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Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer, “for his pivotal role in the development of the mathematical theory of wavelets”.  The actual prize ceremony will be at Oslo in May.

I am actually in Oslo myself currently, having just presented Meyer’s work at the announcement ceremony (and also having written a brief description of some of his work).  The Abel prize has a somewhat unintuitive (and occasionally misunderstood) arrangement in which the presenter of the work of the prize is selected independently of the winner of the prize (I think in part so that the choice of presenter gives no clues as to the identity of the laureate).  In particular, like other presenters before me (which in recent years have included Timothy Gowers, Jordan Ellenberg, and Alex Bellos), I agreed to present the laureate’s work before knowing who the laureate was!  But in this case the task was very easy, because Meyer’s areas of (both pure and applied) harmonic analysis and PDE fell rather squarely within my own area of expertise.  (I had previously written about some other work of Meyer in this blog post.)  Indeed I had learned about Meyer’s wavelet constructions as a graduate student while taking a course from Ingrid Daubechies.   Daubechies also made extremely important contributions to the theory of wavelets, but due to a conflict of interest (as per the guidelines for the prize committee) arising from Daubechies’ presidency of the International Mathematical Union (which nominates the majority of the members of the Abel prize committee, who then serve for two years) from 2011 to 2014 (and her continuing service ex officio on the IMU executive committee from 2015 to 2018), she will not be eligible for the prize until 2021 at the earliest, and so I do not think this prize should be necessarily construed as a judgement on the relative contributions of Meyer and Daubechies to this field.  (In any case I fully agree with the Abel prize committee’s citation of Meyer’s pivotal role in the development of the theory of wavelets.)

[Update, Mar 28: link to prize committee guidelines and clarification of the extent of Daubechies’ conflict of interest added. -T]

Given a function ${f: {\bf N} \rightarrow \{-1,+1\}}$ on the natural numbers taking values in ${+1, -1}$, one can invoke the Furstenberg correspondence principle to locate a measure preserving system ${T \circlearrowright (X, \mu)}$ – a probability space ${(X,\mu)}$ together with a measure-preserving shift ${T: X \rightarrow X}$ (or equivalently, a measure-preserving ${{\bf Z}}$-action on ${(X,\mu)}$) – together with a measurable function (or “observable”) ${F: X \rightarrow \{-1,+1\}}$ that has essentially the same statistics as ${f}$ in the sense that

$\displaystyle \lim \inf_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k)$

$\displaystyle \leq \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x)$

$\displaystyle \leq \lim \sup_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k)$

for any integers ${h_1,\dots,h_k}$. In particular, one has

$\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k) \ \ \ \ \ (1)$

whenever the limit on the right-hand side exists. We will refer to the system ${T \circlearrowright (X,\mu)}$ together with the designated function ${F}$ as a Furstenberg limit ot the sequence ${f}$. These Furstenberg limits capture some, but not all, of the asymptotic behaviour of ${f}$; roughly speaking, they control the typical “local” behaviour of ${f}$, involving correlations such as ${\frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k)}$ in the regime where ${h_1,\dots,h_k}$ are much smaller than ${N}$. However, the control on error terms here is usually only qualitative at best, and one usually does not obtain non-trivial control on correlations in which the ${h_1,\dots,h_k}$ are allowed to grow at some significant rate with ${N}$ (e.g. like some power ${N^\theta}$ of ${N}$).

The correspondence principle is discussed in these previous blog posts. One way to establish the principle is by introducing a Banach limit ${p\!-\!\lim: \ell^\infty({\bf N}) \rightarrow {\bf R}}$ that extends the usual limit functional on the subspace of ${\ell^\infty({\bf N})}$ consisting of convergent sequences while still having operator norm one. Such functionals cannot be constructed explicitly, but can be proven to exist (non-constructively and non-uniquely) using the Hahn-Banach theorem; one can also use a non-principal ultrafilter here if desired. One can then seek to construct a system ${T \circlearrowright (X,\mu)}$ and a measurable function ${F: X \rightarrow \{-1,+1\}}$ for which one has the statistics

$\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x) = p\!-\!\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k) \ \ \ \ \ (2)$

for all ${h_1,\dots,h_k}$. One can explicitly construct such a system as follows. One can take ${X}$ to be the Cantor space ${\{-1,+1\}^{\bf Z}}$ with the product ${\sigma}$-algebra and the shift

$\displaystyle T ( (x_n)_{n \in {\bf Z}} ) := (x_{n+1})_{n \in {\bf Z}}$

with the function ${F: X \rightarrow \{-1,+1\}}$ being the coordinate function at zero:

$\displaystyle F( (x_n)_{n \in {\bf Z}} ) := x_0$

(so in particular ${F( T^h (x_n)_{n \in {\bf Z}} ) = x_h}$ for any ${h \in {\bf Z}}$). The only thing remaining is to construct the invariant measure ${\mu}$. In order to be consistent with (2), one must have

$\displaystyle \mu( \{ (x_n)_{n \in {\bf Z}}: x_{h_j} = \epsilon_j \forall 1 \leq j \leq k \} )$

$\displaystyle = p\!-\!\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N 1_{f(n+h_1)=\epsilon_1} \dots 1_{f(n+h_k)=\epsilon_k}$

for any distinct integers ${h_1,\dots,h_k}$ and signs ${\epsilon_1,\dots,\epsilon_k}$. One can check that this defines a premeasure on the Boolean algebra of ${\{-1,+1\}^{\bf Z}}$ defined by cylinder sets, and the existence of ${\mu}$ then follows from the Hahn-Kolmogorov extension theorem (or the closely related Kolmogorov extension theorem). One can then check that the correspondence (2) holds, and that ${\mu}$ is translation-invariant; the latter comes from the translation invariance of the (Banach-)Césaro averaging operation ${f \mapsto p\!-\!\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n)}$. A variant of this construction shows that the Furstenberg limit is unique up to equivalence if and only if all the limits appearing in (1) actually exist.

One can obtain a slightly tighter correspondence by using a smoother average than the Césaro average. For instance, one can use the logarithmic Césaro averages ${\lim_{N \rightarrow \infty} \frac{1}{\log N}\sum_{n=1}^N \frac{f(n)}{n}}$ in place of the Césaro average ${\sum_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n)}$, thus one replaces (2) by

$\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x)$

$\displaystyle = p\!-\!\lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{f(n+h_1) \dots f(n+h_k)}{n}.$

Whenever the Césaro average of a bounded sequence ${f: {\bf N} \rightarrow {\bf R}}$ exists, then the logarithmic Césaro average exists and is equal to the Césaro average. Thus, a Furstenberg limit constructed using logarithmic Banach-Césaro averaging still obeys (1) for all ${h_1,\dots,h_k}$ when the right-hand side limit exists, but also obeys the more general assertion

$\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x)$

$\displaystyle = \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{f(n+h_1) \dots f(n+h_k)}{n}$

whenever the limit of the right-hand side exists.

In a recent paper of Frantizinakis, the Furstenberg limits of the Liouville function ${\lambda}$ (with logarithmic averaging) were studied. Some (but not all) of the known facts and conjectures about the Liouville function can be interpreted in the Furstenberg limit. For instance, in a recent breakthrough result of Matomaki and Radziwill (discussed previously here), it was shown that the Liouville function exhibited cancellation on short intervals in the sense that

$\displaystyle \lim_{H \rightarrow \infty} \limsup_{X \rightarrow \infty} \frac{1}{X} \int_X^{2X} \frac{1}{H} |\sum_{x \leq n \leq x+H} \lambda(n)|\ dx = 0.$

In terms of Furstenberg limits of the Liouville function, this assertion is equivalent to the assertion that

$\displaystyle \lim_{H \rightarrow \infty} \int_X |\frac{1}{H} \sum_{h=1}^H F(T^h x)|\ d\mu(x) = 0$

for all Furstenberg limits ${T \circlearrowright (X,\mu), F}$ of Liouville (including those without logarithmic averaging). Invoking the mean ergodic theorem (discussed in this previous post), this assertion is in turn equivalent to the observable ${F}$ that corresponds to the Liouville function being orthogonal to the invariant factor ${L^\infty(X,\mu)^{\bf Z} = \{ g \in L^\infty(X,\mu): g \circ T = g \}}$ of ${X}$; equivalently, the first Gowers-Host-Kra seminorm ${\|F\|_{U^1(X)}}$ of ${F}$ (as defined for instance in this previous post) vanishes. The Chowla conjecture, which asserts that

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

for all distinct integers ${h_1,\dots,h_k}$, is equivalent to the assertion that all the Furstenberg limits of Liouville are equivalent to the Bernoulli system (${\{-1,+1\}^{\bf Z}}$ with the product measure arising from the uniform distribution on ${\{-1,+1\}}$, with the shift ${T}$ and observable ${F}$ as before). Similarly, the logarithmically averaged Chowla conjecture

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = 0$

is equivalent to the assertion that all the Furstenberg limits of Liouville with logarithmic averaging are equivalent to the Bernoulli system. Recently, I was able to prove the two-point version

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(n) \lambda(n+h)}{n} = 0 \ \ \ \ \ (3)$

of the logarithmically averaged Chowla conjecture, for any non-zero integer ${h}$; this is equivalent to the perfect strong mixing property

$\displaystyle \int_X F(x) F(T^h x)\ d\mu(x) = 0$

for any Furstenberg limit of Liouville with logarithmic averaging, and any ${h \neq 0}$.

The situation is more delicate with regards to the Sarnak conjecture, which is equivalent to the assertion that

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \lambda(n) f(n) = 0$

for any zero-entropy sequence ${f: {\bf N} \rightarrow {\bf R}}$ (see this previous blog post for more discussion). Morally speaking, this conjecture should be equivalent to the assertion that any Furstenberg limit of Liouville is disjoint from any zero entropy system, but I was not able to formally establish an implication in either direction due to some technical issues regarding the fact that the Furstenberg limit does not directly control long-range correlations, only short-range ones. (There are however ergodic theoretic interpretations of the Sarnak conjecture that involve the notion of generic points; see this paper of El Abdalaoui, Lemancyk, and de la Rue.) But the situation is currently better with the logarithmically averaged Sarnak conjecture

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(n) f(n)}{n} = 0,$

as I was able to show that this conjecture was equivalent to the logarithmically averaged Chowla conjecture, and hence to all Furstenberg limits of Liouville with logarithmic averaging being Bernoulli; I also showed the conjecture was equivalent to local Gowers uniformity of the Liouville function, which is in turn equivalent to the function ${F}$ having all Gowers-Host-Kra seminorms vanishing in every Furstenberg limit with logarithmic averaging. In this recent paper of Frantzikinakis, this analysis was taken further, showing that the logarithmically averaged Chowla and Sarnak conjectures were in fact equivalent to the much milder seeming assertion that all Furstenberg limits with logarithmic averaging were ergodic.

Actually, the logarithmically averaged Furstenberg limits have more structure than just a ${{\bf Z}}$-action on a measure preserving system ${(X,\mu)}$ with a single observable ${F}$. Let ${Aff_+({\bf Z})}$ denote the semigroup of affine maps ${n \mapsto an+b}$ on the integers with ${a,b \in {\bf Z}}$ and ${a}$ positive. Also, let ${\hat {\bf Z}}$ denote the profinite integers (the inverse limit of the cyclic groups ${{\bf Z}/q{\bf Z}}$). Observe that ${Aff_+({\bf Z})}$ acts on ${\hat {\bf Z}}$ by taking the inverse limit of the obvious actions of ${Aff_+({\bf Z})}$ on ${{\bf Z}/q{\bf Z}}$.

Proposition 1 (Enriched logarithmically averaged Furstenberg limit of Liouville) Let ${p\!-\!\lim}$ be a Banach limit. Then there exists a probability space ${(X,\mu)}$ with an action ${\phi \mapsto T^\phi}$ of the affine semigroup ${Aff_+({\bf Z})}$, as well as measurable functions ${F: X \rightarrow \{-1,+1\}}$ and ${M: X \rightarrow \hat {\bf Z}}$, with the following properties:

• (i) (Affine Furstenberg limit) For any ${\phi_1,\dots,\phi_k \in Aff_+({\bf Z})}$, and any congruence class ${a\ (q)}$, one has

$\displaystyle p\!-\!\lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(\phi_1(n)) \dots \lambda(\phi_k(n)) 1_{n = a\ (q)}}{n}$

$\displaystyle = \int_X F( T^{\phi_1}(x) ) \dots F( T^{\phi_k}(x) ) 1_{M(x) = a\ (q)}\ d\mu(x).$

• (ii) (Equivariance of ${M}$) For any ${\phi \in Aff_+({\bf Z})}$, one has

$\displaystyle M( T^\phi(x) ) = \phi( M(x) )$

for ${\mu}$-almost every ${x \in X}$.

• (iii) (Multiplicativity at fixed primes) For any prime ${p}$, one has

$\displaystyle F( T^{p\cdot} x ) = - F(x)$

for ${\mu}$-almost every ${x \in X}$, where ${p \cdot \in Aff_+({\bf Z})}$ is the dilation map ${n \mapsto pn}$.

• (iv) (Measure pushforward) If ${\phi \in Aff_+({\bf Z})}$ is of the form ${\phi(n) = an+b}$ and ${S_\phi \subset X}$ is the set ${S_\phi = \{ x \in X: M(x) \in \phi(\hat {\bf Z}) \}}$, then the pushforward ${T^\phi_* \mu}$ of ${\mu}$ by ${\phi}$ is equal to ${a \mu\downharpoonright_{S_\phi}}$, that is to say one has

$\displaystyle \mu( (T^\phi)^{-1}(E) ) = a \mu( E \cap S_\phi )$

for every measurable ${E \subset X}$.

Note that ${{\bf Z}}$ can be viewed as the subgroup of ${Aff_+({\bf Z})}$ consisting of the translations ${n \mapsto n + b}$. If one only keeps the ${{\bf Z}}$-portion of the ${Aff_+({\bf Z})}$ action and forgets the rest (as well as the function ${M}$) then the action becomes measure-preserving, and we recover an ordinary Furstenberg limit with logarithmic averaging. However, the additional structure here can be quite useful; for instance, one can transfer the proof of (3) to this setting, which we sketch below the fold, after proving the proposition.

The observable ${M}$, roughly speaking, means that points ${x}$ in the Furstenberg limit ${X}$ constructed by this proposition are still “virtual integers” in the sense that one can meaningfully compute the residue class of ${x}$ modulo any natural number modulus ${q}$, by first applying ${M}$ and then reducing mod ${q}$. The action of ${Aff_+({\bf Z})}$ means that one can also meaningfully multiply ${x}$ by any natural number, and translate it by any integer. As with other applications of the correspondence principle, the main advantage of moving to this more “virtual” setting is that one now acquires a probability measure ${\mu}$, so that the tools of ergodic theory can be readily applied.

Given a random variable ${X}$ that takes on only finitely many values, we can define its Shannon entropy by the formula

$\displaystyle H(X) := \sum_x \mathbf{P}(X=x) \log \frac{1}{\mathbf{P}(X=x)}$

with the convention that ${0 \log \frac{1}{0} = 0}$. (In some texts, one uses the logarithm to base ${2}$ rather than the natural logarithm, but the choice of base will not be relevant for this discussion.) This is clearly a nonnegative quantity. Given two random variables ${X,Y}$ taking on finitely many values, the joint variable ${(X,Y)}$ is also a random variable taking on finitely many values, and also has an entropy ${H(X,Y)}$. It obeys the Shannon inequalities

$\displaystyle H(X), H(Y) \leq H(X,Y) \leq H(X) + H(Y)$

so we can define some further nonnegative quantities, the mutual information

$\displaystyle I(X:Y) := H(X) + H(Y) - H(X,Y)$

and the conditional entropies

$\displaystyle H(X|Y) := H(X,Y) - H(Y); \quad H(Y|X) := H(X,Y) - H(X).$

More generally, given three random variables ${X,Y,Z}$, one can define the conditional mutual information

$\displaystyle I(X:Y|Z) := H(X|Z) + H(Y|Z) - H(X,Y|Z)$

and the final of the Shannon entropy inequalities asserts that this quantity is also non-negative.

The mutual information ${I(X:Y)}$ is a measure of the extent to which ${X}$ and ${Y}$ fail to be independent; indeed, it is not difficult to show that ${I(X:Y)}$ vanishes if and only if ${X}$ and ${Y}$ are independent. Similarly, ${I(X:Y|Z)}$ vanishes if and only if ${X}$ and ${Y}$ are conditionally independent relative to ${Z}$. At the other extreme, ${H(X|Y)}$ is a measure of the extent to which ${X}$ fails to depend on ${Y}$; indeed, it is not difficult to show that ${H(X|Y)=0}$ if and only if ${X}$ is determined by ${Y}$ in the sense that there is a deterministic function ${f}$ such that ${X = f(Y)}$. In a related vein, if ${X}$ and ${X'}$ are equivalent in the sense that there are deterministic functional relationships ${X = f(X')}$, ${X' = g(X)}$ between the two variables, then ${X}$ is interchangeable with ${X'}$ for the purposes of computing the above quantities, thus for instance ${H(X) = H(X')}$, ${H(X,Y) = H(X',Y)}$, ${I(X:Y) = I(X':Y)}$, ${I(X:Y|Z) = I(X':Y|Z)}$, etc..

One can get some initial intuition for these information-theoretic quantities by specialising to a simple situation in which all the random variables ${X}$ being considered come from restricting a single random (and uniformly distributed) boolean function ${F: \Omega \rightarrow \{0,1\}}$ on a given finite domain ${\Omega}$ to some subset ${A}$ of ${\Omega}$:

$\displaystyle X = F \downharpoonright_A.$

In this case, ${X}$ has the law of a random uniformly distributed boolean function from ${A}$ to ${\{0,1\}}$, and the entropy here can be easily computed to be ${|A| \log 2}$, where ${|A|}$ denotes the cardinality of ${A}$. If ${X}$ is the restriction of ${F}$ to ${A}$, and ${Y}$ is the restriction of ${F}$ to ${B}$, then the joint variable ${(X,Y)}$ is equivalent to the restriction of ${F}$ to ${A \cup B}$. If one discards the normalisation factor ${\log 2}$, one then obtains the following dictionary between entropy and the combinatorics of finite sets:

 Random variables ${X,Y,Z}$ Finite sets ${A,B,C}$ Entropy ${H(X)}$ Cardinality ${|A|}$ Joint variable ${(X,Y)}$ Union ${A \cup B}$ Mutual information ${I(X:Y)}$ Intersection cardinality ${|A \cap B|}$ Conditional entropy ${H(X|Y)}$ Set difference cardinality ${|A \backslash B|}$ Conditional mutual information ${I(X:Y|Z)}$ ${|(A \cap B) \backslash C|}$ ${X, Y}$ independent ${A, B}$ disjoint ${X}$ determined by ${Y}$ ${A}$ a subset of ${B}$ ${X,Y}$ conditionally independent relative to ${Z}$ ${A \cap B \subset C}$

Every (linear) inequality or identity about entropy (and related quantities, such as mutual information) then specialises to a combinatorial inequality or identity about finite sets that is easily verified. For instance, the Shannon inequality ${H(X,Y) \leq H(X)+H(Y)}$ becomes the union bound ${|A \cup B| \leq |A| + |B|}$, and the definition of mutual information becomes the inclusion-exclusion formula

$\displaystyle |A \cap B| = |A| + |B| - |A \cup B|.$

For a more advanced example, consider the data processing inequality that asserts that if ${X, Z}$ are conditionally independent relative to ${Y}$, then ${I(X:Z) \leq I(X:Y)}$. Specialising to sets, this now says that if ${A, C}$ are disjoint outside of ${B}$, then ${|A \cap C| \leq |A \cap B|}$; this can be made apparent by considering the corresponding Venn diagram. This dictionary also suggests how to prove the data processing inequality using the existing Shannon inequalities. Firstly, if ${A}$ and ${C}$ are not necessarily disjoint outside of ${B}$, then a consideration of Venn diagrams gives the more general inequality

$\displaystyle |A \cap C| \leq |A \cap B| + |(A \cap C) \backslash B|$

and a further inspection of the diagram then reveals the more precise identity

$\displaystyle |A \cap C| + |(A \cap B) \backslash C| = |A \cap B| + |(A \cap C) \backslash B|.$

Using the dictionary in the reverse direction, one is then led to conjecture the identity

$\displaystyle I( X : Z ) + I( X : Y | Z ) = I( X : Y ) + I( X : Z | Y )$

which (together with non-negativity of conditional mutual information) implies the data processing inequality, and this identity is in turn easily established from the definition of mutual information.

On the other hand, not every assertion about cardinalities of sets generalises to entropies of random variables that are not arising from restricting random boolean functions to sets. For instance, a basic property of sets is that disjointness from a given set ${C}$ is preserved by unions:

$\displaystyle A \cap C = B \cap C = \emptyset \implies (A \cup B) \cap C = \emptyset.$

Indeed, one has the union bound

$\displaystyle |(A \cup B) \cap C| \leq |A \cap C| + |B \cap C|. \ \ \ \ \ (1)$

Applying the dictionary in the reverse direction, one might now conjecture that if ${X}$ was independent of ${Z}$ and ${Y}$ was independent of ${Z}$, then ${(X,Y)}$ should also be independent of ${Z}$, and furthermore that

$\displaystyle I(X,Y:Z) \leq I(X:Z) + I(Y:Z)$

but these statements are well known to be false (for reasons related to pairwise independence of random variables being strictly weaker than joint independence). For a concrete counterexample, one can take ${X, Y \in {\bf F}_2}$ to be independent, uniformly distributed random elements of the finite field ${{\bf F}_2}$ of two elements, and take ${Z := X+Y}$ to be the sum of these two field elements. One can easily check that each of ${X}$ and ${Y}$ is separately independent of ${Z}$, but the joint variable ${(X,Y)}$ determines ${Z}$ and thus is not independent of ${Z}$.

From the inclusion-exclusion identities

$\displaystyle |A \cap C| = |A| + |C| - |A \cup C|$

$\displaystyle |B \cap C| = |B| + |C| - |B \cup C|$

$\displaystyle |(A \cup B) \cap C| = |A \cup B| + |C| - |A \cup B \cup C|$

$\displaystyle |A \cap B \cap C| = |A| + |B| + |C| - |A \cup B| - |B \cup C| - |A \cup C|$

$\displaystyle + |A \cup B \cup C|$

one can check that (1) is equivalent to the trivial lower bound ${|A \cap B \cap C| \geq 0}$. The basic issue here is that in the dictionary between entropy and combinatorics, there is no satisfactory entropy analogue of the notion of a triple intersection ${A \cap B \cap C}$. (Even the double intersection ${A \cap B}$ only exists information theoretically in a “virtual” sense; the mutual information ${I(X:Y)}$ allows one to “compute the entropy” of this “intersection”, but does not actually describe this intersection itself as a random variable.)

However, this issue only arises with three or more variables; it is not too difficult to show that the only linear equalities and inequalities that are necessarily obeyed by the information-theoretic quantities ${H(X), H(Y), H(X,Y), I(X:Y), H(X|Y), H(Y|X)}$ associated to just two variables ${X,Y}$ are those that are also necessarily obeyed by their combinatorial analogues ${|A|, |B|, |A \cup B|, |A \cap B|, |A \backslash B|, |B \backslash A|}$. (See for instance the Venn diagram at the Wikipedia page for mutual information for a pictorial summation of this statement.)

One can work with a larger class of special cases of Shannon entropy by working with random linear functions rather than random boolean functions. Namely, let ${S}$ be some finite-dimensional vector space over a finite field ${{\mathbf F}}$, and let ${f: S \rightarrow {\mathbf F}}$ be a random linear functional on ${S}$, selected uniformly among all such functions. Every subspace ${U}$ of ${S}$ then gives rise to a random variable ${X = X_U: U \rightarrow {\mathbf F}}$ formed by restricting ${f}$ to ${U}$. This random variable is also distributed uniformly amongst all linear functions on ${U}$, and its entropy can be easily computed to be ${\mathrm{dim}(U) \log |\mathbf{F}|}$. Given two random variables ${X, Y}$ formed by restricting ${f}$ to ${U, V}$ respectively, the joint random variable ${(X,Y)}$ determines the random linear function ${f}$ on the union ${U \cup V}$ on the two spaces, and thus by linearity on the Minkowski sum ${U+V}$ as well; thus ${(X,Y)}$ is equivalent to the restriction of ${f}$ to ${U+V}$. In particular, ${H(X,Y) = \mathrm{dim}(U+V) \log |\mathbf{F}|}$. This implies that ${I(X:Y) = \mathrm{dim}(U \cap V) \log |\mathbf{F}|}$ and also ${H(X|Y) = \mathrm{dim}(\pi_V(U)) \log |\mathbf{F}|}$, where ${\pi_V: S \rightarrow S/V}$ is the quotient map. After discarding the normalising constant ${\log |\mathbf{F}|}$, this leads to the following dictionary between information theoretic quantities and linear algebra quantities, analogous to the previous dictionary:

 Random variables ${X,Y,Z}$ Subspaces ${U,V,W}$ Entropy ${H(X)}$ Dimension ${\mathrm{dim}(U)}$ Joint variable ${(X,Y)}$ Sum ${U+V}$ Mutual information ${I(X:Y)}$ Dimension of intersection ${\mathrm{dim}(U \cap V)}$ Conditional entropy ${H(X|Y)}$ Dimension of projection ${\mathrm{dim}(\pi_V(U))}$ Conditional mutual information ${I(X:Y|Z)}$ ${\mathrm{dim}(\pi_W(U) \cap \pi_W(V))}$ ${X, Y}$ independent ${U, V}$ transverse (${U \cap V = \{0\}}$) ${X}$ determined by ${Y}$ ${U}$ a subspace of ${V}$ ${X,Y}$ conditionally independent relative to ${Z}$ ${\pi_W(U)}$, ${\pi_W(V)}$ transverse.

The combinatorial dictionary can be regarded as a specialisation of the linear algebra dictionary, by taking ${S}$ to be the vector space ${\mathbf{F}_2^\Omega}$ over the finite field ${\mathbf{F}_2}$ of two elements, and only considering those subspaces ${U}$ that are coordinate subspaces ${U = {\bf F}_2^A}$ associated to various subsets ${A}$ of ${\Omega}$.

As before, every linear inequality or equality that is valid for the information-theoretic quantities discussed above, is automatically valid for the linear algebra counterparts for subspaces of a vector space over a finite field by applying the above specialisation (and dividing out by the normalising factor of ${\log |\mathbf{F}|}$). In fact, the requirement that the field be finite can be removed by applying the compactness theorem from logic (or one of its relatives, such as Los’s theorem on ultraproducts, as done in this previous blog post).

The linear algebra model captures more of the features of Shannon entropy than the combinatorial model. For instance, in contrast to the combinatorial case, it is possible in the linear algebra setting to have subspaces ${U,V,W}$ such that ${U}$ and ${V}$ are separately transverse to ${W}$, but their sum ${U+V}$ is not; for instance, in a two-dimensional vector space ${{\bf F}^2}$, one can take ${U,V,W}$ to be the one-dimensional subspaces spanned by ${(0,1)}$, ${(1,0)}$, and ${(1,1)}$ respectively. Note that this is essentially the same counterexample from before (which took ${{\bf F}}$ to be the field of two elements). Indeed, one can show that any necessarily true linear inequality or equality involving the dimensions of three subspaces ${U,V,W}$ (as well as the various other quantities on the above table) will also be necessarily true when applied to the entropies of three discrete random variables ${X,Y,Z}$ (as well as the corresponding quantities on the above table).

However, the linear algebra model does not completely capture the subtleties of Shannon entropy once one works with four or more variables (or subspaces). This was first observed by Ingleton, who established the dimensional inequality

$\displaystyle \mathrm{dim}(U \cap V) \leq \mathrm{dim}(\pi_W(U) \cap \pi_W(V)) + \mathrm{dim}(\pi_X(U) \cap \pi_X(V)) + \mathrm{dim}(W \cap X) \ \ \ \ \ (2)$

for any subspaces ${U,V,W,X}$. This is easiest to see when the three terms on the right-hand side vanish; then ${\pi_W(U), \pi_W(V)}$ are transverse, which implies that ${U\cap V \subset W}$; similarly ${U \cap V \subset X}$. But ${W}$ and ${X}$ are transverse, and this clearly implies that ${U}$ and ${V}$ are themselves transverse. To prove the general case of Ingleton’s inequality, one can define ${Y := U \cap V}$ and use ${\mathrm{dim}(\pi_W(Y)) \leq \mathrm{dim}(\pi_W(U) \cap \pi_W(V))}$ (and similarly for ${X}$ instead of ${W}$) to reduce to establishing the inequality

$\displaystyle \mathrm{dim}(Y) \leq \mathrm{dim}(\pi_W(Y)) + \mathrm{dim}(\pi_X(Y)) + \mathrm{dim}(W \cap X) \ \ \ \ \ (3)$

which can be rearranged using ${\mathrm{dim}(\pi_W(Y)) = \mathrm{dim}(Y) - \mathrm{dim}(W) + \mathrm{dim}(\pi_Y(W))}$ (and similarly for ${X}$ instead of ${W}$) and ${\mathrm{dim}(W \cap X) = \mathrm{dim}(W) + \mathrm{dim}(X) - \mathrm{dim}(W + X)}$ as

$\displaystyle \mathrm{dim}(W + X ) \leq \mathrm{dim}(\pi_Y(W)) + \mathrm{dim}(\pi_Y(X)) + \mathrm{dim}(Y)$

but this is clear since ${\mathrm{dim}(W + X ) \leq \mathrm{dim}(\pi_Y(W) + \pi_Y(X)) + \mathrm{dim}(Y)}$.

Returning to the entropy setting, the analogue

$\displaystyle H( V ) \leq H( V | Z ) + H(V | W ) + I(Z:W)$

of (3) is true (exercise!), but the analogue

$\displaystyle I(X:Y) \leq I(X:Y|Z) + I(X:Y|W) + I(Z:W) \ \ \ \ \ (4)$

of Ingleton’s inequality is false in general. Again, this is easiest to see when all the terms on the right-hand side vanish; then ${X,Y}$ are conditionally independent relative to ${Z}$, and relative to ${W}$, and ${Z}$ and ${W}$ are independent, and the claim (4) would then be asserting that ${X}$ and ${Y}$ are independent. While there is no linear counterexample to this statement, there are simple non-linear ones: for instance, one can take ${Z,W}$ to be independent uniform variables from ${\mathbf{F}_2}$, and take ${X}$ and ${Y}$ to be (say) ${ZW}$ and ${(1-Z)(1-W)}$ respectively (thus ${X, Y}$ are the indicators of the events ${Z=W=1}$ and ${Z=W=0}$ respectively). Once one conditions on either ${Z}$ or ${W}$, one of ${X,Y}$ has positive conditional entropy and the other has zero entropy, and so ${X, Y}$ are conditionally independent relative to either ${Z}$ or ${W}$; also, ${Z}$ or ${W}$ are independent of each other. But ${X}$ and ${Y}$ are not independent of each other (they cannot be simultaneously equal to ${1}$). Somehow, the feature of the linear algebra model that is not present in general is that in the linear algebra setting, every pair of subspaces ${U, V}$ has a well-defined intersection ${U \cap V}$ that is also a subspace, whereas for arbitrary random variables ${X, Y}$, there does not necessarily exist the analogue of an intersection, namely a “common information” random variable ${V}$ that has the entropy of ${I(X:Y)}$ and is determined either by ${X}$ or by ${Y}$.

I do not know if there is any simpler model of Shannon entropy that captures all the inequalities available for four variables. One significant complication is that there exist some information inequalities in this setting that are not of Shannon type, such as the Zhang-Yeung inequality

$\displaystyle I(X:Y) \leq 2 I(X:Y|Z) + I(X:Z|Y) + I(Y:Z|X)$

$\displaystyle + I(X:Y|W) + I(Z:W).$

One can however still use these simpler models of Shannon entropy to be able to guess arguments that would work for general random variables. An example of this comes from my paper on the logarithmically averaged Chowla conjecture, in which I showed among other things that

$\displaystyle |\sum_{n \leq x} \frac{\lambda(n) \lambda(n+1)}{n}| \leq \varepsilon x \ \ \ \ \ (5)$

whenever ${x}$ was sufficiently large depending on ${\varepsilon>0}$, where ${\lambda}$ is the Liouville function. The information-theoretic part of the proof was as follows. Given some intermediate scale ${H}$ between ${1}$ and ${x}$, one can form certain random variables ${X_H, Y_H}$. The random variable ${X_H}$ is a sign pattern of the form ${(\lambda(n+1),\dots,\lambda(n+H))}$ where ${n}$ is a random number chosen from ${1}$ to ${x}$ (with logarithmic weighting). The random variable ${Y_H}$ was tuple ${(n \hbox{ mod } p)_{p \sim \varepsilon^2 H}}$ of reductions of ${n}$ to primes ${p}$ comparable to ${\varepsilon^2 H}$. Roughly speaking, what was implicitly shown in the paper (after using the multiplicativity of ${\lambda}$, the circle method, and the Matomaki-Radziwill theorem on short averages of multiplicative functions) is that if the inequality (5) fails, then there was a lower bound

$\displaystyle I( X_H : Y_H ) \gg \varepsilon^7 \frac{H}{\log H}$

on the mutual information between ${X_H}$ and ${Y_H}$. From translation invariance, this also gives the more general lower bound

$\displaystyle I( X_{H_0,H} : Y_H ) \gg \varepsilon^7 \frac{H}{\log H} \ \ \ \ \ (6)$

for any ${H_0}$, where ${X_{H_0,H}}$ denotes the shifted sign pattern ${(\lambda(n+H_0+1),\dots,\lambda(n+H_0+H))}$. On the other hand, one had the entropy bounds

$\displaystyle H( X_{H_0,H} ), H(Y_H) \ll H$

and from concatenating sign patterns one could see that ${X_{H_0,H+H'}}$ is equivalent to the joint random variable ${(X_{H_0,H}, X_{H_0+H,H'})}$ for any ${H_0,H,H'}$. Applying these facts and using an “entropy decrement” argument, I was able to obtain a contradiction once ${H}$ was allowed to become sufficiently large compared to ${\varepsilon}$, but the bound was quite weak (coming ultimately from the unboundedness of ${\sum_{\log H_- \leq j \leq \log H_+} \frac{1}{j \log j}}$ as the interval ${[H_-,H_+]}$ of values of ${H}$ under consideration becomes large), something of the order of ${H \sim \exp\exp\exp(\varepsilon^{-7})}$; the quantity ${H}$ needs at various junctures to be less than a small power of ${\log x}$, so the relationship between ${x}$ and ${\varepsilon}$ becomes essentially quadruple exponential in nature, ${x \sim \exp\exp\exp\exp(\varepsilon^{-7})}$. The basic strategy was to observe that the lower bound (6) causes some slowdown in the growth rate ${H(X_{kH})/kH}$ of the mean entropy, in that this quantity decreased by ${\gg \frac{\varepsilon^7}{\log H}}$ as ${k}$ increased from ${1}$ to ${\log H}$, basically by dividing ${X_{kH}}$ into ${k}$ components ${X_{jH, H}}$, ${j=0,\dots,k-1}$ and observing from (6) each of these shares a bit of common information with the same variable ${Y_H}$. This is relatively clear when one works in a set model, in which ${Y_H}$ is modeled by a set ${B_H}$ of size ${O(H)}$, and ${X_{H_0,H}}$ is modeled by a set of the form

$\displaystyle X_{H_0,H} = \bigcup_{H_0 < h \leq H_0+H} A_h$

for various sets ${A_h}$ of size ${O(1)}$ (also there is some translation symmetry that maps ${A_h}$ to a shift ${A_{h+1}}$ while preserving all of the ${B_H}$).

However, on considering the set model recently, I realised that one can be a little more efficient by exploiting the fact (basically the Chinese remainder theorem) that the random variables ${Y_H}$ are basically jointly independent as ${H}$ ranges over dyadic values that are much smaller than ${\log x}$, which in the set model corresponds to the ${B_H}$ all being disjoint. One can then establish a variant

$\displaystyle I( X_{H_0,H} : Y_H | (Y_{H'})_{H' < H}) \gg \varepsilon^7 \frac{H}{\log H} \ \ \ \ \ (7)$

of (6), which in the set model roughly speaking asserts that each ${B_H}$ claims a portion of the ${\bigcup_{H_0 < h \leq H_0+H} A_h}$ of cardinality ${\gg \varepsilon^7 \frac{H}{\log H}}$ that is not claimed by previous choices of ${B_H}$. This leads to a more efficient contradiction (relying on the unboundedness of ${\sum_{\log H_- \leq j \leq \log H_+} \frac{1}{j}}$ rather than ${\sum_{\log H_- \leq j \leq \log H_+} \frac{1}{j \log j}}$) that looks like it removes one order of exponential growth, thus the relationship between ${x}$ and ${\varepsilon}$ is now ${x \sim \exp\exp\exp(\varepsilon^{-7})}$. Returning to the entropy model, one can use (7) and Shannon inequalities to establish an inequality of the form

$\displaystyle \frac{1}{2H} H(X_{2H} | (Y_{H'})_{H' \leq 2H}) \leq \frac{1}{H} H(X_{H} | (Y_{H'})_{H' \leq H}) - \frac{c \varepsilon^7}{\log H}$

for a small constant ${c>0}$, which on iterating and using the boundedness of ${\frac{1}{H} H(X_{H} | (Y_{H'})_{H' \leq H})}$ gives the claim. (A modification of this analysis, at least on the level of the back of the envelope calculation, suggests that the Matomaki-Radziwill theorem is needed only for ranges ${H}$ greater than ${\exp( (\log\log x)^{\varepsilon^{7}} )}$ or so, although at this range the theorem is not significantly simpler than the general case).

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