In the previous set of notes we saw that functions ${f: U \rightarrow {\bf C}}$ that were holomorphic on an open set ${U}$ enjoyed a large number of useful properties, particularly if the domain ${U}$ was simply connected. In many situations, though, we need to consider functions ${f}$ that are only holomorphic (or even well-defined) on most of a domain ${U}$, thus they are actually functions ${f: U \backslash S \rightarrow {\bf C}}$ outside of some small singular set ${S}$ inside ${U}$. (In this set of notes we only consider interior singularities; one can also discuss singular behaviour at the boundary of ${U}$, but this is a whole separate topic and will not be pursued here.) Since we have only defined the notion of holomorphicity on open sets, we will require the singular sets ${S}$ to be closed, so that the domain ${U \backslash S}$ on which ${f}$ remains holomorphic is still open. A typical class of examples are the functions of the form ${\frac{f(z)}{z-z_0}}$ that were already encountered in the Cauchy integral formula; if ${f: U \rightarrow {\bf C}}$ is holomorphic and ${z_0 \in U}$, such a function would be holomorphic save for a singularity at ${z_0}$. Another basic class of examples are the rational functions ${P(z)/Q(z)}$, which are holomorphic outside of the zeroes of the denominator ${Q}$.

Singularities come in varying levels of “badness” in complex analysis. The least harmful type of singularity is the removable singularity – a point ${z_0}$ which is an isolated singularity (i.e., an isolated point of the singular set ${S}$) where the function ${f}$ is undefined, but for which one can extend the function across the singularity in such a fashion that the function becomes holomorphic in a neighbourhood of the singularity. A typical example is that of the complex sinc function ${\frac{\sin(z)}{z}}$, which has a removable singularity at the origin ${0}$, which can be removed by declaring the sinc function to equal ${1}$ at ${0}$. The detection of isolated removable singularities can be accomplished by Riemann’s theorem on removable singularities (Exercise 35 from Notes 3): if a holomorphic function ${f: U \backslash S \rightarrow {\bf C}}$ is bounded near an isolated singularity ${z_0 \in S}$, then the singularity at ${z_0}$ may be removed.

After removable singularities, the mildest form of singularity one can encounter is that of a pole – an isolated singularity ${z_0}$ such that ${f(z)}$ can be factored as ${f(z) = \frac{g(z)}{(z-z_0)^m}}$ for some ${m \geq 1}$ (known as the order of the pole), where ${g}$ has a removable singularity at ${z_0}$ (and is non-zero at ${z_0}$ once the singularity is removed). Such functions have already made a frequent appearance in previous notes, particularly the case of simple poles when ${m=1}$. The behaviour near ${z_0}$ of function ${f}$ with a pole of order ${m}$ is well understood: for instance, ${|f(z)|}$ goes to infinity as ${z}$ approaches ${z_0}$ (at a rate comparable to ${|z-z_0|^{-m}}$). These singularities are not, strictly speaking, removable; but if one compactifies the range ${{\bf C}}$ of the holomorphic function ${f: U \backslash S \rightarrow {\bf C}}$ to a slightly larger space ${{\bf C} \cup \{\infty\}}$ known as the Riemann sphere, then the singularity can be removed. In particular, functions ${f: U \backslash S \rightarrow {\bf C}}$ which only have isolated singularities that are either poles or removable can be extended to holomorphic functions ${f: U \rightarrow {\bf C} \cup \{\infty\}}$ to the Riemann sphere. Such functions are known as meromorphic functions, and are nearly as well-behaved as holomorphic functions in many ways. In fact, in one key respect, the family of meromorphic functions is better: the meromorphic functions on ${U}$ turn out to form a field, in particular the quotient of two meromorphic functions is again meromorphic (if the denominator is not identically zero).

Unfortunately, there are isolated singularities that are neither removable or poles, and are known as essential singularities. A typical example is the function ${f(z) = e^{1/z}}$, which turns out to have an essential singularity at ${z=0}$. The behaviour of such essential singularities is quite wild; we will show here the Casorati-Weierstrass theorem, which shows that the image of ${f}$ near the essential singularity is dense in the complex plane, as well as the more difficult great Picard theorem which asserts that in fact the image can omit at most one point in the complex plane. Nevertheless, around any isolated singularity (even the essential ones) ${z_0}$, it is possible to expand ${f}$ as a variant of a Taylor series known as a Laurent series ${\sum_{n=-\infty}^\infty a_n (z-z_0)^n}$. The ${\frac{1}{z-z_0}}$ coefficient ${a_{-1}}$ of this series is particularly important for contour integration purposes, and is known as the residue of ${f}$ at the isolated singularity ${z_0}$. These residues play a central role in a common generalisation of Cauchy’s theorem and the Cauchy integral formula known as the residue theorem, which is a particularly useful tool for computing (or at least transforming) contour integrals of meromorphic functions, and has proven to be a particularly popular technique to use in analytic number theory. Within complex analysis, one important consequence of the residue theorem is the argument principle, which gives a topological (and analytical) way to control the zeroes and poles of a meromorphic function.

Finally, there are the non-isolated singularities. Little can be said about these singularities in general (for instance, the residue theorem does not directly apply in the presence of such singularities), but certain types of non-isolated singularities are still relatively easy to understand. One particularly common example of such non-isolated singularity arises when trying to invert a non-injective function, such as the complex exponential ${z \mapsto \exp(z)}$ or a power function ${z \mapsto z^n}$, leading to branches of multivalued functions such as the complex logarithm ${z \mapsto \log(z)}$ or the ${n^{th}}$ root function ${z \mapsto z^{1/n}}$ respectively. Such branches will typically have a non-isolated singularity along a branch cut; this branch cut can be moved around the complex domain by switching from one branch to another, but usually cannot be eliminated entirely, unless one is willing to lift up the domain ${U}$ to a more general type of domain known as a Riemann surface. As such, one can view branch cuts as being an “artificial” form of singularity, being an artefact of a choice of local coordinates of a Riemann surface, rather than reflecting any intrinsic singularity of the function itself. The further study of Riemann surfaces is an important topic in complex analysis (as well as the related fields of complex geometry and algebraic geometry), but unfortunately this topic will probably be postponed to the next course in this sequence (which I will not be teaching).

We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute (or at least transform) a large number of contour integrals ${\int_\gamma f(z)\ dz}$ even without knowing any explicit antiderivative of ${f}$. There are many forms and variants of Cauchy’s theorem. To give one such version, we need the basic topological notion of a homotopy:

Definition 1 (Homotopy) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${\gamma_0: [a,b] \rightarrow U}$, ${\gamma_1: [a,b] \rightarrow U}$ be two curves in ${U}$.

• (i) If ${\gamma_0, \gamma_1}$ have the same initial point ${z_0}$ and final point ${z_1}$, we say that ${\gamma_0}$ and ${\gamma_1}$ are homotopic with fixed endpoints in ${U}$ if there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$, and such that ${\gamma(s,a) = z_0}$ and ${\gamma(s,b) = z_1}$ for all ${s \in [0,1]}$.
• (ii) If ${\gamma_0, \gamma_1}$ are closed (but possibly with different initial points), we say that ${\gamma_0}$ and ${\gamma_1}$ are homotopic as closed curves in ${U}$ if there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$, and such that ${\gamma(s,a) = \gamma(s,b)}$ for all ${s \in [0,1]}$.
• (iii) If ${\gamma_2: [c,d] \rightarrow U}$ and ${\gamma_3: [e,f] \rightarrow U}$ are curves with the same initial point and same final point, we say that ${\gamma_2}$ and ${\gamma_3}$ are homotopic with fixed endpoints up to reparameterisation in ${U}$ if there is a reparameterisation ${\tilde \gamma_2: [a,b] \rightarrow U}$ of ${\gamma_2}$ which is homotopic with fixed endpoints in ${U}$ to a reparameterisation ${\tilde \gamma_3: [a,b] \rightarrow U}$ of ${\gamma_3}$.
• (iv) If ${\gamma_2: [c,d] \rightarrow U}$ and ${\gamma_3: [e,f] \rightarrow U}$ are closed curves, we say that ${\gamma_2}$ and ${\gamma_3}$ are homotopic as closed curves up to reparameterisation in ${U}$ if there is a reparameterisation ${\tilde \gamma_2: [a,b] \rightarrow U}$ of ${\gamma_2}$ which is homotopic as closed curves in ${U}$ to a reparameterisation ${\tilde \gamma_3: [a,b] \rightarrow U}$ of ${\gamma_3}$.

In the first two cases, the map ${\gamma}$ will be referred to as a homotopy from ${\gamma_0}$ to ${\gamma_1}$, and we will also say that ${\gamma_0}$ can be continously deformed to ${\gamma_1}$ (either with fixed endpoints, or as closed curves).

Example 2 If ${U}$ is a convex set, that is to say that ${(1-s) z_0 + s z_1 \in U}$ whenever ${z_0,z_1 \in U}$ and ${0 \leq s \leq 1}$, then any two curves ${\gamma_0, \gamma_1: [0,1] \rightarrow U}$ from one point ${z_0}$ to another ${z_1}$ are homotopic, by using the homotopy

$\displaystyle \gamma(s,t) := (1-s) \gamma_0(t) + s \gamma_1(t).$

For a similar reason, in a convex open set ${U}$, any two closed curves will be homotopic to each other as closed curves.

Exercise 3 Let ${U}$ be an open subset of ${{\bf C}}$.

• (i) Prove that the property of being homotopic with fixed endpoints in ${U}$ is an equivalence relation.
• (ii) Prove that the property of being homotopic as closed curves in ${U}$ is an equivalence relation.
• (iii) If ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are closed curves with the same initial point, show that ${\gamma_0}$ is homotopic to ${\gamma_1}$ as closed curves if and only if ${\gamma_0}$ is homotopic to ${\gamma_2 + \gamma_1 + (-\gamma_2)}$ with fixed endpoints for some closed curve ${\gamma_2}$ with the same initial point as ${\gamma_0}$ or ${\gamma_1}$.
• (iv) Define a point in ${U}$ to be a curve ${\gamma_1: [a,b] \rightarrow U}$ of the form ${\gamma_1(t) = z_0}$ for some ${z_0 \in U}$ and all ${t \in [a,b]}$. Let ${\gamma_0: [a,b] \rightarrow U}$ be a closed curve in ${U}$. Show that ${\gamma_0}$ is homotopic with fixed endpoints to a point in ${U}$ if and only if ${\gamma_0}$ is homotopic as a closed curve to a point in ${U}$. (In either case, we will call ${\gamma_0}$ homotopic to a point, null-homotopic, or contractible to a point in ${U}$.)
• (v) If ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are curves with the same initial point and the same terminal point, show that ${\gamma_0}$ is homotopic to ${\gamma_1}$ with fixed endpoints in ${U}$ if and only if ${\gamma_0 + (-\gamma_1)}$ is homotopic to a point in ${U}$.
• (vi) If ${U}$ is connected, and ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are any two curves in ${U}$, show that there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$. Thus the notion of homotopy becomes rather trivial if one does not fix the endpoints or require the curve to be closed.
• (vii) Show that if ${\gamma_1: [a,b] \rightarrow U}$ is a reparameterisation of ${\gamma_0: [a,b] \rightarrow U}$, then ${\gamma_0}$ and ${\gamma_1}$ are homotopic with fixed endpoints in U.
• (viii) Prove that the property of being homotopic with fixed endpoints in ${U}$ up to reparameterisation is an equivalence relation.
• (ix) Prove that the property of being homotopic as closed curves in ${U}$ up to reparameterisation is an equivalence relation.

We can then phrase Cauchy’s theorem as an assertion that contour integration on holomorphic functions is a homotopy invariant. More precisely:

Theorem 4 (Cauchy’s theorem) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic.

• (i) If ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [c,d] \rightarrow U}$ are rectifiable curves that are homotopic in ${U}$ with fixed endpoints up to reparameterisation, then

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.$

• (ii) If ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [c,d] \rightarrow U}$ are closed rectifiable curves that are homotopic in ${U}$ as closed curves up to reparameterisation, then

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.$

This version of Cauchy’s theorem is particularly useful for applications, as it explicitly brings into play the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate. This formulation of Cauchy’s theorem also highlights the close relationship between contour integrals and the algebraic topology of the complex plane (and open subsets ${U}$ thereof). Setting ${\gamma_1}$ to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):

Corollary 5 (Cauchy’s theorem, again) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic. Then for any closed rectifiable curve ${\gamma}$ in ${U}$ that is contractible in ${U}$ to a point, one has ${\int_\gamma f(z)\ dz = 0}$.

Exercise 6 Show that Theorem 4 and Corollary 5 are logically equivalent.

An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on ${f}$. The conclusion of Cauchy’s theorem only involves the values of a function ${f}$ on the images of the two curves ${\gamma_0, \gamma_1}$. However, in order for the hypotheses of Cauchy’s theorem to apply, the function ${f}$ must be holomorphic not only on the images on ${\gamma_0, \gamma_1}$, but on an open set ${U}$ that is large enough (and sufficiently free of “holes”) to support a homotopy between the two curves. This point can be emphasised through the following fundamental near-counterexample to Cauchy’s theorem:

Example 7 Let ${U := {\bf C} \backslash \{0\}}$, and let ${f: U \rightarrow {\bf C}}$ be the holomorphic function ${f(z) := \frac{1}{z}}$. Let ${\gamma_{0,1,\circlearrowleft}: [0,2\pi] \rightarrow {\bf C}}$ be the closed unit circle contour ${\gamma_{0,1,\circlearrowleft}(t) := e^{it}}$. Direct calculation shows that

$\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} f(z)\ dz = 2\pi i \neq 0.$

As a consequence of this and Cauchy’s theorem, we conclude that the contour ${\gamma_{0,1,\circlearrowleft}}$ is not contractible to a point in ${U}$; note that this does not contradict Example 2 because ${U}$ is not convex. Thus we see that the lack of holomorphicity (or singularity) of ${f}$ at the origin can be “blamed” for the non-vanishing of the integral of ${f}$ on the closed contour ${\gamma_{0,1,\circlearrowleft}}$, even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of ${f}$, not just the behaviour in the local neighbourhood of ${\gamma_{0,1,\circlearrowleft}}$, has an impact on the contour integral.

One can of course rewrite this example to involve non-closed contours instead of closed ones. For instance, if we let ${\gamma_0, \gamma_1: [0,\pi] \rightarrow U}$ denote the half-circle contours ${\gamma_0(t) := e^{it}}$ and ${\gamma_1(t) := e^{-it}}$, then ${\gamma_0,\gamma_1}$ are both contours in ${U}$ from ${+1}$ to ${-1}$, but one has

$\displaystyle \int_{\gamma_0} f(z)\ dz = +\pi i$

whereas

$\displaystyle \int_{\gamma_1} f(z)\ dz = -\pi i.$

In order for this to be consistent with Cauchy’s theorem, we conclude that ${\gamma_0}$ and ${\gamma_1}$ are not homotopic in ${U}$ (even after reparameterisation).

In the specific case of functions of the form ${\frac{1}{z}}$, or more generally ${\frac{f(z)}{z-z_0}}$ for some point ${z_0}$ and some ${f}$ that is holomorphic in some neighbourhood of ${z_0}$, we can quantify the precise failure of Cauchy’s theorem through the Cauchy integral formula, and through the concept of a winding number. These turn out to be extremely powerful tools for understanding both the nature of holomorphic functions and the topology of open subsets of the complex plane, as we shall see in this and later notes.

Having discussed differentiation of complex mappings in the preceding notes, we now turn to the integration of complex maps. We first briefly review the situation of integration of (suitably regular) real functions ${f: [a,b] \rightarrow {\bf R}}$ of one variable. Actually there are three closely related concepts of integration that arise in this setting:

• (i) The signed definite integral ${\int_a^b f(x)\ dx}$, which is usually interpreted as the Riemann integral (or equivalently, the Darboux integral), which can be defined as the limit (if it exists) of the Riemann sums

$\displaystyle \sum_{j=1}^n f(x_j^*) (x_j - x_{j-1}) \ \ \ \ \ (1)$

where ${a = x_0 < x_1 < \dots < x_n = b}$ is some partition of ${[a,b]}$, ${x_j^*}$ is an element of the interval ${[x_{j-1},x_j]}$, and the limit is taken as the maximum mesh size ${\max_{1 \leq j \leq n} |x_j - x_{j-1}|}$ goes to zero. It is convenient to adopt the convention that ${\int_b^a f(x)\ dx := - \int_a^b f(x)\ dx}$ for ${a < b}$; alternatively one can interpret ${\int_b^a f(x)\ dx}$ as the limit of the Riemann sums (1), where now the (reversed) partition ${b = x_0 > x_1 > \dots > x_n = a}$ goes leftwards from ${b}$ to ${a}$, rather than rightwards from ${a}$ to ${b}$.

• (ii) The unsigned definite integral ${\int_{[a,b]} f(x)\ dx}$, usually interpreted as the Lebesgue integral. The precise definition of this integral is a little complicated (see e.g. this previous post), but roughly speaking the idea is to approximate ${f}$ by simple functions ${\sum_{i=1}^n c_i 1_{E_i}}$ for some coefficients ${c_i \in {\bf R}}$ and sets ${E_i \subset [a,b]}$, and then approximate the integral ${\int_{[a,b]} f(x)\ dx}$ by the quantities ${\sum_{i=1}^n c_i m(E_i)}$, where ${E_i}$ is the Lebesgue measure of ${E_i}$. In contrast to the signed definite integral, no orientation is imposed or used on the underlying domain of integration, which is viewed as an “undirected” set ${[a,b]}$.
• (iii) The indefinite integral or antiderivative ${\int f(x)\ dx}$, defined as any function ${F: [a,b] \rightarrow {\bf R}}$ whose derivative ${F'}$ exists and is equal to ${f}$ on ${[a,b]}$. Famously, the antiderivative is only defined up to the addition of an arbitrary constant ${C}$, thus for instance ${\int x\ dx = \frac{1}{2} x^2 + C}$.

There are some other variants of the above integrals (e.g. the Henstock-Kurzweil integral, discussed for instance in this previous post), which can handle slightly different classes of functions and have slightly different properties than the standard integrals listed here, but we will not need to discuss such alternative integrals in this course (with the exception of some improper and principal value integrals, which we will encounter in later notes).

The above three notions of integration are closely related to each other. For instance, if ${f: [a,b] \rightarrow {\bf R}}$ is a Riemann integrable function, then the signed definite integral and unsigned definite integral coincide (when the former is oriented correctly), thus

$\displaystyle \int_a^b f(x)\ dx = \int_{[a,b]} f(x)\ dx$

and

$\displaystyle \int_b^a f(x)\ dx = -\int_{[a,b]} f(x)\ dx$

If ${f: [a,b] \rightarrow {\bf R}}$ is continuous, then by the fundamental theorem of calculus, it possesses an antiderivative ${F = \int f(x)\ dx}$, which is well defined up to an additive constant ${C}$, and

$\displaystyle \int_c^d f(x)\ dx = F(d) - F(c)$

for any ${c,d \in [a,b]}$, thus for instance ${\int_a^b F(x)\ dx = F(b) - F(a)}$ and ${\int_b^a F(x)\ dx = F(a) - F(b)}$.

All three of the above integration concepts have analogues in complex analysis. By far the most important notion will be the complex analogue of the signed definite integral, namely the contour integral ${\int_\gamma f(z)\ dz}$, in which the directed line segment from one real number ${a}$ to another ${b}$ is now replaced by a type of curve in the complex plane known as a contour. The contour integral can be viewed as the special case of the more general line integral ${\int_\gamma f(z) dx + g(z) dy}$, that is of particular relevance in complex analysis. There are also analogues of the Lebesgue integral, namely the arclength measure integrals ${\int_\gamma f(z)\ |dz|}$ and the area integrals ${\int_\Omega f(x+iy)\ dx dy}$, but these play only an auxiliary role in the subject. Finally, we still have the notion of an antiderivative ${F(z)}$ (also known as a primitive) of a complex function ${f(z)}$.

As it turns out, the fundamental theorem of calculus continues to hold in the complex plane: under suitable regularity assumptions on a complex function ${f}$ and a primitive ${F}$ of that function, one has

$\displaystyle \int_\gamma f(z)\ dz = F(z_1) - F(z_0)$

whenever ${\gamma}$ is a contour from ${z_0}$ to ${z_1}$ that lies in the domain of ${f}$. In particular, functions ${f}$ that possess a primitive must be conservative in the sense that ${\int_\gamma f(z)\ dz = 0}$ for any closed contour. This property of being conservative is not typical, in that “most” functions ${f}$ will not be conservative. However, there is a remarkable and far-reaching theorem, the Cauchy integral theorem (also known as the Cauchy-Goursat theorem), which asserts that any holomorphic function is conservative, so long as the domain is simply connected (or if one restricts attention to contractible closed contours). We will explore this theorem and several of its consequences the next set of notes.

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the fundamental theorem of calculus (and its higher dimensional generalisations, such as Stokes’ theorem). Similarly, the notion of the complex derivative and the complex line integral (that is to say, the contour integral) lie at the core of any introductory complex analysis course. Once again, they are tied to each other by the fundamental theorem of calculus; but in the complex case there is a further variant of the fundamental theorem, namely Cauchy’s theorem, which endows complex differentiable functions with many important and surprising properties that are often not shared by their real differentiable counterparts. We will give complex differentiable functions another name to emphasise this extra structure, by referring to such functions as holomorphic functions. (This term is also useful to distinguish these functions from the slightly less well-behaved meromorphic functions, which we will discuss in later notes.)

In this set of notes we will focus solely on the concept of complex differentiation, deferring the discussion of contour integration to the next set of notes. To begin with, the theory of complex differentiation will greatly resemble the theory of real differentiation; the definitions look almost identical, and well known laws of differential calculus such as the product rule, quotient rule, and chain rule carry over verbatim to the complex setting, and the theory of complex power series is similarly almost identical to the theory of real power series. However, when one compares the “one-dimensional” differentiation theory of the complex numbers with the “two-dimensional” differentiation theory of two real variables, we find that the dimensional discrepancy forces complex differentiable functions to obey a real-variable constraint, namely the Cauchy-Riemann equations. These equations make complex differentiable functions substantially more “rigid” than their real-variable counterparts; they imply for instance that the imaginary part of a complex differentiable function is essentially determined (up to constants) by the real part, and vice versa. Furthermore, even when considered separately, the real and imaginary components of complex differentiable functions are forced to obey the strong constraint of being harmonic. In later notes we will see these constraints manifest themselves in integral form, particularly through Cauchy’s theorem and the closely related Cauchy integral formula.

Despite all the constraints that holomorphic functions have to obey, a surprisingly large number of the functions of a complex variable that one actually encounters in applications turn out to be holomorphic. For instance, any polynomial ${z \mapsto P(z)}$ with complex coefficients will be holomorphic, as will the complex exponential ${z \mapsto \exp(z)}$. From this and the laws of differential calculus one can then generate many further holomorphic functions. Also, as we will show presently, complex power series will automatically be holomorphic inside their disk of convergence. On the other hand, there are certainly basic complex functions of interest that are not holomorphic, such as the complex conjugation function ${z \mapsto \overline{z}}$, the absolute value function ${z \mapsto |z|}$, or the real and imaginary part functions ${z \mapsto \mathrm{Re}(z), z \mapsto \mathrm{Im}(z)}$. We will also encounter functions that are only holomorphic at some portions of the complex plane, but not on others; for instance, rational functions will be holomorphic except at those few points where the denominator vanishes, and are prime examples of the meromorphic functions mentioned previously. Later on we will also consider functions such as branches of the logarithm or square root, which will be holomorphic outside of a branch cut corresponding to the choice of branch. It is a basic but important skill in complex analysis to be able to quickly recognise which functions are holomorphic and which ones are not, as many of useful theorems available to the former (such as Cauchy’s theorem) break down spectacularly for the latter. Indeed, in my experience, one of the most common “rookie errors” that beginning complex analysis students make is the error of attempting to apply a theorem about holomorphic functions to a function that is not at all holomorphic. This stands in contrast to the situation in real analysis, in which one can often obtain correct conclusions by formally applying the laws of differential or integral calculus to functions that might not actually be differentiable or integrable in a classical sense. (This latter phenomenon, by the way, can be largely explained using the theory of distributions, as covered for instance in this previous post, but this is beyond the scope of the current course.)

Remark 1 In this set of notes it will be convenient to impose some unnecessarily generous regularity hypotheses (e.g. continuous second differentiability) on the holomorphic functions one is studying in order to make the proofs simpler. In later notes, we will discover that these hypotheses are in fact redundant, due to the phenomenon of elliptic regularity that ensures that holomorphic functions are automatically smooth.

Kronecker is famously reported to have said, “God created the natural numbers; all else is the work of man”. The truth of this statement (literal or otherwise) is debatable; but one can certainly view the other standard number systems ${{\bf Z}, {\bf Q}, {\bf R}, {\bf C}}$ as (iterated) completions of the natural numbers ${{\bf N}}$ in various senses. For instance:

• The integers ${{\bf Z}}$ are the additive completion of the natural numbers ${{\bf N}}$ (the minimal additive group that contains a copy of ${{\bf N}}$).
• The rationals ${{\bf Q}}$ are the multiplicative completion of the integers ${{\bf Z}}$ (the minimal field that contains a copy of ${{\bf Z}}$).
• The reals ${{\bf R}}$ are the metric completion of the rationals ${{\bf Q}}$ (the minimal complete metric space that contains a copy of ${{\bf Q}}$).
• The complex numbers ${{\bf C}}$ are the algebraic completion of the reals ${{\bf R}}$ (the minimal algebraically closed field that contains a copy of ${{\bf R}}$).

These descriptions of the standard number systems are elegant and conceptual, but not entirely suitable for constructing the number systems in a non-circular manner from more primitive foundations. For instance, one cannot quite define the reals ${{\bf R}}$ from scratch as the metric completion of the rationals ${{\bf Q}}$, because the definition of a metric space itself requires the notion of the reals! (One can of course construct ${{\bf R}}$ by other means, for instance by using Dedekind cuts or by using uniform spaces in place of metric spaces.) The definition of the complex numbers as the algebraic completion of the reals does not suffer from such a non-circularity issue, but a certain amount of field theory is required to work with this definition initially. For the purposes of quickly constructing the complex numbers, it is thus more traditional to first define ${{\bf C}}$ as a quadratic extension of the reals ${{\bf R}}$, and more precisely as the extension ${{\bf C} = {\bf R}(i)}$ formed by adjoining a square root ${i}$ of ${-1}$ to the reals, that is to say a solution to the equation ${i^2+1=0}$. It is not immediately obvious that this extension is in fact algebraically closed; this is the content of the famous fundamental theorem of algebra, which we will prove later in this course.

The two equivalent definitions of ${{\bf C}}$ – as the algebraic closure, and as a quadratic extension, of the reals respectively – each reveal important features of the complex numbers in applications. Because ${{\bf C}}$ is algebraically closed, all polynomials over the complex numbers split completely, which leads to a good spectral theory for both finite-dimensional matrices and infinite-dimensional operators; in particular, one expects to be able to diagonalise most matrices and operators. Applying this theory to constant coefficient ordinary differential equations leads to a unified theory of such solutions, in which real-variable ODE behaviour such as exponential growth or decay, polynomial growth, and sinusoidal oscillation all become aspects of a single object, the complex exponential ${z \mapsto e^z}$ (or more generally, the matrix exponential ${A \mapsto \exp(A)}$). Applying this theory more generally to diagonalise arbitrary translation-invariant operators over some locally compact abelian group, one arrives at Fourier analysis, which is thus most naturally phrased in terms of complex-valued functions rather than real-valued ones. If one drops the assumption that the underlying group is abelian, one instead discovers the representation theory of unitary representations, which is simpler to study than the real-valued counterpart of orthogonal representations. For closely related reasons, the theory of complex Lie groups is simpler than that of real Lie groups.

Meanwhile, the fact that the complex numbers are a quadratic extension of the reals lets one view the complex numbers geometrically as a two-dimensional plane over the reals (the Argand plane). Whereas a point singularity in the real line disconnects that line, a point singularity in the Argand plane leaves the rest of the plane connected (although, importantly, the punctured plane is no longer simply connected). As we shall see, this fact causes singularities in complex analytic functions to be better behaved than singularities of real analytic functions, ultimately leading to the powerful residue calculus for computing complex integrals. Remarkably, this calculus, when combined with the quintessentially complex-variable technique of contour shifting, can also be used to compute some (though certainly not all) definite integrals of real-valued functions that would be much more difficult to compute by purely real-variable methods; this is a prime example of Hadamard’s famous dictum that “the shortest path between two truths in the real domain passes through the complex domain”.

Another important geometric feature of the Argand plane is the angle between two tangent vectors to a point in the plane. As it turns out, the operation of multiplication by a complex scalar preserves the magnitude and orientation of such angles; the same fact is true for any non-degenerate complex analytic mapping, as can be seen by performing a Taylor expansion to first order. This fact ties the study of complex mappings closely to that of the conformal geometry of the plane (and more generally, of two-dimensional surfaces and domains). In particular, one can use complex analytic maps to conformally transform one two-dimensional domain to another, leading among other things to the famous Riemann mapping theorem, and to the classification of Riemann surfaces.

If one Taylor expands complex analytic maps to second order rather than first order, one discovers a further important property of these maps, namely that they are harmonic. This fact makes the class of complex analytic maps extremely rigid and well behaved analytically; indeed, the entire theory of elliptic PDE now comes into play, giving useful properties such as elliptic regularity and the maximum principle. In fact, due to the magic of residue calculus and contour shifting, we already obtain these properties for maps that are merely complex differentiable rather than complex analytic, which leads to the striking fact that complex differentiable functions are automatically analytic (in contrast to the real-variable case, in which real differentiable functions can be very far from being analytic).

The geometric structure of the complex numbers (and more generally of complex manifolds and complex varieties), when combined with the algebraic closure of the complex numbers, leads to the beautiful subject of complex algebraic geometry, which motivates the much more general theory developed in modern algebraic geometry. However, we will not develop the algebraic geometry aspects of complex analysis here.

Last, but not least, because of the good behaviour of Taylor series in the complex plane, complex analysis is an excellent setting in which to manipulate various generating functions, particularly Fourier series ${\sum_n a_n e^{2\pi i n \theta}}$ (which can be viewed as boundary values of power (or Laurent) series ${\sum_n a_n z^n}$), as well as Dirichlet series ${\sum_n \frac{a_n}{n^s}}$. The theory of contour integration provides a very useful dictionary between the asymptotic behaviour of the sequence ${a_n}$, and the complex analytic behaviour of the Dirichlet or Fourier series, particularly with regard to its poles and other singularities. This turns out to be a particularly handy dictionary in analytic number theory, for instance relating the distribution of the primes to the Riemann zeta function. Nowadays, many of the analytic number theory results first obtained through complex analysis (such as the prime number theorem) can also be obtained by more “real-variable” methods; however the complex-analytic viewpoint is still extremely valuable and illuminating.

We will frequently touch upon many of these connections to other fields of mathematics in these lecture notes. However, these are mostly side remarks intended to provide context, and it is certainly possible to skip most of these tangents and focus purely on the complex analysis material in these notes if desired.

Note: complex analysis is a very visual subject, and one should draw plenty of pictures while learning it. I am however not planning to put too many pictures in these notes, partly as it is somewhat inconvenient to do so on this blog from a technical perspective, but also because pictures that one draws on one’s own are likely to be far more useful to you than pictures that were supplied by someone else.

Next week, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence.  This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular focusing on the basic properties of holomorphic functions such as the Cauchy and residue theorems, the classification of singularities, and the maximum principle, but there will be more of an emphasis on rigour, generalisation and abstraction, and connections with other parts of mathematics.  If time permits I may also cover topics such as factorisation theorems, harmonic functions, conformal mapping, and/or applications to analytic number theory.  The main text I will be using for this course is Stein-Shakarchi (with Ahlfors as a secondary text), but as usual I will also be writing notes for the course on this blog.

This is a postscript to the previous blog post which was concerned with obtaining heuristic asymptotic predictions for the correlation

$\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h), \ \ \ \ \ (1)$

for the divisor function ${\tau(n) := \sum_{d|n} 1}$, in particular recovering the calculation of Ingham that obtained the asymptotic

$\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h) \sim \frac{6}{\pi^2} \sigma_{-1}(h) x \log^2 x \ \ \ \ \ (2)$

when ${h}$ was fixed and non-zero and ${x}$ went to infinity. It is natural to consider the more general correlations

$\displaystyle \sum_{n \leq x} \tau_k(n) \tau_l(n+h)$

for fixed ${k,l \geq 1}$ and non-zero ${h}$, where

$\displaystyle \tau_k(n) := \sum_{d_1 \dots d_k = n} 1$

is the order ${k}$ divisor function. The sum (1) then corresponds to the case ${k=l=2}$. For ${l=1}$, ${\tau_1(n) = 1}$, and a routine application of the Dirichlet hyperbola method (or Perron’s formula) gives the asymptotic

$\displaystyle \sum_{n \leq x} \tau_k(n) \sim \frac{\log^{k-1} x}{(k-1)!} x,$

or more accurately

$\displaystyle \sum_{n \leq x} \tau_k(n) \sim P_k(\log x) x$

where ${P_k(t)}$ is a certain explicit polynomial of degree ${k-1}$ with leading coefficient ${\frac{1}{(k-1)!}}$; see e.g. Exercise 31 of this previous post for a discussion of the ${k=3}$ case (which is already typical). Similarly if ${k=1}$. For more general ${k,l \geq 1}$, there is a conjecture of Conrey and Gonek which predicts that

$\displaystyle \sum_{n \leq x} \tau_k(n) \tau_l(n+h) \sim P_{k,l,h}(\log x) x$

for some polynomial ${P_{k,l,h}(t)}$ of degree ${k+l-2}$ which is explicit but whose form is rather complicated (one has to compute residues of a various complicated products of zeta functions and local factors). This conjecture has been verified when ${k \leq 2}$ or ${l \leq 2}$, by the work of Linnik, Motohashi, Fouvry-Tenenbaum, and others, but all the remaining cases when ${k,l \geq 3}$ are currently open.

In principle, the calculations of the previous post should recover the predictions of Conrey and Gonek. In this post I would like to record this for the top order term:

Conjecture 1 If ${k,l \geq 2}$ and ${h \neq 0}$ are fixed, then

$\displaystyle \sum_{n \leq x} \tau_k(n) \tau_l(n+h) \sim \frac{\log^{k-1} x}{(k-1)!} \frac{\log^{l-1} x}{(l-1)!} x \prod_p {\mathfrak S}_{k,l,p}(h)$

as ${x \rightarrow \infty}$, where the product is over all primes ${p}$, and the local factors ${{\mathfrak S}_{k,l,p}(h)}$ are given by the formula

$\displaystyle {\mathfrak S}_{k,l,p}(h) := (\frac{p-1}{p})^{k+l-2} \sum_{j \geq 0: p^j|h} \frac{1}{p^j} P_{k,l,p}(j) \ \ \ \ \ (3)$

where ${P_{k,l,p}}$ is the degree ${k+l-4}$ polynomial

$\displaystyle P_{k,l,p}(j) := \sum_{k'=2}^k \sum_{l'=2}^l \binom{k-k'+j-1}{k-k'} \binom{l-l'+j-1}{l-l'} \alpha_{k',l',p}$

where

$\displaystyle \alpha_{k',l',p} := (\frac{p}{p-1})^{k'-1} + (\frac{p}{p-1})^{l'-1} - 1$

and one adopts the conventions that ${\binom{-1}{0} = 1}$ and ${\binom{m-1}{m} = 0}$ for ${m \geq 1}$.

For instance, if ${k=l=2}$ then

$\displaystyle P_{2,2,p}(h) = \frac{p}{p-1} + \frac{p}{p-1} - 1 = \frac{p+1}{p-1}$

and hence

$\displaystyle {\mathfrak S}_{2,2,p}(h) = (1 - \frac{1}{p^2}) \sum_{j \geq 0: p^j|h} \frac{1}{p^j}$

and the above conjecture recovers the Ingham formula (2). For ${k=2, l=3}$, we have

$\displaystyle P_{2,3,p}(h) =$

$\displaystyle (\frac{p}{p-1} + (\frac{p}{p-1})^2 - 1) + (\frac{p}{p-1} + \frac{p}{p-1} - 1) j$

$\displaystyle = \frac{p^2+p-1}{(p-1)^2} + \frac{p+1}{p-1} j$

and so we predict

$\displaystyle \sum_{n \leq x} \tau(n) \tau_3(n+h) \sim \frac{x \log^3 x}{2} \prod_p {\mathfrak S}_{2,3,p}(h)$

where

$\displaystyle {\mathfrak S}_{2,3,p}(h) = \sum_{j \geq 0: p^j|h} \frac{\frac{p^3 - 2p + 1}{p^3} + \frac{(p+1)(p-1)^2}{p^3} j}{p^j}.$

Similarly, if ${k=l=3}$ we have

$\displaystyle P_{3,3,p}(h) = ((\frac{p}{p-1})^2 + (\frac{p}{p-1})^2 - 1) + 2 (\frac{p}{p-1} + (\frac{p}{p-1})^2 - 1) j$

$\displaystyle + (\frac{p}{p-1} + \frac{p}{p-1} - 1) j^2$

$\displaystyle = \frac{p^2+2p-1}{(p-1)^2} + 2 \frac{p^2+p-1}{(p-1)^2} j + \frac{p+1}{p-1} j^2$

and so we predict

$\displaystyle \sum_{n \leq x} \tau_3(n) \tau_3(n+h) \sim \frac{x \log^4 x}{4} \prod_p {\mathfrak S}_{3,3,p}(h)$

where

$\displaystyle {\mathfrak S}_{3,3,p}(h) = \sum_{j \geq 0: p^j|h} \frac{\frac{p^4 - 4p^2 + 4p - 1}{p^4} + 2 \frac{(p^2+p-1)(p-1)^2}{p^4} j + \frac{(p+1)(p-1)^3}{p^4} j^2}{p^j}.$

and so forth.

As in the previous blog, the idea is to factorise

$\displaystyle \tau_k(n) = \prod_p \tau_{k,p}(n)$

where the local factors ${\tau_{k,p}(n)}$ are given by

$\displaystyle \tau_{k,p}(n) := \sum_{j_1,\dots,j_k \geq 0: p^{j_1+\dots+j_k} || n} 1$

(where ${p^j || n}$ means that ${p}$ divides ${n}$ precisely ${j}$ times), or in terms of the valuation ${v_p(n)}$ of ${n}$ at ${p}$,

$\displaystyle \tau_{k,p}(n) = \binom{k-1+v_p(n)}{k-1}. \ \ \ \ \ (4)$

We then have the following exact local asymptotics:

Proposition 2 (Local correlations) Let ${{\bf n}}$ be a profinite integer chosen uniformly at random, let ${h}$ be a profinite integer, and let ${k,l \geq 2}$. Then

$\displaystyle {\bf E} \tau_{k,p}({\bf n}) = (\frac{p}{p-1})^{k-1} \ \ \ \ \ (5)$

and

$\displaystyle {\bf E} \tau_{k,p}({\bf n}) \tau_{l,p}({\bf n}+h) = (\frac{p}{p-1})^{k+l-2} {\mathfrak S}_{k,l,p}(h). \ \ \ \ \ (6)$

(For profinite integers it is possible that ${v_p({\bf n})}$ and hence ${\tau_{k,p}({\bf n})}$ are infinite, but this is a probability zero event and so can be ignored.)

Conjecture 1 can then be heuristically justified from the local calculations (2) by various pseudorandomness heuristics, as discussed in the previous post.

I’ll give a short proof of the above proposition below, basically using the recursive methods of the previous post. This short proof actually took be quite a while to find; I spent several hours and a fair bit of scratch paper working out the cases ${k,l = 2,3}$ laboriously by hand (with some assistance and cross-checking from Maple). Here is an unorganised sample of some of this scratch, just to show how the sausage is actually made:

It was only after expending all this effort that I realised that it would be much more efficient to compute the correlations for all values of ${k,l}$ simultaneously by using generating functions. After performing this computation, it then became apparent that there would be a direct combinatorial proof of (6) that was shorter than even the generating function proof. (I will not supply the full generating function calculations here, but will at least show them for the easier correlation (5).)

I am confident that Conjecture 1 is consistent with the explicit asymptotic in the Conrey-Gonek conjecture, but have not yet rigorously established that the leading order term in the latter is indeed identical to the expression provided above.

Let ${\tau(n) := \sum_{d|n} 1}$ be the divisor function. A classical application of the Dirichlet hyperbola method gives the asymptotic

$\displaystyle \sum_{n \leq x} \tau(n) \sim x \log x$

where ${X \sim Y}$ denotes the estimate ${X = (1+o(1))Y}$ as ${x \rightarrow \infty}$. Much better error estimates are possible here, but we will not focus on the lower order terms in this discussion. For somewhat idiosyncratic reasons I will interpret this estimate (and the other analytic number theory estimates discussed here) through the probabilistic lens. Namely, if ${{\bf n} = {\bf n}_x}$ is a random number selected uniformly between ${1}$ and ${x}$, then the above estimate can be written as

$\displaystyle {\bf E} \tau( {\bf n} ) \sim \log x, \ \ \ \ \ (1)$

that is to say the random variable ${\tau({\bf n})}$ has mean approximately ${\log x}$. (But, somewhat paradoxically, this is not the median or mode behaviour of this random variable, which instead concentrates near ${\log^{\log 2} x}$, basically thanks to the Hardy-Ramanujan theorem.)

Now we turn to the pair correlations ${\sum_{n \leq x} \tau(n) \tau(n+h)}$ for a fixed positive integer ${h}$. There is a classical computation of Ingham that shows that

$\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h) \sim \frac{6}{\pi^2} \sigma_{-1}(h) x \log^2 x, \ \ \ \ \ (2)$

where

$\displaystyle \sigma_{-1}(h) := \sum_{d|h} \frac{1}{d}.$

The error term in (2) has been refined by many subsequent authors, as has the uniformity of the estimates in the ${h}$ aspect, as these topics are related to other questions in analytic number theory, such as fourth moment estimates for the Riemann zeta function; but we will not consider these more subtle features of the estimate here. However, we will look at the next term in the asymptotic expansion for (2) below the fold.

Using our probabilistic lens, the estimate (2) can be written as

$\displaystyle {\bf E} \tau( {\bf n} ) \tau( {\bf n} + h ) \sim \frac{6}{\pi^2} \sigma_{-1}(h) \log^2 x. \ \ \ \ \ (3)$

From (1) (and the asymptotic negligibility of the shift by ${h}$) we see that the random variables ${\tau({\bf n})}$ and ${\tau({\bf n}+h)}$ both have a mean of ${\sim \log x}$, so the additional factor of ${\frac{6}{\pi^2} \sigma_{-1}(h)}$ represents some arithmetic coupling between the two random variables.

Ingham’s formula can be established in a number of ways. Firstly, one can expand out ${\tau(n) = \sum_{d|n} 1}$ and use the hyperbola method (splitting into the cases ${d \leq \sqrt{x}}$ and ${n/d \leq \sqrt{x}}$ and removing the overlap). If one does so, one soon arrives at the task of having to estimate sums of the form

$\displaystyle \sum_{n \leq x: d|n} \tau(n+h)$

for various ${d \leq \sqrt{x}}$. For ${d}$ much less than ${\sqrt{x}}$ this can be achieved using a further application of the hyperbola method, but for ${d}$ comparable to ${\sqrt{x}}$ things get a bit more complicated, necessitating the use of non-trivial estimates on Kloosterman sums in order to obtain satisfactory control on error terms. A more modern approach proceeds using automorphic form methods, as discussed in this previous post. A third approach, which unfortunately is only heuristic at the current level of technology, is to apply the Hardy-Littlewood circle method (discussed in this previous post) to express (2) in terms of exponential sums ${\sum_{n \leq x} \tau(n) e(\alpha n)}$ for various frequencies ${\alpha}$. The contribution of “major arc” ${\alpha}$ can be computed after a moderately lengthy calculation which yields the right-hand side of (2) (as well as the correct lower order terms that are currently being suppressed), but there does not appear to be an easy way to show directly that the “minor arc” contributions are of lower order, although the methods discussed previously do indirectly show that this is ultimately the case.

Each of the methods outlined above requires a fair amount of calculation, and it is not obvious while performing them that the factor ${\frac{6}{\pi^2} \sigma_{-1}(h)}$ will emerge at the end. One can at least explain the ${\frac{6}{\pi^2}}$ as a normalisation constant needed to balance the ${\sigma_{-1}(h)}$ factor (at a heuristic level, at least). To see this through our probabilistic lens, introduce an independent copy ${{\bf n}'}$ of ${{\bf n}}$, then

$\displaystyle {\bf E} \tau( {\bf n} ) \tau( {\bf n}' ) = ({\bf E} \tau ({\bf n}))^2 \sim \log^2 x; \ \ \ \ \ (4)$

using symmetry to order ${{\bf n}' > {\bf n}}$ (discarding the diagonal case ${{\bf n} = {\bf n}'}$) and making the change of variables ${{\bf n}' = {\bf n}+h}$, we see that (4) is heuristically consistent with (3) as long as the asymptotic mean of ${\frac{6}{\pi^2} \sigma_{-1}(h)}$ in ${h}$ is equal to ${1}$. (This argument is not rigorous because there was an implicit interchange of limits present, but still gives a good heuristic “sanity check” of Ingham’s formula.) Indeed, if ${{\bf E}_h}$ denotes the asymptotic mean in ${h}$, then we have (heuristically at least)

$\displaystyle {\bf E}_h \sigma_{-1}(h) = \sum_d {\bf E}_h \frac{1}{d} 1_{d|h}$

$\displaystyle = \sum_d \frac{1}{d^2}$

$\displaystyle = \frac{\pi^2}{6}$

and we obtain the desired consistency after multiplying by ${\frac{6}{\pi^2}}$.

This still however does not explain the presence of the ${\sigma_{-1}(h)}$ factor. Intuitively it is reasonable that if ${h}$ has many prime factors, and ${{\bf n}}$ has a lot of factors, then ${{\bf n}+h}$ will have slightly more factors than average, because any common factor to ${h}$ and ${{\bf n}}$ will automatically be acquired by ${{\bf n}+h}$. But how to quantify this effect?

One heuristic way to proceed is through analysis of local factors. Observe from the fundamental theorem of arithmetic that we can factor

$\displaystyle \tau(n) = \prod_p \tau_p(n)$

where the product is over all primes ${p}$, and ${\tau_p(n) := \sum_{p^j|n} 1}$ is the local version of ${\tau(n)}$ at ${p}$ (which in this case, is just one plus the ${p}$valuation ${v_p(n)}$ of ${n}$: ${\tau_p = 1 + v_p}$). Note that all but finitely many of the terms in this product will equal ${1}$, so the infinite product is well-defined. In a similar fashion, we can factor

$\displaystyle \sigma_{-1}(h) = \prod_p \sigma_{-1,p}(h)$

where

$\displaystyle \sigma_{-1,p}(h) := \sum_{p^j|h} \frac{1}{p^j}$

(or in terms of valuations, ${\sigma_{-1,p}(h) = (1 - p^{-v_p(h)-1})/(1-p^{-1})}$). Heuristically, the Chinese remainder theorem suggests that the various factors ${\tau_p({\bf n})}$ behave like independent random variables, and so the correlation between ${\tau({\bf n})}$ and ${\tau({\bf n}+h)}$ should approximately decouple into the product of correlations between the local factors ${\tau_p({\bf n})}$ and ${\tau_p({\bf n}+h)}$. And indeed we do have the following local version of Ingham’s asymptotics:

Proposition 1 (Local Ingham asymptotics) For fixed ${p}$ and integer ${h}$, we have

$\displaystyle {\bf E} \tau_p({\bf n}) \sim \frac{p}{p-1}$

and

$\displaystyle {\bf E} \tau_p({\bf n}) \tau_p({\bf n}+h) \sim (1-\frac{1}{p^2}) \sigma_{-1,p}(h) (\frac{p}{p-1})^2$

$\displaystyle = \frac{p+1}{p-1} \sigma_{-1,p}(h)$

From the Euler formula

$\displaystyle \prod_p (1-\frac{1}{p^2}) = \frac{1}{\zeta(2)} = \frac{6}{\pi^2}$

we see that

$\displaystyle \frac{6}{\pi^2} \sigma_{-1}(h) = \prod_p (1-\frac{1}{p^2}) \sigma_{-1,p}(h)$

and so one can “explain” the arithmetic factor ${\frac{6}{\pi^2} \sigma_{-1}(h)}$ in Ingham’s asymptotic as the product of the arithmetic factors ${(1-\frac{1}{p^2}) \sigma_{-1,p}(h)}$ in the (much easier) local Ingham asymptotics. Unfortunately we have the usual “local-global” problem in that we do not know how to rigorously derive the global asymptotic from the local ones; this problem is essentially the same issue as the problem of controlling the minor arc contributions in the circle method, but phrased in “physical space” language rather than “frequency space”.

Remark 2 The relation between the local means ${\sim \frac{p}{p-1}}$ and the global mean ${\sim \log^2 x}$ can also be seen heuristically through the application

$\displaystyle \prod_{p \leq x^{1/e^\gamma}} \frac{p}{p-1} \sim \log x$

of Mertens’ theorem, where ${1/e^\gamma}$ is Pólya’s magic exponent, which serves as a useful heuristic limiting threshold in situations where the product of local factors is divergent.

Let us now prove this proposition. One could brute-force the computations by observing that for any fixed ${j}$, the valuation ${v_p({\bf n})}$ is equal to ${j}$ with probability ${\sim \frac{p-1}{p} \frac{1}{p^j}}$, and with a little more effort one can also compute the joint distribution of ${v_p({\bf n})}$ and ${v_p({\bf n}+h)}$, at which point the proposition reduces to the calculation of various variants of the geometric series. I however find it cleaner to proceed in a more recursive fashion (similar to how one can prove the geometric series formula by induction); this will also make visible the vague intuition mentioned previously about how common factors of ${{\bf n}}$ and ${h}$ force ${{\bf n}+h}$ to have a factor also.

It is first convenient to get rid of error terms by observing that in the limit ${x \rightarrow \infty}$, the random variable ${{\bf n} = {\bf n}_x}$ converges vaguely to a uniform random variable ${{\bf n}_\infty}$ on the profinite integers ${\hat {\bf Z}}$, or more precisely that the pair ${(v_p({\bf n}_x), v_p({\bf n}_x+h))}$ converges vaguely to ${(v_p({\bf n}_\infty), v_p({\bf n}_\infty+h))}$. Because of this (and because of the easily verified uniform integrability properties of ${\tau_p({\bf n})}$ and their powers), it suffices to establish the exact formulae

$\displaystyle {\bf E} \tau_p({\bf n}_\infty) = \frac{p}{p-1} \ \ \ \ \ (5)$

and

$\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = (1-\frac{1}{p^2}) \sigma_{-1,p}(h) (\frac{p}{p-1})^2 = \frac{p+1}{p-1} \sigma_{-1,p}(h) \ \ \ \ \ (6)$

in the profinite setting (this setting will make it easier to set up the recursion).

We begin with (5). Observe that ${{\bf n}_\infty}$ is coprime to ${p}$ with probability ${\frac{p-1}{p}}$, in which case ${\tau_p({\bf n}_\infty)}$ is equal to ${1}$. Conditioning to the complementary probability ${\frac{1}{p}}$ event that ${{\bf n}_\infty}$ is divisible by ${p}$, we can factor ${{\bf n}_\infty = p {\bf n}'_\infty}$ where ${{\bf n}'_\infty}$ is also uniformly distributed over the profinite integers, in which event we have ${\tau_p( {\bf n}_\infty ) = 1 + \tau_p( {\bf n}'_\infty )}$. We arrive at the identity

$\displaystyle {\bf E} \tau_p({\bf n}_\infty) = \frac{p-1}{p} + \frac{1}{p} ( 1 + {\bf E} \tau_p( {\bf n}'_\infty ) ).$

As ${{\bf n}_\infty}$ and ${{\bf n}'_\infty}$ have the same distribution, the quantities ${{\bf E} \tau_p({\bf n}_\infty)}$ and ${{\bf E} \tau_p({\bf n}'_\infty)}$ are equal, and (5) follows by a brief amount of high-school algebra.

We use a similar method to treat (6). First treat the case when ${h}$ is coprime to ${p}$. Then we see that with probability ${\frac{p-2}{p}}$, ${{\bf n}_\infty}$ and ${{\bf n}_\infty+h}$ are simultaneously coprime to ${p}$, in which case ${\tau_p({\bf n}_\infty) = \tau_p({\bf n}_\infty+h) = 1}$. Furthermore, with probability ${\frac{1}{p}}$, ${{\bf n}_\infty}$ is divisible by ${p}$ and ${{\bf n}_\infty+h}$ is not; in which case we can write ${{\bf n} = p {\bf n}'}$ as before, with ${\tau_p({\bf n}_\infty) = 1 + \tau_p({\bf n}'_\infty)}$ and ${\tau_p({\bf n}_\infty+h)=1}$. Finally, in the remaining event with probability ${\frac{1}{p}}$, ${{\bf n}+h}$ is divisible by ${p}$ and ${{\bf n}}$ is not; we can then write ${{\bf n}_\infty+h = p {\bf n}'_\infty}$, so that ${\tau_p({\bf n}_\infty+h) = 1 + \tau_p({\bf n}'_\infty)}$ and ${\tau_p({\bf n}_\infty) = 1}$. Putting all this together, we obtain

$\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = \frac{p-2}{p} + 2 \frac{1}{p} (1 + {\bf E} \tau_p({\bf n}'_\infty))$

and the claim (6) in this case follows from (5) and a brief computation (noting that ${\sigma_{-1,p}(h)=1}$ in this case).

Now suppose that ${h}$ is divisible by ${p}$, thus ${h=ph'}$ for some integer ${h'}$. Then with probability ${\frac{p-1}{p}}$, ${{\bf n}_\infty}$ and ${{\bf n}_\infty+h}$ are simultaneously coprime to ${p}$, in which case ${\tau_p({\bf n}_\infty) = \tau_p({\bf n}_\infty+h) = 1}$. In the remaining ${\frac{1}{p}}$ event, we can write ${{\bf n}_\infty = p {\bf n}'_\infty}$, and then ${\tau_p({\bf n}_\infty) = 1 + \tau_p({\bf n}'_\infty)}$ and ${\tau_p({\bf n}_\infty+h) = 1 + \tau_p({\bf n}'_\infty+h')}$. Putting all this together we have

$\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = \frac{p-1}{p} + \frac{1}{p} {\bf E} (1+\tau_p({\bf n}'_\infty)(1+\tau_p({\bf n}'_\infty+h)$

which by (5) (and replacing ${{\bf n}'_\infty}$ by ${{\bf n}_\infty}$) leads to the recursive relation

$\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = \frac{p+1}{p-1} + \frac{1}{p} {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h)$

and (6) then follows by induction on the number of powers of ${p}$.

The estimate (2) of Ingham was refined by Estermann, who obtained the more accurate expansion

$\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h) = \frac{6}{\pi^2} \sigma_{-1}(h) x \log^2 x + a_1(h) x \log x + a_2(h) x \ \ \ \ \ (7)$

$\displaystyle + O( x^{11/12+o(1)} )$

for certain complicated but explicit coefficients ${a_1(h), a_2(h)}$. For instance, ${a_1(h)}$ is given by the formula

$\displaystyle a_1(h) = (\frac{12}{\pi^2} (2\gamma-1) + 4 a') \sigma_{-1}(h) - \frac{24}{\pi^2} \sigma'_{-1}(h)$

where ${\gamma}$ is the Euler-Mascheroni constant,

$\displaystyle a' := - \sum_{r=1}^\infty \frac{\mu(r)}{r^2} \log r, \ \ \ \ \ (8)$

and

$\displaystyle \sigma'_{-1}(h) := \sum_{d|h} \frac{\log d}{d}.$

The formula for ${a_2(h)}$ is similar but even more complicated. The error term ${O( x^{11/12+o(1)})}$ was improved by Heath-Brown to ${O( x^{5/6+o(1)})}$; it is conjectured (for instance by Conrey and Gonek) that one in fact has square root cancellation ${O( x^{1/2+o(1)})}$ here, but this is well out of reach of current methods.

These lower order terms are traditionally computed either from a Dirichlet series approach (using Perron’s formula) or a circle method approach. It turns out that a refinement of the above heuristics can also predict these lower order terms, thus keeping the calculation purely in physical space as opposed to the “multiplicative frequency space” of the Dirichlet series approach, or the “additive frequency space” of the circle method, although the computations are arguably as messy as the latter computations for the purposes of working out the lower order terms. We illustrate this just for the ${a_1(h) x \log x}$ term below the fold.

Fifteen years ago, I wrote a paper entitled Global regularity of wave maps. II. Small energy in two dimensions, in which I established global regularity of wave maps from two spatial dimensions to the unit sphere, assuming that the initial data had small energy. Recently, Hao Jia (personal communication) discovered a small gap in the argument that requires a slightly non-trivial fix. The issue does not really affect the subsequent literature, because the main result has since been reproven and extended by methods that avoid the gap (see in particular this subsequent paper of Tataru), but I have decided to describe the gap and its fix on this blog.

I will assume familiarity with the notation of my paper. In Section 10, some complicated spaces ${S[k] = S[k]({\bf R}^{1+n})}$ are constructed for each frequency scale ${k}$, and then a further space ${S(c) = S(c)({\bf R}^{1+n})}$ is constructed for a given frequency envelope ${c}$ by the formula

$\displaystyle \| \phi \|_{S(c)({\bf R}^{1+n})} := \|\phi \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})} + \sup_k c_k^{-1} \| \phi_k \|_{S[k]({\bf R}^{1+n})} \ \ \ \ \ (1)$

where ${\phi_k := P_k \phi}$ is the Littlewood-Paley projection of ${\phi}$ to frequency magnitudes ${\sim 2^k}$. Then, given a spacetime slab ${[-T,T] \times {\bf R}^n}$, we define the restrictions

$\displaystyle \| \phi \|_{S(c)([-T,T] \times {\bf R}^n)} := \inf \{ \| \tilde \phi \|_{S(c)({\bf R}^{1+n})}: \tilde \phi \downharpoonright_{[-T,T] \times {\bf R}^n} = \phi \}$

where the infimum is taken over all extensions ${\tilde \phi}$ of ${\phi}$ to the Minkowski spacetime ${{\bf R}^{1+n}}$; similarly one defines

$\displaystyle \| \phi_k \|_{S_k([-T,T] \times {\bf R}^n)} := \inf \{ \| \tilde \phi_k \|_{S_k({\bf R}^{1+n})}: \tilde \phi_k \downharpoonright_{[-T,T] \times {\bf R}^n} = \phi_k \}.$

The gap in the paper is as follows: it was implicitly assumed that one could restrict (1) to the slab ${[-T,T] \times {\bf R}^n}$ to obtain the equality

$\displaystyle \| \phi \|_{S(c)([-T,T] \times {\bf R}^n)} = \|\phi \|_{L^\infty_t L^\infty_x([-T,T] \times {\bf R}^n)} + \sup_k c_k^{-1} \| \phi_k \|_{S[k]([-T,T] \times {\bf R}^n)}.$

(This equality is implicitly used to establish the bound (36) in the paper.) Unfortunately, (1) only gives the lower bound, not the upper bound, and it is the upper bound which is needed here. The problem is that the extensions ${\tilde \phi_k}$ of ${\phi_k}$ that are optimal for computing ${\| \phi_k \|_{S[k]([-T,T] \times {\bf R}^n)}}$ are not necessarily the Littlewood-Paley projections of the extensions ${\tilde \phi}$ of ${\phi}$ that are optimal for computing ${\| \phi \|_{S(c)([-T,T] \times {\bf R}^n)}}$.

To remedy the problem, one has to prove an upper bound of the form

$\displaystyle \| \phi \|_{S(c)([-T,T] \times {\bf R}^n)} \lesssim \|\phi \|_{L^\infty_t L^\infty_x([-T,T] \times {\bf R}^n)} + \sup_k c_k^{-1} \| \phi_k \|_{S[k]([-T,T] \times {\bf R}^n)}$

for all Schwartz ${\phi}$ (actually we need affinely Schwartz ${\phi}$, but one can easily normalise to the Schwartz case). Without loss of generality we may normalise the RHS to be ${1}$. Thus

$\displaystyle \|\phi \|_{L^\infty_t L^\infty_x([-T,T] \times {\bf R}^n)} \leq 1 \ \ \ \ \ (2)$

and

$\displaystyle \|P_k \phi \|_{S[k]([-T,T] \times {\bf R}^n)} \leq c_k \ \ \ \ \ (3)$

for each ${k}$, and one has to find a single extension ${\tilde \phi}$ of ${\phi}$ such that

$\displaystyle \|\tilde \phi \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})} \lesssim 1 \ \ \ \ \ (4)$

and

$\displaystyle \|P_k \tilde \phi \|_{S[k]({\bf R}^{1+n})} \lesssim c_k \ \ \ \ \ (5)$

for each ${k}$. Achieving a ${\tilde \phi}$ that obeys (4) is trivial (just extend ${\phi}$ by zero), but such extensions do not necessarily obey (5). On the other hand, from (3) we can find extensions ${\tilde \phi_k}$ of ${P_k \phi}$ such that

$\displaystyle \|\tilde \phi_k \|_{S[k]({\bf R}^{1+n})} \lesssim c_k; \ \ \ \ \ (6)$

the extension ${\tilde \phi := \sum_k \tilde \phi_k}$ will then obey (5) (here we use Lemma 9 from my paper), but unfortunately is not guaranteed to obey (4) (the ${S[k]}$ norm does control the ${L^\infty_t L^\infty_x}$ norm, but a key point about frequency envelopes for the small energy regularity problem is that the coefficients ${c_k}$, while bounded, are not necessarily summable).

This can be fixed as follows. For each ${k}$ we introduce a time cutoff ${\eta_k}$ supported on ${[-T-2^{-k}, T+2^{-k}]}$ that equals ${1}$ on ${[-T-2^{-k-1},T+2^{-k+1}]}$ and obeys the usual derivative estimates in between (the ${j^{th}}$ time derivative of size ${O_j(2^{jk})}$ for each ${j}$). Later we will prove the truncation estimate

$\displaystyle \| \eta_k \tilde \phi_k \|_{S[k]({\bf R}^{1+n})} \lesssim \| \tilde \phi_k \|_{S[k]({\bf R}^{1+n})}. \ \ \ \ \ (7)$

Assuming this estimate, then if we set ${\tilde \phi := \sum_k \eta_k \tilde \phi_k}$, then using Lemma 9 in my paper and (6), (7) (and the local stability of frequency envelopes) we have the required property (5). (There is a technical issue arising from the fact that ${\tilde \phi}$ is not necessarily Schwartz due to slow decay at temporal infinity, but by considering partial sums in the ${k}$ summation and taking limits we can check that ${\tilde \phi}$ is the strong limit of Schwartz functions, which suffices here; we omit the details for sake of exposition.) So the only issue is to establish (4), that is to say that

$\displaystyle \| \sum_k \eta_k(t) \tilde \phi_k(t) \|_{L^\infty_x({\bf R}^n)} \lesssim 1$

for all ${t \in {\bf R}}$.

For ${t \in [-T,T]}$ this is immediate from (2). Now suppose that ${t \in [T+2^{k_0-1}, T+2^{k_0}]}$ for some integer ${k_0}$ (the case when ${t \in [-T-2^{k_0}, -T-2^{k_0-1}]}$ is treated similarly). Then we can split

$\displaystyle \sum_k \eta_k(t) \tilde \phi_k(t) = \Phi_1 + \Phi_2 + \Phi_3$

where

$\displaystyle \Phi_1 := \sum_{k < k_0} \tilde \phi_k(T)$

$\displaystyle \Phi_2 := \sum_{k < k_0} \tilde \phi_k(t) - \tilde \phi_k(T)$

$\displaystyle \Phi_3 := \eta_{k_0}(t) \tilde \phi_{k_0}(t).$

The contribution of the ${\Phi_3}$ term is acceptable by (6) and estimate (82) from my paper. The term ${\Phi_1}$ sums to ${P_{ which is acceptable by (2). So it remains to control the ${L^\infty_x}$ norm of ${\Phi_2}$. By the triangle inequality and the fundamental theorem of calculus, we can bound

$\displaystyle \| \Phi_2 \|_{L^\infty_x} \leq (t-T) \sum_{k < k_0} \| \partial_t \tilde \phi_k \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})}.$

By hypothesis, ${t-T \leq 2^{-k_0}}$. Using the first term in (79) of my paper and Bernstein’s inequality followed by (6) we have

$\displaystyle \| \partial_t \tilde \phi_k \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})} \lesssim 2^k \| \tilde \phi_k \|_{S[k]({\bf R}^{1+n})} \lesssim 2^k;$

and then we are done by summing the geometric series in ${k}$.

It remains to prove the truncation estimate (7). This estimate is similar in spirit to the algebra estimates already in my paper, but unfortunately does not seem to follow immediately from these estimates as written, and so one has to repeat the somewhat lengthy decompositions and case checkings used to prove these estimates. We do this below the fold.

[This blog post was written jointly by Terry Tao and Will Sawin.]

In the previous blog post, one of us (Terry) implicitly introduced a notion of rank for tensors which is a little different from the usual notion of tensor rank, and which (following BCCGNSU) we will call “slice rank”. This notion of rank could then be used to encode the Croot-Lev-Pach-Ellenberg-Gijswijt argument that uses the polynomial method to control capsets.

Afterwards, several papers have applied the slice rank method to further problems – to control tri-colored sum-free sets in abelian groups (BCCGNSU, KSS) and from there to the triangle removal lemma in vector spaces over finite fields (FL), to control sunflowers (NS), and to bound progression-free sets in ${p}$-groups (P).

In this post we investigate the notion of slice rank more systematically. In particular, we show how to give lower bounds for the slice rank. In many cases, we can show that the upper bounds on slice rank given in the aforementioned papers are sharp to within a subexponential factor. This still leaves open the possibility of getting a better bound for the original combinatorial problem using the slice rank of some other tensor, but for very long arithmetic progressions (at least eight terms), we show that the slice rank method cannot improve over the trivial bound using any tensor.

It will be convenient to work in a “basis independent” formalism, namely working in the category of abstract finite-dimensional vector spaces over a fixed field ${{\bf F}}$. (In the applications to the capset problem one takes ${{\bf F}={\bf F}_3}$ to be the finite field of three elements, but most of the discussion here applies to arbitrary fields.) Given ${k}$ such vector spaces ${V_1,\dots,V_k}$, we can form the tensor product ${\bigotimes_{i=1}^k V_i}$, generated by the tensor products ${v_1 \otimes \dots \otimes v_k}$ with ${v_i \in V_i}$ for ${i=1,\dots,k}$, subject to the constraint that the tensor product operation ${(v_1,\dots,v_k) \mapsto v_1 \otimes \dots \otimes v_k}$ is multilinear. For each ${1 \leq j \leq k}$, we have the smaller tensor products ${\bigotimes_{1 \leq i \leq k: i \neq j} V_i}$, as well as the ${j^{th}}$ tensor product

$\displaystyle \otimes_j: V_j \times \bigotimes_{1 \leq i \leq k: i \neq j} V_i \rightarrow \bigotimes_{i=1}^k V_i$

defined in the obvious fashion. Elements of ${\bigotimes_{i=1}^k V_i}$ of the form ${v_j \otimes_j v_{\hat j}}$ for some ${v_j \in V_j}$ and ${v_{\hat j} \in \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$ will be called rank one functions, and the slice rank (or rank for short) ${\hbox{rank}(v)}$ of an element ${v}$ of ${\bigotimes_{i=1}^k V_i}$ is defined to be the least nonnegative integer ${r}$ such that ${v}$ is a linear combination of ${r}$ rank one functions. If ${V_1,\dots,V_k}$ are finite-dimensional, then the rank is always well defined as a non-negative integer (in fact it cannot exceed ${\min( \hbox{dim}(V_1), \dots, \hbox{dim}(V_k))}$. It is also clearly subadditive:

$\displaystyle \hbox{rank}(v+w) \leq \hbox{rank}(v) + \hbox{rank}(w). \ \ \ \ \ (1)$

For ${k=1}$, ${\hbox{rank}(v)}$ is ${0}$ when ${v}$ is zero, and ${1}$ otherwise. For ${k=2}$, ${\hbox{rank}(v)}$ is the usual rank of the ${2}$-tensor ${v \in V_1 \otimes V_2}$ (which can for instance be identified with a linear map from ${V_1}$ to the dual space ${V_2^*}$). The usual notion of tensor rank for higher order tensors uses complete tensor products ${v_1 \otimes \dots \otimes v_k}$, ${v_i \in V_i}$ as the rank one objects, rather than ${v_j \otimes_j v_{\hat j}}$, giving a rank that is greater than or equal to the slice rank studied here.

From basic linear algebra we have the following equivalences:

Lemma 1 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$, let ${v}$ be an element of ${V_1 \otimes \dots \otimes V_k}$, and let ${r}$ be a non-negative integer. Then the following are equivalent:

• (i) One has ${\hbox{rank}(v) \leq r}$.
• (ii) One has a representation of the form

$\displaystyle v = \sum_{j=1}^k \sum_{s \in S_j} v_{j,s} \otimes_j v_{\hat j,s}$

where ${S_1,\dots,S_k}$ are finite sets of total cardinality ${|S_1|+\dots+|S_k|}$ at most ${r}$, and for each ${1 \leq j \leq k}$ and ${s \in S_j}$, ${v_{j,s} \in V_j}$ and ${v_{\hat j,s} \in \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$.

• (iii) One has

$\displaystyle v \in \sum_{j=1}^k U_j \otimes_j \bigotimes_{1 \leq i \leq k: i \neq j} V_i$

where for each ${j=1,\dots,k}$, ${U_j}$ is a subspace of ${V_j}$ of total dimension ${\hbox{dim}(U_1)+\dots+\hbox{dim}(U_k)}$ at most ${r}$, and we view ${U_j \otimes_j \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$ as a subspace of ${\bigotimes_{i=1}^k V_i}$ in the obvious fashion.

• (iv) (Dual formulation) There exist subspaces ${W_j}$ of the dual space ${V_j^*}$ for ${j=1,\dots,k}$, of total dimension at least ${\hbox{dim}(V_1)+\dots+\hbox{dim}(V_k) - r}$, such that ${v}$ is orthogonal to ${\bigotimes_{j=1}^k W_j}$, in the sense that one has the vanishing

$\displaystyle \langle \bigotimes_{j=1}^k w_j, v \rangle = 0$

for all ${w_j \in W_j}$, where ${\langle, \rangle: \bigotimes_{j=1}^k V_j^* \times \bigotimes_{j=1}^k V_j \rightarrow {\bf F}}$ is the obvious pairing.

Proof: The equivalence of (i) and (ii) is clear from definition. To get from (ii) to (iii) one simply takes ${U_j}$ to be the span of the ${v_{j,s}}$, and conversely to get from (iii) to (ii) one takes the ${v_{j,s}}$ to be a basis of the ${U_j}$ and computes ${v_{\hat j,s}}$ by using a basis for the tensor product ${\bigotimes_{j=1}^k U_j \otimes_j \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$ consisting entirely of functions of the form ${v_{j,s} \otimes_j e}$ for various ${e}$. To pass from (iii) to (iv) one takes ${W_j}$ to be the annihilator ${\{ w_j \in V_j: \langle w_j, v_j \rangle = 0 \forall v_j \in U_j \}}$ of ${U_j}$, and conversely to pass from (iv) to (iii). $\Box$

One corollary of the formulation (iv), is that the set of tensors of slice rank at most ${r}$ is Zariski closed (if the field ${{\mathbf F}}$ is algebraically closed), and so the slice rank itself is a lower semi-continuous function. This is in contrast to the usual tensor rank, which is not necessarily semicontinuous.

Corollary 2 Let ${V_1,\dots, V_k}$ be finite-dimensional vector spaces over an algebraically closed field ${{\bf F}}$. Let ${r}$ be a nonnegative integer. The set of elements of ${V_1 \otimes \dots \otimes V_k}$ of slice rank at most ${r}$ is closed in the Zariski topology.

Proof: In view of Lemma 1(i and iv), this set is the union over tuples of integers ${d_1,\dots,d_k}$ with ${d_1 + \dots + d_k \geq \hbox{dim}(V_1)+\dots+\hbox{dim}(V_k) - r}$ of the projection from ${\hbox{Gr}(d_1, V_1) \times \dots \times \hbox{Gr}(d_k, V_k) \times ( V_1 \otimes \dots \otimes V_k)}$ of the set of tuples ${(W_1,\dots,W_k, v)}$ with ${ v}$ orthogonal to ${W_1 \times \dots \times W_k}$, where ${\hbox{Gr}(d,V)}$ is the Grassmanian parameterizing ${d}$-dimensional subspaces of ${V}$.

One can check directly that the set of tuples ${(W_1,\dots,W_k, v)}$ with ${ v}$ orthogonal to ${W_1 \times \dots \times W_k}$ is Zariski closed in ${\hbox{Gr}(d_1, V_1) \times \dots \times \hbox{Gr}(d_k, V_k) \times V_1 \otimes \dots \otimes V_k}$ using a set of equations of the form ${\langle \bigotimes_{j=1}^k w_j, v \rangle = 0}$ locally on ${\hbox{Gr}(d_1, V_1) \times \dots \times \hbox{Gr}(d_k, V_k) }$. Hence because the Grassmanian is a complete variety, the projection of this set to ${V_1 \otimes \dots \otimes V_k}$ is also Zariski closed. So the finite union over tuples ${d_1,\dots,d_k}$ of these projections is also Zariski closed.

$\Box$

We also have good behaviour with respect to linear transformations:

Lemma 3 Let ${V_1,\dots,V_k, W_1,\dots,W_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$, let ${v}$ be an element of ${V_1 \otimes \dots \otimes V_k}$, and for each ${1 \leq j \leq k}$, let ${\phi_j: V_j \rightarrow W_j}$ be a linear transformation, with ${\bigotimes_{j=1}^k \phi_j: \bigotimes_{j=1}^k V_k \rightarrow \bigotimes_{j=1}^k W_k}$ the tensor product of these maps. Then

$\displaystyle \hbox{rank}( (\bigotimes_{j=1}^k \phi_j)(v) ) \leq \hbox{rank}(v). \ \ \ \ \ (2)$

Furthermore, if the ${\phi_j}$ are all injective, then one has equality in (2).

Thus, for instance, the rank of a tensor ${v \in \bigotimes_{j=1}^k V_k}$ is intrinsic in the sense that it is unaffected by any enlargements of the spaces ${V_1,\dots,V_k}$.

Proof: The bound (2) is clear from the formulation (ii) of rank in Lemma 1. For equality, apply (2) to the injective ${\phi_j}$, as well as to some arbitrarily chosen left inverses ${\phi_j^{-1}: W_j \rightarrow V_j}$ of the ${\phi_j}$. $\Box$

Computing the rank of a tensor is difficult in general; however, the problem becomes a combinatorial one if one has a suitably sparse representation of that tensor in some basis, where we will measure sparsity by the property of being an antichain.

Proposition 4 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$. For each ${1 \leq j \leq k}$, let ${(v_{j,s})_{s \in S_j}}$ be a linearly independent set in ${V_j}$ indexed by some finite set ${S_j}$. Let ${\Gamma}$ be a subset of ${S_1 \times \dots \times S_k}$.

Let ${v \in \bigotimes_{j=1}^k V_j}$ be a tensor of the form

$\displaystyle v = \sum_{(s_1,\dots,s_k) \in \Gamma} c_{s_1,\dots,s_k} v_{1,s_1} \otimes \dots \otimes v_{k,s_k} \ \ \ \ \ (3)$

where for each ${(s_1,\dots,s_k)}$, ${c_{s_1,\dots,s_k}}$ is a coefficient in ${{\bf F}}$. Then one has

$\displaystyle \hbox{rank}(v) \leq \min_{\Gamma = \Gamma_1 \cup \dots \cup \Gamma_k} |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)| \ \ \ \ \ (4)$

where the minimum ranges over all coverings of ${\Gamma}$ by sets ${\Gamma_1,\dots,\Gamma_k}$, and ${\pi_j: S_1 \times \dots \times S_k \rightarrow S_j}$ for ${j=1,\dots,k}$ are the projection maps.

Now suppose that the coefficients ${c_{s_1,\dots,s_k}}$ are all non-zero, that each of the ${S_j}$ are equipped with a total ordering ${\leq_j}$, and ${\Gamma'}$ is the set of maximal elements of ${\Gamma}$, thus there do not exist distinct ${(s_1,\dots,s_k) \in \Gamma'}$, ${(t_1,\dots,t_k) \in \Gamma}$ such that ${s_j \leq t_j}$ for all ${j=1,\dots,k}$. Then one has

$\displaystyle \hbox{rank}(v) \geq \min_{\Gamma' = \Gamma_1 \cup \dots \cup \Gamma_k} |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)|. \ \ \ \ \ (5)$

In particular, if ${\Gamma}$ is an antichain (i.e. every element is maximal), then equality holds in (4).

Proof: By Lemma 3 (or by enlarging the bases ${v_{j,s_j}}$), we may assume without loss of generality that each of the ${V_j}$ is spanned by the ${v_{j,s_j}}$. By relabeling, we can also assume that each ${S_j}$ is of the form

$\displaystyle S_j = \{1,\dots,|S_j|\}$

with the usual ordering, and by Lemma 3 we may take each ${V_j}$ to be ${{\bf F}^{|S_j|}}$, with ${v_{j,s_j} = e_{s_j}}$ the standard basis.

Let ${r}$ denote the rank of ${v}$. To show (4), it suffices to show the inequality

$\displaystyle r \leq |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)| \ \ \ \ \ (6)$

for any covering of ${\Gamma}$ by ${\Gamma_1,\dots,\Gamma_k}$. By removing repeated elements we may assume that the ${\Gamma_i}$ are disjoint. For each ${1 \leq j \leq k}$, the tensor

$\displaystyle \sum_{(s_1,\dots,s_k) \in \Gamma_j} c_{s_1,\dots,s_k} e_{s_1} \otimes \dots \otimes e_{s_k}$

can (after collecting terms) be written as

$\displaystyle \sum_{s_j \in \pi_j(\Gamma_j)} e_{s_j} \otimes_j v_{\hat j,s_j}$

for some ${v_{\hat j, s_j} \in \bigotimes_{1 \leq i \leq k: i \neq j} {\bf F}^{|S_i|}}$. Summing and using (1), we conclude the inequality (6).

Now assume that the ${c_{s_1,\dots,s_k}}$ are all non-zero and that ${\Gamma'}$ is the set of maximal elements of ${\Gamma}$. To conclude the proposition, it suffices to show that the reverse inequality

$\displaystyle r \geq |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)| \ \ \ \ \ (7)$

holds for some ${\Gamma_1,\dots,\Gamma_k}$ covering ${\Gamma'}$. By Lemma 1(iv), there exist subspaces ${W_j}$ of ${({\bf F}^{|S_j|})^*}$ whose dimension ${d_j := \hbox{dim}(W_j)}$ sums to

$\displaystyle \sum_{j=1}^k d_j = \sum_{j=1}^k |S_j| - r \ \ \ \ \ (8)$

such that ${v}$ is orthogonal to ${\bigotimes_{j=1}^k W_j}$.

Let ${1 \leq j \leq k}$. Using Gaussian elimination, one can find a basis ${w_{j,1},\dots,w_{j,d_j}}$ of ${W_j}$ whose representation in the standard dual basis ${e^*_{1},\dots,e^*_{|S_j|}}$ of ${({\bf F}^{|S_j|})^*}$ is in row-echelon form. That is to say, there exist natural numbers

$\displaystyle 1 \leq s_{j,1} < \dots < s_{j,d_j} \leq |S_j|$

such that for all ${1 \leq t \leq d_j}$, ${w_{j,t}}$ is a linear combination of the dual vectors ${e^*_{s_{j,t}},\dots,e^*_{|S_j|}}$, with the ${e^*_{s_{j,t}}}$ coefficient equal to one.

We now claim that ${\prod_{j=1}^k \{ s_{j,t}: 1 \leq t \leq d_j \}}$ is disjoint from ${\Gamma'}$. Suppose for contradiction that this were not the case, thus there exists ${1 \leq t_j \leq d_j}$ for each ${1 \leq j \leq k}$ such that

$\displaystyle (s_{1,t_1}, \dots, s_{k,t_k}) \in \Gamma'.$

As ${\Gamma'}$ is the set of maximal elements of ${\Gamma}$, this implies that

$\displaystyle (s'_1,\dots,s'_k) \not \in \Gamma$

for any tuple ${(s'_1,\dots,s'_k) \in \prod_{j=1}^k \{ s_{j,t_j}, \dots, |S_j|\}}$ other than ${(s_{1,t_1}, \dots, s_{k,t_k})}$. On the other hand, we know that ${w_{j,t_j}}$ is a linear combination of ${e^*_{s_{j,t_j}},\dots,e^*_{|S_j|}}$, with the ${e^*_{s_{j,t_j}}}$ coefficient one. We conclude that the tensor product ${\bigotimes_{j=1}^k w_{j,t_j}}$ is equal to

$\displaystyle \bigotimes_{j=1}^k e^*_{s_{j,t_j}}$

plus a linear combination of other tensor products ${\bigotimes_{j=1}^k e^*_{s'_j}}$ with ${(s'_1,\dots,s'_k)}$ not in ${\Gamma}$. Taking inner products with (3), we conclude that ${\langle v, \bigotimes_{j=1}^k w_{j,t_j}\rangle = c_{s_{1,t_1},\dots,s_{k,t_k}} \neq 0}$, contradicting the fact that ${v}$ is orthogonal to ${\prod_{j=1}^k W_j}$. Thus we have ${\prod_{j=1}^k \{ s_{j,t}: 1 \leq t \leq d_j \}}$ disjoint from ${\Gamma'}$.

For each ${1 \leq j \leq k}$, let ${\Gamma_j}$ denote the set of tuples ${(s_1,\dots,s_k)}$ in ${\Gamma'}$ with ${s_j}$ not of the form ${\{ s_{j,t}: 1 \leq t \leq d_j \}}$. From the previous discussion we see that the ${\Gamma_j}$ cover ${\Gamma'}$, and we clearly have ${\pi_j(\Gamma_j) \leq |S_j| - d_j}$, and hence from (8) we have (7) as claimed. $\Box$

As an instance of this proposition, we recover the computation of diagonal rank from the previous blog post:

Example 5 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$ for some ${k \geq 2}$. Let ${d}$ be a natural number, and for ${1 \leq j \leq k}$, let ${e_{j,1},\dots,e_{j,d}}$ be a linearly independent set in ${V_j}$. Let ${c_1,\dots,c_d}$ be non-zero coefficients in ${{\bf F}}$. Then

$\displaystyle \sum_{t=1}^d c_t e_{1,t} \otimes \dots \otimes e_{k,t}$

has rank ${d}$. Indeed, one applies the proposition with ${S_1,\dots,S_k}$ all equal to ${\{1,\dots,d\}}$, with ${\Gamma}$ the diagonal in ${S_1 \times \dots \times S_k}$; this is an antichain if we give one of the ${S_i}$ the standard ordering, and another of the ${S_i}$ the opposite ordering (and ordering the remaining ${S_i}$ arbitrarily). In this case, the ${\pi_j}$ are all bijective, and so it is clear that the minimum in (4) is simply ${d}$.

The combinatorial minimisation problem in the above proposition can be solved asymptotically when working with tensor powers, using the notion of the Shannon entropy ${h(X)}$ of a discrete random variable ${X}$.

Proposition 6 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$. For each ${1 \leq j \leq k}$, let ${(v_{j,s})_{s \in S_j}}$ be a linearly independent set in ${V_j}$ indexed by some finite set ${S_j}$. Let ${\Gamma}$ be a non-empty subset of ${S_1 \times \dots \times S_k}$.

Let ${v \in \bigotimes_{j=1}^k V_j}$ be a tensor of the form (3) for some coefficients ${c_{s_1,\dots,s_k}}$. For each natural number ${n}$, let ${v^{\otimes n}}$ be the tensor power of ${n}$ copies of ${v}$, viewed as an element of ${\bigotimes_{j=1}^k V_j^{\otimes n}}$. Then

$\displaystyle \hbox{rank}(v^{\otimes n}) \leq \exp( (H + o(1)) n ) \ \ \ \ \ (9)$

as ${n \rightarrow \infty}$, where ${H}$ is the quantity

$\displaystyle H = \hbox{sup}_{(X_1,\dots,X_k)} \hbox{min}( h(X_1), \dots, h(X_k) ) \ \ \ \ \ (10)$

and ${(X_1,\dots,X_k)}$ range over the random variables taking values in ${\Gamma}$.

Now suppose that the coefficients ${c_{s_1,\dots,s_k}}$ are all non-zero and that each of the ${S_j}$ are equipped with a total ordering ${\leq_j}$. Let ${\Gamma'}$ be the set of maximal elements of ${\Gamma}$ in the product ordering, and let ${H' = \hbox{sup}_{(X_1,\dots,X_k)} \hbox{min}( h(X_1), \dots, h(X_k) ) }$ where ${(X_1,\dots,X_k)}$ range over random variables taking values in ${\Gamma'}$. Then

$\displaystyle \hbox{rank}(v^{\otimes n}) \geq \exp( (H' + o(1)) n ) \ \ \ \ \ (11)$

as ${n \rightarrow \infty}$. In particular, if the maximizer in (10) is supported on the maximal elements of ${\Gamma}$ (which always holds if ${\Gamma}$ is an antichain in the product ordering), then equality holds in (9).

Proof:

It will suffice to show that

$\displaystyle \min_{\Gamma^n = \Gamma_{n,1} \cup \dots \cup \Gamma_{n,k}} |\pi_{n,1}(\Gamma_{n,1})| + \dots + |\pi_{n,k}(\Gamma_{n,k})| = \exp( (H + o(1)) n ) \ \ \ \ \ (12)$

as ${n \rightarrow \infty}$, where ${\pi_{n,j}: \prod_{i=1}^k S_i^n \rightarrow S_j^n}$ is the projection map. Then the same thing will apply to ${\Gamma'}$ and ${H'}$. Then applying Proposition 4, using the lexicographical ordering on ${S_j^n}$ and noting that, if ${\Gamma'}$ are the maximal elements of ${\Gamma}$, then ${\Gamma'^n}$ are the maximal elements of ${\Gamma^n}$, we obtain both (9) and (11).

We first prove the lower bound. By compactness (and the continuity properties of entropy), we can find a random variable ${(X_1,\dots,X_k)}$ taking values in ${\Gamma}$ such that

$\displaystyle H = \hbox{min}( h(X_1), \dots, h(X_k) ). \ \ \ \ \ (13)$

Let ${\varepsilon = o(1)}$ be a small positive quantity that goes to zero sufficiently slowly with ${n}$. Let ${\Sigma = \Sigma_{X_1,\dots,X_k} \subset \Gamma^n}$ denote the set of all tuples ${(a_1, \dots, \vec a_n)}$ in ${\Gamma^n}$ that are within ${\varepsilon}$ of being distributed according to the law of ${(X_1,\dots,X_k)}$, in the sense that for all ${a \in \Gamma}$, one has

$\displaystyle |\frac{|\{ 1 \leq l \leq n: a_l = a \}|}{n} - {\bf P}( (X_1,\dots,X_k) = a )| \leq \varepsilon.$

By the asymptotic equipartition property, the cardinality of ${\Sigma}$ can be computed to be

$\displaystyle |\Sigma| = \exp( (h( X_1,\dots,X_k)+o(1)) n ) \ \ \ \ \ (14)$

if ${\varepsilon}$ goes to zero slowly enough. Similarly one has

$\displaystyle |\pi_{n,j}(\Sigma)| = \exp( (h( X_j)+o(1)) n ),$

and for each ${s_{n,j} \in \pi_{n,j}(\Sigma)}$, one has

$\displaystyle |\{ \sigma \in \Sigma: \pi_{n,j}(\sigma) = s_{n,j} \}| \leq \exp( (h( X_1,\dots,X_k)-h(X_j)+o(1)) n ). \ \ \ \ \ (15)$

Now let ${\Gamma^n = \Gamma_{n,1} \cup \dots \cup \Gamma_{n,k}}$ be an arbitrary covering of ${\Gamma^n}$. By the pigeonhole principle, there exists ${1 \leq j \leq k}$ such that

$\displaystyle |\Gamma_{n,j} \cap \Sigma| \geq \frac{1}{k} |\Sigma|$

and hence by (14), (15)

$\displaystyle |\pi_{n,j}( \Gamma_{n,j} \cap \Sigma)| \geq \frac{1}{k} \exp( (h( X_j)+o(1)) n )$

which by (13) implies that

$\displaystyle |\pi_{n,1}(\Gamma_{n,1})| + \dots + |\pi_{n,k}(\Gamma_{n,k})| \geq \exp( (H + o(1)) n )$

noting that the ${\frac{1}{k}}$ factor can be absorbed into the ${o(1)}$ error). This gives the lower bound in (12).

Now we prove the upper bound. We can cover ${\Gamma^n}$ by ${O(\exp(o(n))}$ sets of the form ${\Sigma_{X_1,\dots,X_k}}$ for various choices of random variables ${(X_1,\dots,X_k)}$ taking values in ${\Gamma}$. For each such random variable ${(X_1,\dots,X_k)}$, we can find ${1 \leq j \leq k}$ such that ${h(X_j) \leq H}$; we then place all of ${\Sigma_{X_1,\dots,X_k}}$ in ${\Gamma_j}$. It is then clear that the ${\Gamma_j}$ cover ${\Gamma}$ and that

$\displaystyle |\Gamma_j| \leq \exp( (H+o(1)) n )$

for all ${j=1,\dots,n}$, giving the required upper bound. $\Box$

It is of interest to compute the quantity ${H}$ in (10). We have the following criterion for when a maximiser occurs:

Proposition 7 Let ${S_1,\dots,S_k}$ be finite sets, and ${\Gamma \subset S_1 \times \dots \times S_k}$ be non-empty. Let ${H}$ be the quantity in (10). Let ${(X_1,\dots,X_k)}$ be a random variable taking values in ${\Gamma}$, and let ${\Gamma^* \subset \Gamma}$ denote the essential range of ${(X_1,\dots,X_k)}$, that is to say the set of tuples ${(t_1,\dots,t_k)\in \Gamma}$ such that ${{\bf P}( X_1=t_1, \dots, X_k = t_k)}$ is non-zero. Then the following are equivalent:

• (i) ${(X_1,\dots,X_k)}$ attains the maximum in (10).
• (ii) There exist weights ${w_1,\dots,w_k \geq 0}$ and a finite quantity ${D \geq 0}$, such that ${w_j=0}$ whenever ${h(X_j) > \min(h(X_1),\dots,h(X_k))}$, and such that

$\displaystyle \sum_{j=1}^k w_j \log \frac{1}{{\bf P}(X_j = t_j)} \leq D \ \ \ \ \ (16)$

for all ${(t_1,\dots,t_k) \in \Gamma}$, with equality if ${(t_1,\dots,t_k) \in \Gamma^*}$. (In particular, ${w_j}$ must vanish if there exists a ${t_j \in \pi_i(\Gamma)}$ with ${{\bf P}(X_j=t_j)=0}$.)

Furthermore, when (i) and (ii) holds, one has

$\displaystyle D = H \sum_{j=1}^k w_j. \ \ \ \ \ (17)$

Proof: We first show that (i) implies (ii). The function ${p \mapsto p \log \frac{1}{p}}$ is concave on ${[0,1]}$. As a consequence, if we define ${C}$ to be the set of tuples ${(h_1,\dots,h_k) \in [0,+\infty)^k}$ such that there exists a random variable ${(Y_1,\dots,Y_k)}$ taking values in ${\Gamma}$ with ${h(Y_j)=h_j}$, then ${C}$ is convex. On the other hand, by (10), ${C}$ is disjoint from the orthant ${(H,+\infty)^k}$. Thus, by the hyperplane separation theorem, we conclude that there exists a half-space

$\displaystyle \{ (h_1,\dots,h_k) \in {\bf R}^k: w_1 h_1 + \dots + w_k h_k \geq c \},$

where ${w_1,\dots,w_k}$ are reals that are not all zero, and ${c}$ is another real, which contains ${(h(X_1),\dots,h(X_k))}$ on its boundary and ${(H,+\infty)^k}$ in its interior, such that ${C}$ avoids the interior of the half-space. Since ${(h(X_1),\dots,h(X_k))}$ is also on the boundary of ${(H,+\infty)^k}$, we see that the ${w_j}$ are non-negative, and that ${w_j = 0}$ whenever ${h(X_j) \neq H}$.

By construction, the quantity

$\displaystyle w_1 h(Y_1) + \dots + w_k h(Y_k)$

is maximised when ${(Y_1,\dots,Y_k) = (X_1,\dots,X_k)}$. At this point we could use the method of Lagrange multipliers to obtain the required constraints, but because we have some boundary conditions on the ${(Y_1,\dots,Y_k)}$ (namely, that the probability that they attain a given element of ${\Gamma}$ has to be non-negative) we will work things out by hand. Let ${t = (t_1,\dots,t_k)}$ be an element of ${\Gamma}$, and ${s = (s_1,\dots,s_k)}$ an element of ${\Gamma^*}$. For ${\varepsilon>0}$ small enough, we can form a random variable ${(Y_1,\dots,Y_k)}$ taking values in ${\Gamma}$, whose probability distribution is the same as that for ${(X_1,\dots,X_k)}$ except that the probability of attaining ${(t_1,\dots,t_k)}$ is increased by ${\varepsilon}$, and the probability of attaining ${(s_1,\dots,s_k)}$ is decreased by ${\varepsilon}$. If there is any ${j}$ for which ${{\bf P}(X_j = t_j)=0}$ and ${w_j \neq 0}$, then one can check that

$\displaystyle w_1 h(Y_1) + \dots + w_k h(Y_k) - (w_1 h(X_1) + \dots + w_k h(X_k)) \gg \varepsilon \log \frac{1}{\varepsilon}$

for sufficiently small ${\varepsilon}$, contradicting the maximality of ${(X_1,\dots,X_k)}$; thus we have ${{\bf P}(X_j = t_j) > 0}$ whenever ${w_j \neq 0}$. Taylor expansion then gives

$\displaystyle w_1 h(Y_1) + \dots + w_k h(Y_k) - (w_1 h(X_1) + \dots + w_k h(X_k)) = (A_t - A_s) \varepsilon + O(\varepsilon^2)$

for small ${\varepsilon}$, where

$\displaystyle A_t := \sum_{j=1}^k w_j \log \frac{1}{{\bf P}(X_j = t_j)}$

and similarly for ${A_s}$. We conclude that ${A_t \leq A_s}$ for all ${s \in \Gamma^*}$ and ${t \in \Gamma}$, thus there exists a quantity ${D}$ such that ${A_s = D}$ for all ${s \in \Gamma^*}$, and ${A_t \leq D}$ for all ${t \in \Gamma}$. By construction ${D}$ must be nonnegative. Sampling ${(t_1,\dots,t_k)}$ using the distribution of ${(X_1,\dots,X_k)}$, one has

$\displaystyle \sum_{j=1}^k w_j \log \frac{1}{{\bf P}(X_j = t_j)} = D$

almost surely; taking expectations we conclude that

$\displaystyle \sum_{j=1}^k w_j \sum_{t_j \in S_j} {\bf P}( X_j = t_j) \log \frac{1}{{\bf P}(X_j = t_j)} = D.$

The inner sum is ${h(X_j)}$, which equals ${H}$ when ${w_j}$ is non-zero, giving (17).

Now we show conversely that (ii) implies (i). As noted previously, the function ${p \mapsto p \log \frac{1}{p}}$ is concave on ${[0,1]}$, with derivative ${\log \frac{1}{p} - 1}$. This gives the inequality

$\displaystyle q \log \frac{1}{q} \leq p \log \frac{1}{p} + (q-p) ( \log \frac{1}{p} - 1 ) \ \ \ \ \ (18)$

for any ${0 \leq p,q \leq 1}$ (note the right-hand side may be infinite when ${p=0}$ and ${q>0}$). Let ${(Y_1,\dots,Y_k)}$ be any random variable taking values in ${\Gamma}$, then on applying the above inequality with ${p = {\bf P}(X_j = t_j)}$ and ${q = {\bf P}( Y_j = t_j )}$, multiplying by ${w_j}$, and summing over ${j=1,\dots,k}$ and ${t_j \in S_j}$ gives

$\displaystyle \sum_{j=1}^k w_j h(Y_j) \leq \sum_{j=1}^k w_j h(X_j)$

$\displaystyle + \sum_{j=1}^k \sum_{t_j \in S_j} w_j ({\bf P}(Y_j = t_j) - {\bf P}(X_j = t_j)) ( \log \frac{1}{{\bf P}(X_j=t_j)} - 1 ).$

By construction, one has

$\displaystyle \sum_{j=1}^k w_j h(X_j) = \min(h(X_1),\dots,h(X_k)) \sum_{j=1}^k w_j$

and

$\displaystyle \sum_{j=1}^k w_j h(Y_j) \geq \min(h(Y_1),\dots,h(Y_k)) \sum_{j=1}^k w_j$

so to prove that ${\min(h(Y_1),\dots,h(Y_k)) \leq \min(h(X_1),\dots,h(X_k))}$ (which would give (i)), it suffices to show that

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j ({\bf P}(Y_j = t_j) - {\bf P}(X_j = t_j)) ( \log \frac{1}{{\bf P}(X_j=t_j)} - 1 ) \leq 0,$

or equivalently that the quantity

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j {\bf P}(Y_j = t_j) ( \log \frac{1}{{\bf P}(X_j=t_j)} - 1 )$

is maximised when ${(Y_1,\dots,Y_k) = (X_1,\dots,X_k)}$. Since

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j {\bf P}(Y_j = t_j) = \sum_{j=1}^k w_j$

it suffices to show this claim for the quantity

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j {\bf P}(Y_j = t_j) \log \frac{1}{{\bf P}(X_j=t_j)}.$

One can view this quantity as

$\displaystyle {\bf E}_{(Y_1,\dots,Y_k)} \sum_{j=1}^k w_j \log \frac{1}{{\bf P}_{X_j}(X_j=Y_j)}.$

By (ii), this quantity is bounded by ${D}$, with equality if ${(Y_1,\dots,Y_k)}$ is equal to ${(X_1,\dots,X_k)}$ (and is in particular ranging in ${\Gamma^*}$), giving the claim. $\Box$

The second half of the proof of Proposition 7 only uses the marginal distributions ${{{\bf P}(X_j=t_j)}}$ and the equation(16), not the actual distribution of ${(X_1,\dots,X_k)}$, so it can also be used to prove an upper bound on ${H}$ when the exact maximizing distribution is not known, given suitable probability distributions in each variable. The logarithm of the probability distribution here plays the role that the weight functions do in BCCGNSU.

Remark 8 Suppose one is in the situation of (i) and (ii) above; assume the nondegeneracy condition that ${H}$ is positive (or equivalently that ${D}$ is positive). We can assign a “degree” ${d_j(t_j)}$ to each element ${t_j \in S_j}$ by the formula

$\displaystyle d_j(t_j) := w_j \log \frac{1}{{\bf P}(X_j = t_j)}, \ \ \ \ \ (19)$

then every tuple ${(t_1,\dots,t_k)}$ in ${\Gamma}$ has total degree at most ${D}$, and those tuples in ${\Gamma^*}$ have degree exactly ${D}$. In particular, every tuple in ${\Gamma^n}$ has degree at most ${nD}$, and hence by (17), each such tuple has a ${j}$-component of degree less than or equal to ${nHw_j}$ for some ${j}$ with ${w_j>0}$. On the other hand, we can compute from (19) and the fact that ${h(X_j) = H}$ for ${w_j > 0}$ that ${Hw_j = {\bf E} d_j(X_j)}$. Thus, by asymptotic equipartition, and assuming ${w_j \neq 0}$, the number of “monomials” in ${S_j^n}$ of total degree at most ${nHw_j}$ is at most ${\exp( (h(X_j)+o(1)) n )}$; one can in fact use (19) and (18) to show that this is in fact an equality. This gives a direct way to cover ${\Gamma^n}$ by sets ${\Gamma_{n,1},\dots,\Gamma_{n,k}}$ with ${|\pi_j(\Gamma_{n,j})| \leq \exp( (H+o(1)) n)}$, which is in the spirit of the Croot-Lev-Pach-Ellenberg-Gijswijt arguments from the previous post.

We can now show that the rank computation for the capset problem is sharp:

Proposition 9 Let ${V_1^{\otimes n} = V_2^{\otimes n} = V_3^{\otimes n}}$ denote the space of functions from ${{\bf F}_3^n}$ to ${{\bf F}_3}$. Then the function ${(x,y,z) \mapsto \delta_{0^n}(x,y,z)}$ from ${{\bf F}_3^n \times {\bf F}_3^n \times {\bf F}_3^n}$ to ${{\bf F}}$, viewed as an element of ${V_1^{\otimes n} \otimes V_2^{\otimes n} \otimes V_3^{\otimes n}}$, has rank ${\exp( (H^*+o(1)) n )}$ as ${n \rightarrow \infty}$, where ${H^* \approx 1.013445}$ is given by the formula

$\displaystyle H^* = \alpha \log \frac{1}{\alpha} + \beta \log \frac{1}{\beta} + \gamma \log \frac{1}{\gamma} \ \ \ \ \ (20)$

with

$\displaystyle \alpha = \frac{32}{3(15 + \sqrt{33})} \approx 0.51419$

$\displaystyle \beta = \frac{4(\sqrt{33}-1)}{3(15+\sqrt{33})} \approx 0.30495$

$\displaystyle \gamma = \frac{(\sqrt{33}-1)^2}{6(15+\sqrt{33})} \approx 0.18086.$

Proof: In ${{\bf F}_3 \times {\bf F}_3 \times {\bf F}_3}$, we have

$\displaystyle \delta_0(x+y+z) = 1 - (x+y+z)^2$

$\displaystyle = (1-x^2) - y^2 - z^2 + xy + yz + zx.$

Thus, if we let ${V_1=V_2=V_3}$ be the space of functions from ${{\bf F}_3}$ to ${{\bf F}_3}$ (with domain variable denoted ${x,y,z}$ respectively), and define the basis functions

$\displaystyle v_{1,0} := 1; v_{1,1} := x; v_{1,2} := x^2$

$\displaystyle v_{2,0} := 1; v_{2,1} := y; v_{2,2} := y^2$

$\displaystyle v_{3,0} := 1; v_{3,1} := z; v_{3,2} := z^2$

of ${V_1,V_2,V_3}$ indexed by ${S_1=S_2=S_3 := \{ 0,1,2\}}$ (with the usual ordering), respectively, and set ${\Gamma \subset S_1 \times S_2 \times S_3}$ to be the set

$\displaystyle \{ (2,0,0), (0,2,0), (0,0,2), (1,1,0), (0,1,1), (1,0,1),(0,0,0) \}$

then ${\delta_0(x,y,z)}$ is a linear combination of the ${v_{1,t_1} \otimes v_{1,t_2} \otimes v_{1,t_3}}$ with ${(t_1,t_2,t_3) \in \Gamma}$, and all coefficients non-zero. Then we have ${\Gamma'= \{ (2,0,0), (0,2,0), (0,0,2), (1,1,0), (0,1,1), (1,0,1) \}}$. We will show that the quantity ${H}$ of (10) agrees with the quantity ${H^*}$ of (20), and that the optimizing distribution is supported on ${\Gamma'}$, so that by Proposition 6 the rank of ${\delta_{0^n}(x,y,z)}$ is ${\exp( (H+o(1)) n)}$.

To compute the quantity at (10), we use the criterion in Proposition 7. We take ${(X_1,X_2,X_3)}$ to be the random variable taking values in ${\Gamma}$ that attains each of the values ${(2,0,0), (0,2,0), (0,0,2)}$ with a probability of ${\gamma \approx 0.18086}$, and each of ${(1,1,0), (0,1,1), (1,0,1)}$ with a probability of ${\alpha - 2\gamma = \beta/2 \approx 0.15247}$; then each of the ${X_j}$ attains the values of ${0,1,2}$ with probabilities ${\alpha,\beta,\gamma}$ respectively, so in particular ${h(X_1)=h(X_2)=h(X_3)}$ is equal to the quantity ${H'}$ in (20). If we now set ${w_1 = w_2 = w_3 := 1}$ and

$\displaystyle D := 2\log \frac{1}{\alpha} + \log \frac{1}{\gamma} = \log \frac{1}{\alpha} + 2 \log \frac{1}{\beta} = 3H^* \approx 3.04036$

we can verify the condition (16) with equality for all ${(t_1,t_2,t_3) \in \Gamma'}$, which from (17) gives ${H=H'=H^*}$ as desired. $\Box$

This statement already follows from the result of Kleinberg-Sawin-Speyer, which gives a “tri-colored sum-free set” in ${\mathbb F_3^n}$ of size ${\exp((H'+o(1))n)}$, as the slice rank of this tensor is an upper bound for the size of a tri-colored sum-free set. If one were to go over the proofs more carefully to evaluate the subexponential factors, this argument would give a stronger lower bound than KSS, as it does not deal with the substantial loss that comes from Behrend’s construction. However, because it actually constructs a set, the KSS result rules out more possible approaches to give an exponential improvement of the upper bound for capsets. The lower bound on slice rank shows that the bound cannot be improved using only the slice rank of this particular tensor, whereas KSS shows that the bound cannot be improved using any method that does not take advantage of the “single-colored” nature of the problem.

We can also show that the slice rank upper bound in a result of Naslund-Sawin is similarly sharp:

Proposition 10 Let ${V_1^{\otimes n} = V_2^{\otimes n} = V_3^{\otimes n}}$ denote the space of functions from ${\{0,1\}^n}$ to ${\mathbb C}$. Then the function ${(x,y,z) \mapsto \prod_{i=1}^n (x_i+y_i+z_i)-1}$ from ${\{0,1\}^n \times \{0,1\}^n \times \{0,1\}^n \rightarrow \mathbb C}$, viewed as an element of ${V_1^{\otimes n} \otimes V_2^{\otimes n} \otimes V_3^{\otimes n}}$, has slice rank ${(3/2^{2/3})^n e^{o(n)}}$

Proof: Let ${v_{1,0}=1}$ and ${v_{1,1}=x}$ be a basis for the space ${V_1}$ of functions on ${\{0,1\}}$, itself indexed by ${S_1=\{0,1\}}$. Choose similar bases for ${V_2}$ and ${V_3}$, with ${v_{2,0}=1, v_{2,1}=y}$ and ${v_{3,0}=1,v_{3,1}=z-1}$.

Set ${\Gamma = \{(1,0,0),(0,1,0),(0,0,1)\}}$. Then ${x+y+z-1}$ is a linear combination of the ${v_{1,t_1} \otimes v_{1,t_2} \otimes v_{1,t_3}}$ with ${(t_1,t_2,t_3) \in \Gamma}$, and all coefficients non-zero. Order ${S_1,S_2,S_3}$ the usual way so that ${\Gamma}$ is an antichain. We will show that the quantity ${H}$ of (10) is ${\log(3/2^{2/3})}$, so that applying the last statement of Proposition 6, we conclude that the rank of ${\delta_{0^n}(x,y,z)}$ is ${\exp( (\log(3/2^{2/3})+o(1)) n)= (3/2^{2/3})^n e^{o(n)}}$ ,

Let ${(X_1,X_2,X_3)}$ be the random variable taking values in ${\Gamma}$ that attains each of the values ${(1,0,0),(0,1,0),(0,0,1)}$ with a probability of ${1/3}$. Then each of the ${X_i}$ attains the value ${1}$ with probability ${1/3}$ and ${0}$ with probability ${2/3}$, so

$\displaystyle h(X_1)=h(X_2)=h(X_3) = (1/3) \log (3) + (2/3) \log(3/2) = \log 3 - (2/3) \log 2= \log (3/2^{2/3})$

Setting ${w_1=w_2=w_3=1}$ and ${D=3 \log(3/2^{2/3})=3 \log 3 - 2 \log 2}$, we can verify the condition (16) with equality for all ${(t_1,t_2,t_3) \in \Gamma'}$, which from (17) gives ${H=\log (3/2^{2/3})}$ as desired. $\Box$

We used a slightly different method in each of the last two results. In the first one, we use the most natural bases for all three vector spaces, and distinguish ${\Gamma}$ from its set of maximal elements ${\Gamma'}$. In the second one we modify one basis element slightly, with ${v_{3,1}=z-1}$ instead of the more obvious choice ${z}$, which allows us to work with ${\Gamma = \{(1,0,0),(0,1,0),(0,0,1)\}}$ instead of ${\Gamma=\{(1,0,0),(0,1,0),(0,0,1),(0,0,0)\}}$. Because ${\Gamma}$ is an antichain, we do not need to distinguish ${\Gamma}$ and ${\Gamma'}$. Both methods in fact work with either problem, and they are both about equally difficult, but we include both as either might turn out to be substantially more convenient in future work.

Proposition 11 Let ${k \geq 8}$ be a natural number and let ${G}$ be a finite abelian group. Let ${{\bf F}}$ be any field. Let ${V_1 = \dots = V_k}$ denote the space of functions from ${G}$ to ${{\bf F}}$.

Let ${F}$ be any ${{\bf F}}$-valued function on ${G^k}$ that is nonzero only when the ${k}$ elements of ${G^n}$ form a ${k}$-term arithmetic progression, and is nonzero on every ${k}$-term constant progression.

Then the slice rank of ${F}$ is ${|G|}$.

Proof: We apply Proposition 4, using the standard bases of ${V_1,\dots,V_k}$. Let ${\Gamma}$ be the support of ${F}$. Suppose that we have ${k}$ orderings on ${H}$ such that the constant progressions are maximal elements of ${\Gamma}$ and thus all constant progressions lie in ${\Gamma'}$. Then for any partition ${\Gamma_1,\dots, \Gamma_k}$ of ${\Gamma'}$, ${\Gamma_j}$ can contain at most ${|\pi_j(\Gamma_j)|}$ constant progressions, and as all ${|G|}$ constant progressions must lie in one of the ${\Gamma_j}$, we must have ${\sum_{j=1}^k |\pi_j(\Gamma_j)| \geq |G|}$. By Proposition 4, this implies that the slice rank of ${F}$ is at least ${|G|}$. Since ${F}$ is a ${|G| \times \dots \times |G|}$ tensor, the slice rank is at most ${|G|}$, hence exactly ${|G|}$.

So it is sufficient to find ${k}$ orderings on ${G}$ such that the constant progressions are maximal element of ${\Gamma}$. We make several simplifying reductions: We may as well assume that ${\Gamma}$ consists of all the ${k}$-term arithmetic progressions, because if the constant progressions are maximal among the set of all progressions then they are maximal among its subset ${\Gamma}$. So we are looking for an ordering in which the constant progressions are maximal among all ${k}$-term arithmetic progressions. We may as well assume that ${G}$ is cyclic, because if for each cyclic group we have an ordering where constant progressions are maximal, on an arbitrary finite abelian group the lexicographic product of these orderings is an ordering for which the constant progressions are maximal. We may assume ${k=8}$, as if we have an ${8}$-tuple of orderings where constant progressions are maximal, we may add arbitrary orderings and the constant progressions will remain maximal.

So it is sufficient to find ${8}$ orderings on the cyclic group ${\mathbb Z/n}$ such that the constant progressions are maximal elements of the set of ${8}$-term progressions in ${\mathbb Z/n}$ in the ${8}$-fold product ordering. To do that, let the first, second, third, and fifth orderings be the usual order on ${\{0,\dots,n-1\}}$ and let the fourth, sixth, seventh, and eighth orderings be the reverse of the usual order on ${\{0,\dots,n-1\}}$.

Then let ${(c,c,c,c,c,c,c,c)}$ be a constant progression and for contradiction assume that ${(a,a+b,a+2b,a+3b,a+4b,a+5b,a+6b,a+7b)}$ is a progression greater than ${(c,c,c,c,c,c,c,c)}$ in this ordering. We may assume that ${c \in [0, (n-1)/2]}$, because otherwise we may reverse the order of the progression, which has the effect of reversing all eight orderings, and then apply the transformation ${x \rightarrow n-1-x}$, which again reverses the eight orderings, bringing us back to the original problem but with ${c \in [0,(n-1)/2]}$.

Take a representative of the residue class ${b}$ in the interval ${[-n/2,n/2]}$. We will abuse notation and call this ${b}$. Observe that ${a+b, a+2b,}$ ${a+3b}$, and ${a+5b}$ are all contained in the interval ${[0,c]}$ modulo ${n}$. Take a representative of the residue class ${a}$ in the interval ${[0,c]}$. Then ${a+b}$ is in the interval ${[mn,mn+c]}$ for some ${m}$. The distance between any distinct pair of intervals of this type is greater than ${n/2}$, but the distance between ${a}$ and ${a+b}$ is at most ${n/2}$, so ${a+b}$ is in the interval ${[0,c]}$. By the same reasoning, ${a+2b}$ is in the interval ${[0,c]}$. Therefore ${|b| \leq c/2< n/4}$. But then the distance between ${a+2b}$ and ${a+4b}$ is at most ${n/2}$, so by the same reasoning ${a+4b}$ is in the interval ${[0,c]}$. Because ${a+3b}$ is between ${a+2b}$ and ${a+4b}$, it also lies in the interval ${[0,c]}$. Because ${a+3b}$ is in the interval ${[0,c]}$, and by assumption it is congruent mod ${n}$ to a number in the set ${\{0,\dots,n-1\}}$ greater than or equal to ${c}$, it must be exactly ${c}$. Then, remembering that ${a+2b}$ and ${a+4b}$ lie in ${[0,c]}$, we have ${c-b \leq b}$ and ${c+b \leq b}$, so ${b=0}$, hence ${a=c}$, thus ${(a,\dots,a+7b)=(c,\dots,c)}$, which contradicts the assumption that ${(a,\dots,a+7b)>(c,\dots,c)}$. $\Box$

In fact, given a ${k}$-term progressions mod ${n}$ and a constant, we can form a ${k}$-term binary sequence with a ${1}$ for each step of the progression that is greater than the constant and a ${0}$ for each step that is less. Because a rotation map, viewed as a dynamical system, has zero topological entropy, the number of ${k}$-term binary sequences that appear grows subexponentially in ${k}$. Hence there must be, for large enough ${k}$, at least one sequence that does not appear. In this proof we exploit a sequence that does not appear for ${k=8}$.