 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 {d=1} 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 {d=2} case was settled more recently by Bhattacharya, but the higher dimensional cases {d > 2} 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 {|F|} of the tile is bounded:

Theorem 2 (Quantitative periodic tiling in {{\bf Z}^2}) If a finite tile {F \subset {\bf Z}^2} has at least one tiling set, then it has a tiling set which is {M{\bf Z}^2}-periodic for some {M \ll_{|F|} \mathrm{diam}(F)^{O(|F|^4)}}.

Among other things, this shows that the problem of deciding whether a given subset of {{\bf Z}^2} of bounded cardinality tiles {{\bf Z}^2} or not is in the NP complexity class with respect to the diameter {\mathrm{diam}(F)}. (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 {{\bf Z}^2}) Every tiling set of a finite tile {F \subset {\bf Z}^2} is weakly periodic. In fact, the tiling set is the union of at most {|F|-1} disjoint sets, each of which is periodic in a direction of magnitude {O_{|F|}( \mathrm{diam}(F)^{O(|F|^2)})}.

We also have a new bound for the periodicity of tilings in {{\bf Z}}:

Theorem 4 (Universal period for tilings in {{\bf Z}}) Let {F \subset {\bf Z}} be finite, and normalized so that {0 \in F}. Then every tiling set of {F} is {qn}-periodic, where {q} is the least common multiple of all primes up to {2|F|}, and {n} is the least common multiple of the magnitudes {|f|} of all {f \in F \backslash \{0\}}.

We remark that the current best complexity bound of determining whether a subset of {{\bf Z}} tiles {{\bf Z}} or not is {O( \exp(\mathrm{diam}(F)^{1/3+o(1)}))}, 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 {F \subset {\bf Z}^2} and a level {4} tiling {1_F * 1_A = 4} such that {A} is not weakly periodic.

We do not know if there is a corresponding counterexample to the higher level periodic tiling conjecture (that if {F} tiles {{\bf Z}^d} at level {k}, then there is a periodic tiling at the same level {k}). Note that it is important to keep the level fixed, since one trivially always has a periodic tiling at level {|F|} from the identity {1_F * 1 = |F|}.

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 {A} is a tiling of a finite tile {F \subset {\bf Z}^d}. Then {A} is also a tiling of the dilated tile {rF} for any {r} coprime to {n}, where {n} is the least common multiple of all the primes up to {|F|}.

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 {r} is a prime {p>|F|}. We need to show that {1_{pF} * 1_A = 1}. It suffices to show the claim {1_{pF} * 1_A = 1 \hbox{ mod } p}, since both sides take values in {\{0,\dots,|F|\} \subset \{0,\dots,p-1\}}. The convolution algebra {{\bf F}_p[{\bf Z}^d]} (or group algebra) of finitely supported functions from {{\bf Z}^d} to {{\bf F}_p} is a commutative algebra of characteristic {p}, so we have the Frobenius identity {(f+g)^{*p} = f^{*p} + g^{*p}} for any {f,g}. As a consequence we see that {1_{pF} = 1_F^{*p} \hbox{ mod } p}. The claim now follows by convolving the identity {1_F * 1_A = 1 \hbox{ mod } p} by {p-1} further copies of {1_F}.

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 {r} 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 {A} is a tiling of a finite tile {F \subset {\bf Z}^d}, where we normalize {0 \in F}. Then we have a decomposition

\displaystyle  1_A = 1 - \sum_{f \in F \backslash 0} \varphi_f \ \ \ \ \ (1)

where each {\varphi_f: {\bf Z}^d \rightarrow [0,1]} is a function that is periodic in the direction {nf}, where {n} is the least common multiple of all the primes up to {|F|}.

Proof: From Lemma 6 we have {1_A = 1 - \sum_{f \in F \backslash 0} \delta_{rf} * 1_A} for any {r = 1 \hbox{ mod } n}, where {\delta_{rf}} is the Kronecker delta at {rf}. Now average over {r} (extracting a weak limit or generalised limit as necessary) to obtain the conclusion. \Box

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

\displaystyle  \sum_{f \in F \backslash 0} \varphi_f = 0 \hbox{ mod } 1

which in two dimensions in particular puts a lot of structure on each individual {\varphi_f} (roughly speaking it makes the {\varphi_f \hbox{ mod } 1} behave in a polynomial fashion, after collecting commensurable terms). On the other hand we have the inequality

\displaystyle  \sum_{f \in F \backslash 0} \varphi_f \leq 1 \ \ \ \ \ (2)

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 {k} tilings the analogue of (2) becomes

\displaystyle  \sum_{f \in F \backslash 0} \varphi_f \leq k

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.

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