A sequence {a: {\bf Z} \rightarrow {\bf C}} of complex numbers is said to be quasiperiodic if it is of the form

\displaystyle  a(n) = F( \alpha_1 n \hbox{ mod } 1, \dots, \alpha_k n \hbox{ mod } 1)

for some real numbers {\alpha_1,\dots,\alpha_k} and continuous function {F: ({\bf R}/{\bf Z})^k \rightarrow {\bf C}}. For instance, linear phases such as {n \mapsto e(\alpha n + \beta)} (where {e(\theta) := e^{2\pi i \theta}}) are examples of quasiperiodic sequences; the top order coefficient {\alpha} (modulo {1}) can be viewed as a “frequency” of the integers, and an element of the Pontryagin dual {\hat {\bf Z} \equiv {\bf R}/{\bf Z}} of the integers. Any periodic sequence is also quasiperiodic (taking {k=1} and {\alpha_1} to be the reciprocal of the period). A sequence is said to be almost periodic if it is the uniform limit of quasiperiodic sequences. For instance any Fourier series of the form

\displaystyle  a(n) = \sum_{j=1}^\infty c_j e(\alpha_j n)

with {\alpha_1,\alpha_2,\dots} real numbers and {c_1,c_2,\dots} an absolutely summable sequence of complex coefficients, will be almost periodic.

These sequences arise in various “complexity one” problems in arithmetic combinatorics and ergodic theory. For instance, if {(X, \mu, T)} is a measure-preserving system – a probability space {(X,\mu)} equipped with a measure-preserving shift, and {f_1,f_2 \in L^\infty(X)} are bounded measurable functions, then the correlation sequence

\displaystyle  a(n) := \int_X f_1(x) f_2(T^n x)\ d\mu(x)

can be shown to be an almost periodic sequence, plus an error term {b_n} which is “null” in the sense that it has vanishing uniform density:

\displaystyle  \sup_N \frac{1}{M} \sum_{n=N+1}^{n+M} |b_n| \rightarrow 0 \hbox{ as } M \rightarrow \infty.

This can be established in a number of ways, for instance by writing {a(n)} as the Fourier coefficients of the spectral measure of the shift {T} with respect to the functions {f_1,f_2}, and then decomposing that measure into pure point and continuous components.

In the last two decades or so, it has become clear that there are natural higher order versions of these concepts, in which linear polynomials such as {\alpha n + \beta} are replaced with higher degree counterparts. The most obvious candidates for these counterparts would be the polynomials {\alpha_d n^d + \dots + \alpha_0}, but this turns out to not be a complete set of higher degree objects needed for the theory. Instead, the higher order versions of quasiperiodic and almost periodic sequences are now known as basic nilsequences and nilsequences respectively, while the higher order version of a linear phase is a nilcharacter; each nilcharacter then has a symbol that is a higher order generalisation of the concept of a frequency (and the collection of all symbols forms a group that can be viewed as a higher order version of the Pontryagin dual of {{\bf Z}}). The theory of these objects is spread out in the literature across a number of papers; in particular, the theory of nilcharacters is mostly developed in Appendix E of this 116-page paper of Ben Green, Tamar Ziegler, and myself, and is furthermore written using nonstandard analysis and treating the more general setting of higher dimensional sequences. I therefore decided to rewrite some of that material in this blog post, in the simpler context of the qualitative asymptotic theory of one-dimensional nilsequences and nilcharacters rather than the quantitative single-scale theory that is needed for combinatorial applications (and which necessitated the use of nonstandard analysis in the previous paper).

For technical reasons (having to do with the non-trivial topological structure on nilmanifolds), it will be convenient to work with vector-valued sequences, that take values in a finite-dimensional complex vector space {{\bf C}^m} rather than in {{\bf C}}. By doing the space of sequences is now, technically, no longer a ring, as the operations of addition and multiplication on vector-valued sequences become ill-defined. However, we can still take complex conjugates of a sequence, and add sequences taking values in the same vector space {{\bf C}^m}, and for sequences taking values in different vector spaces {{\bf C}^m}, {{\bf C}^{m'}}, we may utilise the tensor product {\otimes: {\bf C}^m \times {\bf C}^{m'} \rightarrow {\bf C}^{mm'}}, which we will normalise by defining

\displaystyle  (z_1,\dots,z_m) \otimes (w_1,\dots,w_{m'}) = (z_1 w_1, \dots, z_1 w_{m'}, \dots, z_m w_1, \dots, z_m w_{m'} ).

This product is associative and bilinear, and also commutative up to permutation of the indices. It also interacts well with the Hermitian norm

\displaystyle  \| (z_1,\dots,z_m) \| := \sqrt{|z_1|^2 + \dots + |z_m|^2}

since we have {\|z \otimes w\| = \|z\| \|w\|}.

The traditional definition of a basic nilsequence (as defined for instance by Bergelson, Host, and Kra) is as follows:

Definition 1 (Basic nilsequence, first definition) A nilmanifold of step at most {d} is a quotient {G/\Gamma}, where {G} is a connected, simply connected nilpotent Lie group of step at most {d} (thus, all {d+1}-fold commutators vanish) and {\Gamma} is a discrete cocompact lattice in {G}. A basic nilsequence of degree at most {d} is a sequence of the form {n \mapsto F(g^n g_0 \Gamma)}, where {g_0 \Gamma \in G/\Gamma}, {g \in G}, and {F: G/\Gamma \rightarrow {\bf C}^m} is a continuous function.

For instance, it is not difficult using this definition to show that a sequence is a basic nilsequence of degree at most {1} if and only if it is quasiperiodic. The requirement that {G} be simply connected can be easily removed if desired by passing to a universal cover, but it is technically convenient to assume it (among other things, it allows for a well-defined logarithm map that obeys the Baker-Campbell-Hausdorff formula). When one wishes to perform a more quantitative analysis of nilsequences (particularly when working on a “single scale”. sich as on a single long interval {\{ N+1, \dots, N+M\}}), it is common to impose additional regularity conditions on the function {F}, such as Lipschitz continuity or smoothness, but ordinary continuity will suffice for the qualitative discussion in this blog post.

Nowadays, and particularly when one needs to understand the “single-scale” equidistribution properties of nilsequences, it is more convenient (as is for instance done in this ICM paper of Green) to use an alternate definition of a nilsequence as follows.

Definition 2 Let {d \geq 0}. A filtered group of degree at most {d} is a group {G} together with a sequence {G_\bullet = (G_0,G_1,G_2,\dots)} of subgroups {G \geq G_0 \geq G_1 \geq \dots} with {G_{d+1}=\{\hbox{id}\}} and {[G_i,G_j] \subset G_{i+j}} for {i,j \geq 0}. A polynomial sequence {g: {\bf Z} \rightarrow G} into a filtered group is a function such that {\partial_{h_i} \dots \partial_{h_1} g(n) \in G_i} for all {i \geq 0} and {n,h_1,\dots,h_i \in{\bf Z}}, where {\partial_h g(n) := g(n+h) g(n)^{-1}} is the difference operator. A filtered nilmanifold of degree at most {s} is a quotient {G/\Gamma}, where {G} is a filtered group of degree at most {s} such that {G} and all of the subgroups {G_i} are connected, simply connected nilpotent filtered Lie group, and {\Gamma} is a discrete cocompact subgroup of {G} such that {\Gamma_i := \Gamma \cap G_i} is a discrete cocompact subgroup of {G_i}. A basic nilsequence of degree at most {d} is a sequence of the form {n \mapsto F(g(n))}, where {g: {\bf Z} \rightarrow G} is a polynomial sequence, {G/\Gamma} is a filtered nilmanifold of degree at most {d}, and {F: G \rightarrow {\bf C}^m} is a continuous function which is {\Gamma}-automorphic, in the sense that {F(g \gamma) = F(g)} for all {g \in G} and {\gamma \in \Gamma}.

One can easily identify a {\Gamma}-automorphic function on {G} with a function on {G/\Gamma}, but there are some (very minor) advantages to working on the group {G} instead of the quotient {G/\Gamma}, as it becomes slightly easier to modify the automorphy group {\Gamma} when needed. (But because the action of {\Gamma} on {G} is free, one can pass from {\Gamma}-automorphic functions on {G} to functions on {G/\Gamma} with very little difficulty.) The main reason to work with polynomial sequences {n \mapsto g(n)} rather than geometric progressions {n \mapsto g^n g_0 \Gamma} is that they form a group, a fact essentially established by by Lazard and Leibman; see Corollary B.4 of this paper of Green, Ziegler, and myself for a proof in the filtered group setting.

It is easy to see that any sequence that is a basic nilsequence of degree at most {d} in the sense of the first definition, is also a basic nilsequence of degree at most {d} in the second definition, since a nilmanifold of degree at most {d} can be filtered using the lower central series, and any linear sequence {n \mapsto g^n g_0} will be a polynomial sequence with respect to that filtration. The converse implication is a little trickier, but still not too hard to show: see Appendix C of this paper of Ben Green, Tamar Ziegler, and myself. There are two key examples of basic nilsequences to keep in mind. The first are the polynomially quasiperiodic sequences

\displaystyle  a(n) = F( P_1(n), \dots, P_k(n) ),

where {P_1,\dots,P_k: {\bf Z} \rightarrow {\bf R}} are polynomials of degree at most {d}, and {F: {\bf R}^k \rightarrow {\bf C}^m} is a {{\bf Z}^k}-automorphic (i.e., {{\bf Z}^k}-periodic) continuous function. The map {P: {\bf Z} \rightarrow {\bf R}^k} defined by {P(n) := (P_1(n),\dots,P_k(n))} is a polynomial map of degree at most {d}, if one filters {{\bf R}^k} by defining {({\bf R}^k)_i} to equal {{\bf R}^k} when {i \leq d}, and {\{0\}} for {i > d}. The torus {{\bf R}^k/{\bf Z}^k} then becomes a filtered nilmanifold of degree at most {d}, and {a(n)} is thus a basic nilsequence of degree at most {d} as per the second definition. It is also possible explicitly describe {a_n} as a basic nilsequence of degree at most {d} as per the first definition, for instance (in the {k=1} case) by taking {G} to be the space of upper triangular unipotent {d+1 \times d+1} real matrices, and {\Gamma} the subgroup with integer coefficients; we leave the details to the interested reader.

The other key example is a sequence of the form

\displaystyle  a(n) = F( \alpha n, \{ \alpha n \} \beta n )

where {\alpha,\beta} are real numbers, {\{ \alpha n \} = \alpha n - \lfloor \alpha n \rfloor} denotes the fractional part of {\alpha n}, and and {F: {\bf R}^2 \rightarrow {\bf C}^m} is a {{\bf Z}^2}-automorphic continuous function that vanishes in a neighbourhood of {{\bf Z} \times {\bf R}}. To describe this as a nilsequence, we use the nilpotent connected, simply connected degree {2}, Heisenberg group

\displaystyle  G := \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix}

with the lower central series filtration {G_0=G_1=G}, {G_2= [G,G] = \begin{pmatrix} 1 &0 & {\bf R} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}}, and {G_i = \{ \mathrm{id} \}} for {i > 2}, {\Gamma} to be the discrete compact subgroup

\displaystyle  \Gamma := \begin{pmatrix} 1 & {\bf Z} & {\bf Z} \\ 0 & 1 & {\bf Z} \\ 0 & 0 & 1 \end{pmatrix},

{g: {\bf Z} \rightarrow G} to be the polynomial sequence

\displaystyle  g(n) := \begin{pmatrix} 1 & \beta n & \alpha \beta n^2 \\ 0 & 1 & \alpha n \\ 0 & 0 & 1 \end{pmatrix}

and {\tilde F: G \rightarrow {\bf C}^m} to be the {\Gamma}-automorphic function

\displaystyle  \tilde F( \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} ) = F( \{ z \}, y - \lfloor z \rfloor x );

one easily verifies that this function is indeed {\Gamma}-automorphic, and it is continuous thanks to the vanishing properties of {F}. Also we have {a(n) = \tilde F(g(n))}, so {a} is a basic nilsequence of degree at most {2}. One can concoct similar examples with {\{ \alpha n \} \beta n} replaced by other “bracket polynomials” of {n}; for instance

\displaystyle  a(n) = F( \alpha n, \{ \alpha n - \frac{1}{2} \} \beta n )

will be a basic nilsequence if {F} now vanishes in a neighbourhood of {(\frac{1}{2}+{\bf Z}) \times {\bf R}} rather than {{\bf Z} \times {\bf R}}. See this paper of Bergelson and Leibman for more discussion of bracket polynomials (also known as generalised polynomials) and their relationship to nilsequences.

A nilsequence of degree at most {d} is defined to be a sequence that is the uniform limit of basic nilsequences of degree at most {d}. Thus for instance a sequence is a nilsequence of degree at most {1} if and only if it is almost periodic, while a sequence is a nilsequence of degree at most {0} if and only if it is constant. Such objects arise in higher order recurrence: for instance, if {h_0,\dots,h_d} are integers, {(X,\mu,T)} is a measure-preserving system, and {f_0,\dots,f_d \in L^\infty(X)}, then it was shown by Leibman that the sequence

\displaystyle  n \mapsto \int_X f_0(T^{h_0 n} x) \dots f_d(T^{h_d n} x)\ d\mu(x)

is equal to a nilsequence of degree at most {d}, plus a null sequence. (The special case when the measure-preserving system was ergodic and {h_i = i} for {i=0,\dots,d} was previously established by Bergelson, Host, and Kra.) Nilsequences also arise in the inverse theory of the Gowers uniformity norms, as discussed for instance in this previous post.

It is easy to see that a sequence {a: {\bf Z} \rightarrow {\bf C}^m} is a basic nilsequence of degree at most {d} if and only if each of its {m} components are. The scalar basic nilsequences {a: {\bf Z} \rightarrow {\bf C}} of degree {d} are easily seen to form a {*}-algebra (that is to say, they are a complex vector space closed under pointwise multiplication and complex conjugation), which implies similarly that vector-valued basic nilsequences {a: {\bf Z} \rightarrow {\bf C}^m} of degree at most {d} form a complex vector space closed under complex conjugation for each {m}, and that the tensor product of any two basic nilsequences of degree at most {d} is another basic nilsequence of degree at most {d}. Similarly with “basic nilsequence” replaced by “nilsequence” throughout.

Now we turn to the notion of a nilcharacter, as defined in this paper of Ben Green, Tamar Ziegler, and myself:

Definition 3 (Nilcharacters) Let {d \geq 1}. A sub-nilcharacter of degree {d} is a basic nilsequence {\chi: n \mapsto F(g(n))} of degree at most {d}, such that {F} obeys the additional modulation property

\displaystyle  F( g_d g ) = e( \xi \cdot g_d ) F(g) \ \ \ \ \ (1)

for all {g \in G} and {g_d \in G_d}, where {\xi: G_d \rightarrow {\bf R}} is a continuous homomorphism {g_d \mapsto \xi \cdot g_d}. (Note from (1) and {\Gamma}-automorphy that unless {F} vanishes identically, {\xi} must map {\Gamma_d} to {{\bf Z}}, thus without loss of generality one can view {\xi} as an element of the Pontryagial dual of the torus {G_d / \Gamma_d}.) If in addition one has {\|F(g)\|=1} for all {g \in G}, we call {\chi} a nilcharacter of degree {d \geq 1}.

In the degree one case {d=1}, the only sub-nilcharacters are of the form {\chi(n) = e(\alpha n)} for some vector {c \in {\bf C}^m} and {\alpha \in {\bf R}}, and this is a nilcharacter if {c} is a unit vector. Similarly, in higher degree, any sequence of the form {\chi(n) = c e(P(n))}, where {c \in {\bf C}^m} is a vector and {P: {\bf Z} \rightarrow {\bf R}} is a polynomial of degree at most {d}, is a sub-nilcharacter of degree {d}, and a character if {c} is a unit vector. A nilsequence of degree at most {d-1} is automatically a sub-nilcharacter of degree {d}, and a nilcharacter if it is of magnitude {1}. A further example of a nilcharacter is provided by the two-dimensional sequence {\chi: {\bf Z} \rightarrow {\bf C}^2} defined by

\displaystyle  \chi(n) := ( F_0( \alpha n ) e( \{ \alpha n \} \beta n ), F_{1/2}( \alpha n ) e( \{ \alpha n - \frac{1}{2} \} \beta n ) ) \ \ \ \ \ (2)

where {F_0, F_{1/2}: {\bf R} \rightarrow {\bf C}} are continuous, {{\bf Z}}-automorphic functions that vanish on a neighbourhood of {{\bf Z}} and {\frac{1}{2}+{\bf Z}} respectively, and which form a partition of unity in the sense that

\displaystyle  |F_0(x)|^2 + |F_{1/2}(x)|^2 = 1

for all {x \in {\bf R}}. Note that one needs both {F_0} and {F_{1/2}} to be not identically zero in order for all these conditions to be satisfied; it turns out (for topological reasons) that there is no scalar nilcharacter that is “equivalent” to this nilcharacter in a sense to be defined shortly. In some literature, one works exclusively with sub-nilcharacters rather than nilcharacters, however the former space contains zero-divisors, which is a little annoying technically. Nevertheless, both nilcharacters and sub-nilcharacters generate the same set of “symbols” as we shall see later.

We claim that every degree {d} sub-nilcharacter {f: {\bf Z} \rightarrow {\bf C}^m} can be expressed in the form {f = c \chi}, where {\chi: {\bf Z} \rightarrow {\bf C}^{m'}} is a degree {d} nilcharacter, and {c: {\bf C}^{m'} \rightarrow {\bf C}^m} is a linear transformation. Indeed, by scaling we may assume {f(n) = F(g(n))} where {|F| < 1} uniformly. Using partitions of unity, one can find further functions {F_1,\dots,F_m} also obeying (1) for the same character {\xi} such that {|F_1|^2 + \dots + |F_m|^2} is non-zero; by dividing out the {F_1,\dots,F_m} by the square root of this quantity, and then multiplying by {\sqrt{1-|F|^2}}, we may assume that

\displaystyle  |F|^2 + |F_1|^2 + \dots + |F_m|^2 = 1,

and then

\displaystyle \chi(n) := (F(g(n)), F_1(g(n)), \dots, F_m(g(n)))

becomes a degree {d} nilcharacter that contains {f(n)} amongst its components, giving the claim.

As we shall show below, nilsequences can be approximated uniformly by linear combinations of nilcharacters, in much the same way that quasiperiodic or almost periodic sequences can be approximated uniformly by linear combinations of linear phases. In particular, nilcharacters can be used as “obstructions to uniformity” in the sense of the inverse theory of the Gowers uniformity norms.

The space of degree {d} nilcharacters forms a semigroup under tensor product, with the constant sequence {1} as the identity. One can upgrade this semigroup to an abelian group by quotienting nilcharacters out by equivalence:

Definition 4 Let {d \geq 1}. We say that two degree {d} nilcharacters {\chi: {\bf Z} \rightarrow {\bf C}^m}, {\chi': {\bf Z} \rightarrow {\bf C}^{m'}} are equivalent if {\chi \otimes \overline{\chi'}: {\bf Z} \rightarrow {\bf C}^{mm'}} is equal (as a sequence) to a basic nilsequence of degree at most {d-1}. (We will later show that this is indeed an equivalence relation.) The equivalence class {[\chi]_{\mathrm{Symb}^d({\bf Z})}} of such a nilcharacter will be called the symbol of that nilcharacter (in analogy to the symbol of a differential or pseudodifferential operator), and the collection of such symbols will be denoted {\mathrm{Symb}^d({\bf Z})}.

As we shall see below the fold, {\mathrm{Symb}^d({\bf Z})} has the structure of an abelian group, and enjoys some nice “symbol calculus” properties; also, one can view symbols as precisely describing the obstruction to equidistribution for nilsequences. For {d=1}, the group is isomorphic to the Ponytragin dual {\hat {\bf Z} = {\bf R}/{\bf Z}} of the integers, and {\mathrm{Symb}^d({\bf Z})} for {d > 1} should be viewed as higher order generalisations of this Pontryagin dual. In principle, this group can be explicitly described for all {d}, but the theory rapidly gets complicated as {d} increases (much as the classification of nilpotent Lie groups or Lie algebras of step {d} rapidly gets complicated even for medium-sized {d} such as {d=3} or {d=4}). We will give an explicit description of the {d=2} case here. There is however one nice (and non-trivial) feature of {\mathrm{Symb}^d({\bf Z})} for {d \geq 2} – it is not just an abelian group, but is in fact a vector space over the rationals {{\bf Q}}!

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How many groups of order four are there? Technically, there are an enormous number, so much so, in fact, that the class of groups of order four is not even a set, but merely a proper class. This is because any four objects {a,b,c,d} can be turned into a group {\{a,b,c,d\}} by designating one of the four objects, say {a}, to be the group identity, and imposing a suitable multiplication table (and inversion law) on the four elements in a manner that obeys the usual group axioms. Since all sets are themselves objects, the class of four-element groups is thus at least as large as the class of all sets, which by Russell’s paradox is known not to itself be a set (assuming the usual ZFC axioms of set theory).

A much better question is to ask how many groups of order four there are up to isomorphism, counting each isomorphism class of groups exactly once. Now, as one learns in undergraduate group theory classes, the answer is just “two”: the cyclic group {C_4} and the Klein four-group {C_2 \times C_2}.

More generally, given a class of objects {X} and some equivalence relation {\sim} on {X} (which one should interpret as describing the property of two objects in {X} being “isomorphic”), one can consider the number {|X / \sim|} of objects in {X} “up to isomorphism”, which is simply the cardinality of the collection {X/\sim} of equivalence classes {[x]:=\{y\in X:x \sim {}y \}} of {X}. In the case where {X} is finite, one can express this cardinality by the formula

\displaystyle |X/\sim| = \sum_{x \in X} \frac{1}{|[x]|}, \ \ \ \ \ (1)

thus one counts elements in {X}, weighted by the reciprocal of the number of objects they are isomorphic to.

Example 1 Let {X} be the five-element set {\{-2,-1,0,1,2\}} of integers between {-2} and {2}. Let us say that two elements {x, y} of {X} are isomorphic if they have the same magnitude: {x \sim y \iff |x| = |y|}. Then the quotient space {X/\sim} consists of just three equivalence classes: {\{-2,2\} = [2] = [-2]}, {\{-1,1\} = [-1] = [1]}, and {\{0\} = [0]}. Thus there are three objects in {X} up to isomorphism, and the identity (1) is then just

\displaystyle 3 = \frac{1}{2} + \frac{1}{2} + 1 + \frac{1}{2} + \frac{1}{2}.

Thus, to count elements in {X} up to equivalence, the elements {-2,-1,1,2} are given a weight of {1/2} because they are each isomorphic to two elements in {X}, while the element {0} is given a weight of {1} because it is isomorphic to just one element in {X} (namely, itself).

Given a finite probability set {X}, there is also a natural probability distribution on {X}, namely the uniform distribution, according to which a random variable {\mathbf{x} \in X} is set equal to any given element {x} of {X} with probability {\frac{1}{|X|}}:

\displaystyle {\bf P}( \mathbf{x} = x ) = \frac{1}{|X|}.

Given a notion {\sim} of isomorphism on {X}, one can then define the random equivalence class {[\mathbf{x}] \in X/\sim} that the random element {\mathbf{x}} belongs to. But if the isomorphism classes are unequal in size, we now encounter a biasing effect: even if {\mathbf{x}} was drawn uniformly from {X}, the equivalence class {[\mathbf{x}]} need not be uniformly distributed in {X/\sim}. For instance, in the above example, if {\mathbf{x}} was drawn uniformly from {\{-2,-1,0,1,2\}}, then the equivalence class {[\mathbf{x}]} will not be uniformly distributed in the three-element space {X/\sim}, because it has a {2/5} probability of being equal to the class {\{-2,2\}} or to the class {\{-1,1\}}, and only a {1/5} probability of being equal to the class {\{0\}}.

However, it is possible to remove this bias by changing the weighting in (1), and thus changing the notion of what cardinality means. To do this, we generalise the previous situation. Instead of considering sets {X} with an equivalence relation {\sim} to capture the notion of isomorphism, we instead consider groupoids, which are sets {X} in which there are allowed to be multiple isomorphisms between elements in {X} (and in particular, there are allowed to be multiple automorphisms from an element to itself). More precisely:

Definition 2 A groupoid is a set (or proper class) {X}, together with a (possibly empty) collection {\mathrm{Iso}(x \rightarrow y)} of “isomorphisms” between any pair {x,y} of elements of {X}, and a composition map {f, g \mapsto g \circ f} from isomorphisms {f \in \mathrm{Iso}(x \rightarrow y)}, {g \in \mathrm{Iso}(y \rightarrow z)} to isomorphisms in {\mathrm{Iso}(x \rightarrow z)} for every {x,y,z \in X}, obeying the following group-like axioms:

  • (Identity) For every {x \in X}, there is an identity isomorphism {\mathrm{id}_x \in \mathrm{Iso}(x \rightarrow x)}, such that {f \circ \mathrm{id}_x = \mathrm{id}_y \circ f = f} for all {f \in \mathrm{Iso}(x \rightarrow y)} and {x,y \in X}.
  • (Associativity) If {f \in \mathrm{Iso}(x \rightarrow y)}, {g \in \mathrm{Iso}(y \rightarrow z)}, and {h \in \mathrm{Iso}(z \rightarrow w)} for some {x,y,z,w \in X}, then {h \circ (g \circ f) = (h \circ g) \circ f}.
  • (Inverse) If {f \in \mathrm{Iso}(x \rightarrow y)} for some {x,y \in X}, then there exists an inverse isomorphism {f^{-1} \in \mathrm{Iso}(y \rightarrow x)} such that {f \circ f^{-1} = \mathrm{id}_y} and {f^{-1} \circ f = \mathrm{id}_x}.

We say that two elements {x,y} of a groupoid are isomorphic, and write {x \sim y}, if there is at least one isomorphism from {x} to {y}.

Example 3 Any category gives a groupoid by taking {X} to be the set (or class) of objects, and {\mathrm{Iso}(x \rightarrow y)} to be the collection of invertible morphisms from {x} to {y}. For instance, in the category {\mathbf{Set}} of sets, {\mathrm{Iso}(x \rightarrow y)} would be the collection of bijections from {x} to {y}; in the category {\mathbf{Vec}/k} of linear vector spaces over some given base field {k}, {\mathrm{Iso}(x \rightarrow y)} would be the collection of invertible linear transformations from {x} to {y}; and so forth.

Every set {X} equipped with an equivalence relation {\sim} can be turned into a groupoid by assigning precisely one isomorphism {\iota_{x \rightarrow y}} from {x} to {y} for any pair {x,y \in X} with {x \sim y}, and no isomorphisms from {x} to {y} when {x \not \sim y}, with the groupoid operations of identity, composition, and inverse defined in the only way possible consistent with the axioms. We will call this the simply connected groupoid associated with this equivalence relation. For instance, with {X = \{-2,-1,0,1,2\}} as above, if we turn {X} into a simply connected groupoid, there will be precisely one isomorphism from {2} to {-2}, and also precisely one isomorphism from {2} to {2}, but no isomorphisms from {2} to {-1}, {0}, or {1}.

However, one can also form multiply-connected groupoids in which there can be multiple isomorphisms from one element of {X} to another. For instance, one can view {X = \{-2,-1,0,1,2\}} as a space that is acted on by multiplication by the two-element group {\{-1,+1\}}. This gives rise to two types of isomorphisms, an identity isomorphism {(+1)_x} from {x} to {x} for each {x \in X}, and a negation isomorphism {(-1)_x} from {x} to {-x} for each {x \in X}; in particular, there are two automorphisms of {0} (i.e., isomorphisms from {0} to itself), namely {(+1)_0} and {(-1)_0}, whereas the other four elements of {X} only have a single automorphism (the identity isomorphism). One defines composition, identity, and inverse in this groupoid in the obvious fashion (using the group law of the two-element group {\{-1,+1\}}); for instance, we have {(-1)_{-2} \circ (-1)_2 = (+1)_2}.

For a finite multiply-connected groupoid, it turns out that the natural notion of “cardinality” (or as I prefer to call it, “cardinality up to isomorphism”) is given by the variant

\displaystyle \sum_{x \in X} \frac{1}{|\{ f: f \in \mathrm{Iso}(x \rightarrow y) \hbox{ for some } y\}|}

of (1). That is to say, in the multiply connected case, the denominator is no longer the number of objects {y} isomorphic to {x}, but rather the number of isomorphisms from {x} to other objects {y}. Grouping together all summands coming from a single equivalence class {[x]} in {X/\sim}, we can also write this expression as

\displaystyle \sum_{[x] \in X/\sim} \frac{1}{|\mathrm{Aut}(x)|} \ \ \ \ \ (2)

where {\mathrm{Aut}(x) := \mathrm{Iso}(x \rightarrow x)} is the automorphism group of {x}, that is to say the group of isomorphisms from {x} to itself. (Note that if {x,x'} belong to the same equivalence class {[x]}, then the two groups {\mathrm{Aut}(x)} and {\mathrm{Aut}(x')} will be isomorphic and thus have the same cardinality, and so the above expression is well-defined.

For instance, if we take {X} to be the simply connected groupoid on {\{-2,-1,0,1,2\}}, then the number of elements of {X} up to isomorphism is

\displaystyle \frac{1}{2} + \frac{1}{2} + 1 + \frac{1}{2} + \frac{1}{2} = 1 + 1 + 1 = 3

exactly as before. If however we take the multiply connected groupoid on {\{-2,-1,0,1,2\}}, in which {0} has two automorphisms, the number of elements of {X} up to isomorphism is now the smaller quantity

\displaystyle \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 1 + \frac{1}{2} + 1 = \frac{5}{2};

the equivalence class {[0]} is now counted with weight {1/2} rather than {1} due to the two automorphisms on {0}. Geometrically, one can think of this groupoid as being formed by taking the five-element set {\{-2,-1,0,1,2\}}, and “folding it in half” around the fixed point {0}, giving rise to two “full” quotient points {[1], [2]} and one “half” point {[0]}. More generally, given a finite group {G} acting on a finite set {X}, and forming the associated multiply connected groupoid, the cardinality up to isomorphism of this groupoid will be {|X|/|G|}, since each element {x} of {X} will have {|G|} isomorphisms on it (whether they be to the same element {x}, or to other elements of {X}).

The definition (2) can also make sense for some infinite groupoids; to my knowledge this was first explicitly done in this paper of Baez and Dolan. Consider for instance the category {\mathbf{FinSet}} of finite sets, with isomorphisms given by bijections as in Example 3. Every finite set is isomorphic to {\{1,\dots,n\}} for some natural number {n}, so the equivalence classes of {\mathbf{FinSet}} may be indexed by the natural numbers. The automorphism group {S_n} of {\{1,\dots,n\}} has order {n!}, so the cardinality of {\mathbf{FinSet}} up to isomorphism is

\displaystyle \sum_{n=0}^\infty \frac{1}{n!} = e.

(This fact is sometimes loosely stated as “the number of finite sets is {e}“, but I view this statement as somewhat misleading if the qualifier “up to isomorphism” is not added.) Similarly, when one allows for multiple isomorphisms from a group to itself, the number of groups of order four up to isomorphism is now

\displaystyle \frac{1}{2} + \frac{1}{6} = \frac{2}{3}

because the cyclic group {C_4} has two automorphisms, whereas the Klein four-group {C_2 \times C_2} has six.

In the case that the cardinality of a groupoid {X} up to isomorphism is finite and non-empty, one can now define the notion of a random isomorphism class {[\mathbf{x}]} in {X/\sim} drawn “uniformly up to isomorphism”, by requiring the probability of attaining any given isomorphism class {[x]} to be

\displaystyle {\mathbf P}([\mathbf{x}] = [x]) = \frac{1 / |\mathrm{Aut}(x)|}{\sum_{[y] \in X/\sim} 1/|\mathrm{Aut}(y)|},

thus the probability of being isomorphic to a given element {x} will be inversely proportional to the number of automorphisms that {x} has. For instance, if we take {X} to be the set {\{-2,-1,0,1,2\}} with the simply connected groupoid, {[\mathbf{x}]} will be drawn uniformly from the three available equivalence classes {[0], [1], [2]}, with a {1/3} probability of attaining each; but if instead one uses the multiply connected groupoid coming from the action of {\{-1,+1\}}, and draws {[\mathbf{x}]} uniformly up to isomorphism, then {[1]} and {[2]} will now be selected with probability {2/5} each, and {[0]} will be selected with probability {1/5}. Thus this distribution has accounted for the bias mentioned previously: if a finite group {G} acts on a finite space {X}, and {\mathbf{x}} is drawn uniformly from {X}, then {[\mathbf{x}]} now still be drawn uniformly up to isomorphism from {X/G}, if we use the multiply connected groupoid coming from the {G} action, rather than the simply connected groupoid coming from just the {G}-orbit structure on {X}.

Using the groupoid of finite sets, we see that a finite set chosen uniformly up to isomorphism will have a cardinality that is distributed according to the Poisson distribution of parameter {1}, that is to say it will be of cardinality {n} with probability {\frac{e^{-1}}{n!}}.

One important source of groupoids are the fundamental groupoids {\pi_1(M)} of a manifold {M} (one can also consider more general topological spaces than manifolds, but for simplicity we will restrict this discussion to the manifold case), in which the underlying space is simply {M}, and the isomorphisms from {x} to {y} are the equivalence classes of paths from {x} to {y} up to homotopy; in particular, the automorphism group of any given point is just the fundamental group of {M} at that base point. The equivalence class {[x]} of a point in {M} is then the connected component of {x} in {M}. The cardinality up to isomorphism of the fundamental groupoid is then

\displaystyle \sum_{M' \in \pi_0(M)} \frac{1}{|\pi_1(M')|}

where {\pi_0(M)} is the collection of connected components {M'} of {M}, and {|\pi_1(M')|} is the order of the fundamental group of {M'}. Thus, simply connected components of {M} count for a full unit of cardinality, whereas multiply connected components (which can be viewed as quotients of their simply connected cover by their fundamental group) will count for a fractional unit of cardinality, inversely to the order of their fundamental group.

This notion of cardinality up to isomorphism of a groupoid behaves well with respect to various basic notions. For instance, suppose one has an {n}-fold covering map {\pi: X \rightarrow Y} of one finite groupoid {Y} by another {X}. This means that {\pi} is a functor that is surjective, with all preimages of cardinality {n}, with the property that given any pair {y,y'} in the base space {Y} and any {x} in the preimage {\pi^{-1}(\{y\})} of {y}, every isomorphism {f \in \mathrm{Iso}(y \rightarrow y')} has a unique lift {\tilde f \in \mathrm{Iso}(x \rightarrow x')} from the given initial point {x} (and some {x'} in the preimage of {y'}). Then one can check that the cardinality up to isomorphism of {X} is {n} times the cardinality up to isomorphism of {Y}, which fits well with the geometric picture of {X} as the {n}-fold cover of {Y}. (For instance, if one covers a manifold {M} with finite fundamental group by its universal cover, this is a {|\pi_1(M)|}-fold cover, the base has cardinality {1/|\pi_1(M)|} up to isomorphism, and the universal cover has cardinality one up to isomorphism.) Related to this, if one draws an equivalence class {[\mathrm{x}]} of {X} uniformly up to isomorphism, then {\pi([\mathrm{x}])} will be an equivalence class of {Y} drawn uniformly up to isomorphism also.

Indeed, one can show that this notion of cardinality up to isomorphism for groupoids is uniquely determined by a small number of axioms such as these (similar to the axioms that determine Euler characteristic); see this blog post of Qiaochu Yuan for details.

The probability distributions on isomorphism classes described by the above recipe seem to arise naturally in many applications. For instance, if one draws a profinite abelian group up to isomorphism at random in this fashion (so that each isomorphism class {[G]} of a profinite abelian group {G} occurs with probability inversely proportional to the number of automorphisms of this group), then the resulting distribution is known as the Cohen-Lenstra distribution, and seems to emerge as the natural asymptotic distribution of many randomly generated profinite abelian groups in number theory and combinatorics, such as the class groups of random quadratic fields; see this previous blog post for more discussion. For a simple combinatorial example, the set of fixed points of a random permutation on {n} elements will have a cardinality that converges in distribution to the Poisson distribution of rate {1} (as discussed in this previous post), thus we see that the fixed points of a large random permutation asymptotically are distributed uniformly up to isomorphism. I’ve been told that this notion of cardinality up to isomorphism is also particularly compatible with stacks (which are a good framework to describe such objects as moduli spaces of algebraic varieties up to isomorphism), though I am not sufficiently acquainted with this theory to say much more than this.

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.

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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).

Daniel Kane and I have just uploaded to the arXiv our paper “A bound on partitioning clusters“, submitted to the Electronic Journal of Combinatorics. In this short and elementary paper, we consider a question that arose from biomathematical applications: given a finite family {X} of sets (or “clusters”), how many ways can there be of partitioning a set {A \in X} in this family as the disjoint union {A = A_1 \uplus A_2} of two other sets {A_1, A_2} in this family? That is to say, what is the best upper bound one can place on the quantity

\displaystyle | \{ (A,A_1,A_2) \in X^3: A = A_1 \uplus A_2 \}|

in terms of the cardinality {|X|} of {X}? A trivial upper bound would be {|X|^2}, since this is the number of possible pairs {(A_1,A_2)}, and {A_1,A_2} clearly determine {A}. In our paper, we establish the improved bound

\displaystyle | \{ (A,A_1,A_2) \in X^3: A = A_1 \uplus A_2 \}| \leq |X|^{3/p}

where {p} is the somewhat strange exponent

\displaystyle p := \log_3 \frac{27}{4} = 1.73814\dots, \ \ \ \ \ (1)

 

so that {3/p = 1.72598\dots}. Furthermore, this exponent is best possible!

Actually, the latter claim is quite easy to show: one takes {X} to be all the subsets of {\{1,\dots,n\}} of cardinality either {n/3} or {2n/3}, for {n} a multiple of {3}, and the claim follows readily from Stirling’s formula. So it is perhaps the former claim that is more interesting (since many combinatorial proof techniques, such as those based on inequalities such as the Cauchy-Schwarz inequality, tend to produce exponents that are rational or at least algebraic). We follow the common, though unintuitive, trick of generalising a problem to make it simpler. Firstly, one generalises the bound to the “trilinear” bound

\displaystyle | \{ (A_1,A_2,A_3) \in X_1 \times X_2 \times X_3: A_3 = A_1 \uplus A_2 \}|

\displaystyle \leq |X_1|^{1/p} |X_2|^{1/p} |X_3|^{1/p}

for arbitrary finite collections {X_1,X_2,X_3} of sets. One can place all the sets in {X_1,X_2,X_3} inside a single finite set such as {\{1,\dots,n\}}, and then by replacing every set {A_3} in {X_3} by its complement in {\{1,\dots,n\}}, one can phrase the inequality in the equivalent form

\displaystyle | \{ (A_1,A_2,A_3) \in X_1 \times X_2 \times X_3: \{1,\dots,n\} =A_1 \uplus A_2 \uplus A_3 \}|

\displaystyle \leq |X_1|^{1/p} |X_2|^{1/p} |X_3|^{1/p}

for arbitrary collections {X_1,X_2,X_3} of subsets of {\{1,\dots,n\}}. We generalise further by turning sets into functions, replacing the estimate with the slightly stronger convolution estimate

\displaystyle f_1 * f_2 * f_3 (1,\dots,1) \leq \|f_1\|_{\ell^p(\{0,1\}^n)} \|f_2\|_{\ell^p(\{0,1\}^n)} \|f_3\|_{\ell^p(\{0,1\}^n)}

for arbitrary functions {f_1,f_2,f_3} on the Hamming cube {\{0,1\}^n}, where the convolution is on the integer lattice {\bf Z}^n rather than on the finite field vector space {\bf F}_2^n. The advantage of working in this general setting is that it becomes very easy to apply induction on the dimension {n}; indeed, to prove this estimate for arbitrary {n} it suffices to do so for {n=1}. This reduces matters to establishing the elementary inequality

\displaystyle (ab(1-c))^{1/p} + (bc(1-a))^{1/p} + (ca(1-b))^{1/p} \leq 1

for all {0 \leq a,b,c \leq 1}, which can be done by a combination of undergraduate multivariable calculus and a little bit of numerical computation. (The left-hand side turns out to have local maxima at {(1,1,0), (1,0,1), (0,1,1), (2/3,2/3,2/3)}, with the latter being the cause of the numerology (1).)

The same sort of argument also gives an energy bound

\displaystyle E(A,A) \leq |A|^{\log_2 6}

for any subset {A \subset \{0,1\}^n} of the Hamming cube, where

\displaystyle E(A,A) := |\{(a_1,a_2,a_3,a_4) \in A^4: a_1+a_2 = a_3 + a_4 \}|

is the additive energy of {A}. The example {A = \{0,1\}^n} shows that the exponent {\log_2 6} cannot be improved.

The self-chosen remit of my blog is “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”.  Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not related to mathematics or the community of mathematicians in any significant fashion.

This is not one of the 1%.

Mathematical research is clearly an international activity.  But actually a stronger claim is true: mathematical research is a transnational activity, in that the specific nationality of individual members of a research team or research community are (or should be) of no appreciable significance for the purpose of advancing mathematics.  For instance, even during the height of the Cold War, there was no movement in (say) the United States to boycott Soviet mathematicians or theorems, or to only use results from Western literature (though the latter did sometimes happen by default, due to the limited avenues of information exchange between East and West, and former did occasionally occur for political reasons, most notably with the Soviet Union preventing Gregory Margulis from traveling to receive his Fields Medal in 1978 EDIT: and also Sergei Novikov in 1970).    The national origin of even the most fundamental components of mathematics, whether it be the geometry (γεωμετρία) of the ancient Greeks, the algebra (الجبر) of the Islamic world, or the Hindu-Arabic numerals 0,1,\dots,9, are primarily of historical interest, and have only a negligible impact on the worldwide adoption of these mathematical tools. While it is true that individual mathematicians or research teams sometimes compete with each other to be the first to solve some desired problem, and that a citizen could take pride in the mathematical achievements of researchers from their country, one did not see any significant state-sponsored “space races” in which it was deemed in the national interest that a particular result ought to be proven by “our” mathematicians and not “theirs”.   Mathematical research ability is highly non-fungible, and the value added by foreign students and faculty to a mathematics department cannot be completely replaced by an equivalent amount of domestic students and faculty, no matter how large and well educated the country (though a state can certainly work at the margins to encourage and support more domestic mathematicians).  It is no coincidence that all of the top mathematics department worldwide actively recruit the best mathematicians regardless of national origin, and often retain immigration counsel to assist with situations in which these mathematicians come from a country that is currently politically disfavoured by their own.

Of course, mathematicians cannot ignore the political realities of the modern international order altogether.  Anyone who has organised an international conference or program knows that there will inevitably be visa issues to resolve because the host country makes it particularly difficult for certain nationals to attend the event.  I myself, like many other academics working long-term in the United States, have certainly experienced my own share of immigration bureaucracy, starting with various glitches in the renewal or application of my J-1 and O-1 visas, then to the lengthy vetting process for acquiring permanent residency (or “green card”) status, and finally to becoming naturalised as a US citizen (retaining dual citizenship with Australia).  Nevertheless, while the process could be slow and frustrating, there was at least an order to it.  The rules of the game were complicated, but were known in advance, and did not abruptly change in the middle of playing it (save in truly exceptional situations, such as the days after the September 11 terrorist attacks).  One just had to study the relevant visa regulations (or hire an immigration lawyer to do so), fill out the paperwork and submit to the relevant background checks, and remain in good standing until the application was approved in order to study, work, or participate in a mathematical activity held in another country.  On rare occasion, some senior university administrator may have had to contact a high-ranking government official to approve some particularly complicated application, but for the most part one could work through normal channels in order to ensure for instance that the majority of participants of a conference could actually be physically present at that conference, or that an excellent mathematician hired by unanimous consent by a mathematics department could in fact legally work in that department.

With the recent and highly publicised executive order on immigration, many of these fundamental assumptions have been seriously damaged, if not destroyed altogether.  Even if the order was withdrawn immediately, there is no longer an assurance, even for nationals not initially impacted by that order, that some similar abrupt and major change in the rules for entry to the United States could not occur, for instance for a visitor who has already gone through the lengthy visa application process and background checks, secured the appropriate visa, and is already in flight to the country.  This is already affecting upcoming or ongoing mathematical conferences or programs in the US, with many international speakers (including those from countries not directly affected by the order) now cancelling their visit, either in protest or in concern about their ability to freely enter and leave the country.  Even some conferences outside the US are affected, as some mathematicians currently in the US with a valid visa or even permanent residency are uncertain if they could ever return back to their place of work if they left the country to attend a meeting.  In the slightly longer term, it is likely that the ability of elite US institutions to attract the best students and faculty will be seriously impacted.  Again, the losses would be strongest regarding candidates that were nationals of the countries affected by the current executive order, but I fear that many other mathematicians from other countries would now be much more concerned about entering and living in the US than they would have previously.

It is still possible for this sort of long-term damage to the mathematical community (both within the US and abroad) to be reversed or at least contained, but at present there is a real risk of the damage becoming permanent.  To prevent this, it seems insufficient for me for the current order to be rescinded, as desirable as that would be; some further legislative or judicial action would be needed to begin restoring enough trust in the stability of the US immigration and visa system that the international travel that is so necessary to modern mathematical research becomes “just” a bureaucratic headache again.

Of course, the impact of this executive order is far, far broader than just its effect on mathematicians and mathematical research.  But there are countless other venues on the internet and elsewhere to discuss these other aspects (or politics in general).  (For instance, discussion of the qualifications, or lack thereof, of the current US president can be carried out at this previous post.) I would therefore like to open this post to readers to discuss the effects or potential effects of this order on the mathematical community; I particularly encourage mathematicians who have been personally affected by this order to share their experiences.  As per the rules of the blog, I request that “the discussions are kept constructive, polite, and at least tangentially relevant to the topic at hand”.

Some relevant links (please feel free to suggest more, either through comments or by email):

I’ve just uploaded to the arXiv my paper “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest than I had hoped, but they did at least seem to indicate a potential strategy to make further progress on the problem, and also highlight some of the difficulties of the problem.

One can rephrase the lonely runner conjecture as the following covering problem. Given any integer “velocity” {v} and radius {0 < \delta < 1/2}, define the Bohr set {B(v,\delta)} to be the subset of the unit circle {{\bf R}/{\bf Z}} given by the formula

\displaystyle B(v,\delta) := \{ t \in {\bf R}/{\bf Z}: \|vt\| \leq \delta \},

where {\|x\|} denotes the distance of {x} to the nearest integer. Thus, for {v} positive, {B(v,\delta)} is simply the union of the {v} intervals {[\frac{a-\delta}{v}, \frac{a+\delta}{v}]} for {a=0,\dots,v-1}, projected onto the unit circle {{\bf R}/{\bf Z}}; in the language of the usual formulation of the lonely runner conjecture, {B(v,\delta)} represents those times in which a runner moving at speed {v} returns to within {\delta} of his or her starting position. For any non-zero integers {v_1,\dots,v_n}, let {\delta(v_1,\dots,v_n)} be the smallest radius {\delta} such that the {n} Bohr sets {B(v_1,\delta),\dots,B(v_n,\delta)} cover the unit circle:

\displaystyle {\bf R}/{\bf Z} = \bigcup_{i=1}^n B(v_i,\delta). \ \ \ \ \ (1)

 

Then define {\delta_n} to be the smallest value of {\delta(v_1,\dots,v_n)}, as {v_1,\dots,v_n} ranges over tuples of distinct non-zero integers. The Dirichlet approximation theorem quickly gives that

\displaystyle \delta(1,\dots,n) = \frac{1}{n+1}

and hence

\displaystyle \delta_n \leq \frac{1}{n+1}

for any {n \geq 1}. The lonely runner conjecture is equivalent to the assertion that this bound is in fact optimal:

Conjecture 1 (Lonely runner conjecture) For any {n \geq 1}, one has {\delta_n = \frac{1}{n+1}}.

This conjecture is currently known for {n \leq 6} (see this paper of Barajas and Serra), but remains open for higher {n}.

It is natural to try to attack the problem by establishing lower bounds on the quantity {\delta_n}. We have the following “trivial” bound, that gets within a factor of two of the conjecture:

Proposition 2 (Trivial bound) For any {n \geq 1}, one has {\delta_n \geq \frac{1}{2n}}.

Proof: It is not difficult to see that for any non-zero velocity {v} and any {0 < \delta < 1/2}, the Bohr set {B(v,\delta)} has Lebesgue measure {m(B(v,\delta)) = 2\delta}. In particular, by the union bound

\displaystyle m(\bigcup_{i=1}^n B(v_i,\delta)) \leq \sum_{i=1}^n m(B(v_i,\delta)) \ \ \ \ \ (2)

 

we see that the covering (1) is only possible if {1 \leq 2 n \delta}, giving the claim. \Box

So, in some sense, all the difficulty is coming from the need to improve upon the trivial union bound (2) by a factor of two.

Despite the crudeness of the union bound (2), it has proven surprisingly hard to make substantial improvements on the trivial bound {\delta_n \geq \frac{1}{2n}}. In 1994, Chen obtained the slight improvement

\displaystyle \delta_n \geq \frac{1}{2n - 1 + \frac{1}{2n-3}}

which was improved a little by Chen and Cusick in 1999 to

\displaystyle \delta_n \geq \frac{1}{2n-3}

when {2n-3} was prime. In a recent paper of Perarnau and Serra, the bound

\displaystyle \delta_n \geq \frac{1}{2n-2+o(1)}

was obtained for arbitrary {n}. These bounds only improve upon the trivial bound by a multiplicative factor of {1+O(1/n)}. Heuristically, one reason for this is as follows. The union bound (2) would of course be sharp if the Bohr sets {B(v_i,\delta)} were all disjoint. Strictly speaking, such disjointness is not possible, because all the Bohr sets {B(v_i,\delta)} have to contain the origin as an interior point. However, it is possible to come up with a large number of Bohr sets {B(v_i,\delta)} which are almost disjoint. For instance, suppose that we had velocities {v_1,\dots,v_s} that were all prime numbers between {n/4} and {n/2}, and that {\delta} was equal to {\delta_n} (and in particular was between {1/2n} and {1/(n+1)}. Then each set {B(v_i,\delta)} can be split into a “kernel” interval {[-\frac{\delta}{v_i}, \frac{\delta}{v_i}]}, together with the “petal” intervals {\bigcup_{a=1}^{v_i-1} [\frac{a-\delta}{v_i}, \frac{a+\delta}{v_i}]}. Roughly speaking, as the prime {v_i} varies, the kernel interval stays more or less fixed, but the petal intervals range over disjoint sets, and from this it is not difficult to show that

\displaystyle m(\bigcup_{i=1}^s B(v_i,\delta)) = (1-O(\frac{1}{n})) \sum_{i=1}^s m(B(v_i,\delta)),

so that the union bound is within a multiplicative factor of {1+O(\frac{1}{n})} of the truth in this case.

This does not imply that {\delta_n} is within a multiplicative factor of {1+O(1/n)} of {\frac{1}{2n}}, though, because there are not enough primes between {n/4} and {n/2} to assign to {n} distinct velocities; indeed, by the prime number theorem, there are only about {\frac{n}{4\log n}} such velocities that could be assigned to a prime. So, while the union bound could be close to tight for up to {\asymp n/\log n} Bohr sets, the above counterexamples don’t exclude improvements to the union bound for larger collections of Bohr sets. Following this train of thought, I was able to obtain a logarithmic improvement to previous lower bounds:

Theorem 3 For sufficiently large {n}, one has {\delta_n \geq \frac{1}{2n} + \frac{c \log n}{n^2 (\log\log n)^2}} for some absolute constant {c>0}.

The factors of {\log\log n} in the denominator are for technical reasons and might perhaps be removable by a more careful argument. However it seems difficult to adapt the methods to improve the {\log n} in the numerator, basically because of the obstruction provided by the near-counterexample discussed above.

Roughly speaking, the idea of the proof of this theorem is as follows. If we have the covering (1) for {\delta} very close to {1/2n}, then the multiplicity function {\sum_{i=1}^n 1_{B(v_i,\delta)}} will then be mostly equal to {1}, but occasionally be larger than {1}. On the other hand, one can compute that the {L^2} norm of this multiplicity function is significantly larger than {1} (in fact it is at least {(3/2-o(1))^{1/2}}). Because of this, the {L^3} norm must be very large, which means that the triple intersections {B(v_i,\delta) \cap B(v_j,\delta) \cap B(v_k,\delta)} must be quite large for many triples {(i,j,k)}. Using some basic Fourier analysis and additive combinatorics, one can deduce from this that the velocities {v_1,\dots,v_n} must have a large structured component, in the sense that there exists an arithmetic progression of length {\asymp n} that contains {\asymp n} of these velocities. For simplicity let us take the arithmetic progression to be {\{1,\dots,n\}}, thus {\asymp n} of the velocities {v_1,\dots,v_n} lie in {\{1,\dots,n\}}. In particular, from the prime number theorem, most of these velocities will not be prime, and will in fact likely have a “medium-sized” prime factor (in the precise form of the argument, “medium-sized” is defined to be “between {\log^{10} n} and {n^{1/10}}“). Using these medium-sized prime factors, one can show that many of the {B(v_i,\delta)} will have quite a large overlap with many of the other {B(v_j,\delta)}, and this can be used after some elementary arguments to obtain a more noticeable improvement on the union bound (2) than was obtained previously.

A modification of the above argument also allows for the improved estimate

\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1+c-o(1)}{2n} \ \ \ \ \ (3)

 

if one knows that all of the velocities {v_1,\dots,v_n} are of size {O(n)}.

In my previous blog post, I showed that in order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that all of the velocities {v_1,\dots,v_n} are of size {O(n^{O(n^2)})}; I reproduce this argument (slightly cleaned up for publication) in the current preprint. There is unfortunately a huge gap between {O(n)} and {O(n^{O(n^2)})}, so the above bound (3) does not immediately give any new bounds for {\delta_n}. However, one could perhaps try to start attacking the lonely runner conjecture by increasing the range {O(n)} for which one has good results, and by decreasing the range {O(n^{O(n^2)})} that one can reduce to. For instance, in the current preprint I give an elementary argument (using a certain amount of case-checking) that shows that the lonely runner bound

\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1}{n+1} \ \ \ \ \ (4)

 

holds if all the velocities {v_1,\dots,v_n} are assumed to lie between {1} and {1.2 n}. This upper threshold of {1.2 n} is only a tiny improvement over the trivial threshold of {n}, but it seems to be an interesting sub-problem of the lonely runner conjecture to increase this threshold further. One key target would be to get up to {2n}, as there are actually a number of {n}-tuples {(v_1,\dots,v_n)} in this range for which (4) holds with equality. The Dirichlet approximation theorem of course gives the tuple {(1,2,\dots,n)}, but there is also the double {(2,4,\dots,2n)} of this tuple, and furthermore there is an additional construction of Goddyn and Wong that gives some further examples such as {(1,2,3,4,5,7,12)}, or more generally one can start with the standard tuple {(1,\dots,n)} and accelerate one of the velocities {v} to {2v}; this turns out to work as long as {v} shares a common factor with every integer between {n-v+1} and {2n-2v+1}. There are a few more examples of this type in the paper of Goddyn and Wong, but all of them can be placed in an arithmetic progression of length {O(n \log n)} at most, so if one were very optimistic, one could perhaps envision a strategy in which the upper bound of {O(n^{O(n^2)})} mentioned earlier was reduced all the way to something like {O( n \log n )}, and then a separate argument deployed to treat this remaining case, perhaps isolating the constructions of Goddyn and Wong (and possible variants thereof) as the only extreme cases.

I just learned (from Emmanuel Kowalski’s blog) that the AMS has just started a repository of open-access mathematics lecture notes.  There are only a few such sets of notes there at present, but hopefully it will grow in the future; I just submitted some old lecture notes of mine from an undergraduate linear algebra course I taught in 2002 (with some updating of format and fixing of various typos).

 

[Update, Dec 22: my own notes are now on the repository.]

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear Schrödinger system, submitted to Analysis and PDE. This paper is an analogue of a recent paper of mine in which I constructed a supercritical defocusing nonlinear wave (NLW) system {-\partial_{tt} u + \Delta u = (\nabla F)(u)} which exhibited smooth solutions that developed singularities in finite time. Here, we achieve essentially the same conclusion for the (inhomogeneous) supercritical defocusing nonlinear Schrödinger (NLS) equation

\displaystyle  i \partial_t u + \Delta u = (\nabla F)(u) + G \ \ \ \ \ (1)

where {u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}^m} is now a system of scalar fields, {F: {\bf C}^m \rightarrow {\bf R}} is a potential which is strictly positive and homogeneous of degree {p+1} (and invariant under phase rotations {u \mapsto e^{i\theta} u}), and {G: {\bf R} \times {\bf R}^d \rightarrow {\bf C}^m} is a smooth compactly supported forcing term, needed for technical reasons.

To oversimplify somewhat, the equation (1) is known to be globally regular in the energy-subcritical case when {d \leq 2}, or when {d \geq 3} and {p < 1+\frac{4}{d-2}}; global regularity is also known (but is significantly more difficult to establish) in the energy-critical case when {d \geq 3} and {p = 1 +\frac{4}{d-2}}. (This is an oversimplification for a number of reasons, in particular in higher dimensions one only knows global well-posedness instead of global regularity. See this previous post for some exploration of this issue in the context of nonlinear wave equations.) The main result of this paper is to show that global regularity can break down in the remaining energy-supercritical case when {d \geq 3} and {p > 1 + \frac{4}{d-2}}, at least when the target dimension {m} is allowed to be sufficiently large depending on the spatial dimension {d} (I did not try to achieve the optimal value of {m} here, but the argument gives a value of {m} that grows quadratically in {d}). Unfortunately, this result does not directly impact the most interesting case of the defocusing scalar NLS equation

\displaystyle  i \partial_t u + \Delta u = |u|^{p-1} u \ \ \ \ \ (2)

in which {m=1}; however it does establish a rigorous barrier to any attempt to prove global regularity for the scalar NLS equation, in that such an attempt needs to crucially use some property of the scalar NLS that is not shared by the more general systems in (1). For instance, any approach that is primarily based on the conservation laws of mass, momentum, and energy (which are common to both (1) and (2)) will not be sufficient to establish global regularity of supercritical defocusing scalar NLS.

The method of proof in this paper is broadly similar to that in the previous paper for NLW, but with a number of additional technical complications. Both proofs begin by reducing matters to constructing a discretely self-similar solution. In the case of NLW, this solution lived on a forward light cone {\{ (t,x): |x| \leq t \}} and obeyed a self-similarity

\displaystyle  u(2t, 2x) = 2^{-\frac{2}{p-1}} u(t,x).

The ability to restrict to a light cone arose from the finite speed of propagation properties of NLW. For NLS, the solution will instead live on the domain

\displaystyle  H_d := ([0,+\infty) \times {\bf R}^d) \backslash \{(0,0)\}

and obey a parabolic self-similarity

\displaystyle  u(4t, 2x) = 2^{-\frac{2}{p-1}} u(t,x)

and solve the homogeneous version {G=0} of (1). (The inhomogeneity {G} emerges when one truncates the self-similar solution so that the initial data is compactly supported in space.) A key technical point is that {u} has to be smooth everywhere in {H_d}, including the boundary component {\{ (0,x): x \in {\bf R}^d \backslash \{0\}\}}. This unfortunately rules out many of the existing constructions of self-similar solutions, which typically will have some sort of singularity at the spatial origin.

The remaining steps of the argument can broadly be described as quantifier elimination: one systematically eliminates each of the degrees of freedom of the problem in turn by locating the necessary and sufficient conditions required of the remaining degrees of freedom in order for the constraints of a particular degree of freedom to be satisfiable. The first such degree of freedom to eliminate is the potential function {F}. The task here is to determine what constraints must exist on a putative solution {u} in order for there to exist a (positive, homogeneous, smooth away from origin) potential {F} obeying the homogeneous NLS equation

\displaystyle  i \partial_t u + \Delta u = (\nabla F)(u).

Firstly, the requirement that {F} be homogeneous implies the Euler identity

\displaystyle  \langle (\nabla F)(u), u \rangle = (p+1) F(u)

(where {\langle,\rangle} denotes the standard real inner product on {{\bf C}^m}), while the requirement that {F} be phase invariant similarly yields the variant identity

\displaystyle  \langle (\nabla F)(u), iu \rangle = 0,

so if one defines the potential energy field to be {V = F(u)}, we obtain from the chain rule the equations

\displaystyle  \langle i \partial_t u + \Delta u, u \rangle = (p+1) V

\displaystyle  \langle i \partial_t u + \Delta u, iu \rangle = 0

\displaystyle  \langle i \partial_t u + \Delta u, \partial_t u \rangle = \partial_t V

\displaystyle  \langle i \partial_t u + \Delta u, \partial_{x_j} u \rangle = \partial_{x_j} V.

Conversely, it turns out (roughly speaking) that if one can locate fields {u} and {V} obeying the above equations (as well as some other technical regularity and non-degeneracy conditions), then one can find an {F} with all the required properties. The first of these equations can be thought of as a definition of the potential energy field {V}, and the other three equations are basically disguised versions of the conservation laws of mass, energy, and momentum respectively. The construction of {F} relies on a classical extension theorem of Seeley that is a relative of the Whitney extension theorem.

Now that the potential {F} is eliminated, the next degree of freedom to eliminate is the solution field {u}. One can observe that the above equations involving {u} and {V} can be expressed instead in terms of {V} and the Gram-type matrix {G[u,u]} of {u}, which is a {(2d+4) \times (2d+4)} matrix consisting of the inner products {\langle D_1 u, D_2 u \rangle} where {D_1,D_2} range amongst the {2d+4} differential operators

\displaystyle  D_1,D_2 \in \{ 1, i, \partial_t, i\partial_t, \partial_{x_1},\dots,\partial_{x_d}, i\partial_{x_1}, \dots, i\partial_{x_d}\}.

To eliminate {u}, one thus needs to answer the question of what properties are required of a {(2d+4) \times (2d+4)} matrix {G} for it to be the Gram-type matrix {G = G[u,u]} of a field {u}. Amongst some obvious necessary conditions are that {G} needs to be symmetric and positive semi-definite; there are also additional constraints coming from identities such as

\displaystyle  \partial_t \langle u, u \rangle = 2 \langle u, \partial_t u \rangle

\displaystyle  \langle i u, \partial_t u \rangle = - \langle u, i \partial_t u \rangle

and

\displaystyle  \partial_{x_j} \langle iu, \partial_{x_k} u \rangle - \partial_{x_k} \langle iu, \partial_{x_j} u \rangle = 2 \langle i \partial_{x_j} u, \partial_{x_k} u \rangle.

Ideally one would like a theorem that asserts (for {m} large enough) that as long as {G} obeys all of the “obvious” constraints, then there exists a suitably non-degenerate map {u} such that {G = G[u,u]}. In the case of NLW, the analogous claim was basically a consequence of the Nash embedding theorem (which can be viewed as a theorem about the solvability of the system of equations {\langle \partial_{x_j} u, \partial_{x_k} u \rangle = g_{jk}} for a given positive definite symmetric set of fields {g_{jk}}). However, the presence of the complex structure in the NLS case poses some significant technical challenges (note for instance that the naive complex version of the Nash embedding theorem is false, due to obstructions such as Liouville’s theorem that prevent a compact complex manifold from being embeddable holomorphically in {{\bf C}^m}). Nevertheless, by adapting the proof of the Nash embedding theorem (in particular, the simplified proof of Gunther that avoids the need to use the Nash-Moser iteration scheme) we were able to obtain a partial complex analogue of the Nash embedding theorem that sufficed for our application; it required an artificial additional “curl-free” hypothesis on the Gram-type matrix {G[u,u]}, but fortunately this hypothesis ends up being automatic in our construction. Also, this version of the Nash embedding theorem is unable to prescribe the component {\langle \partial_t u, \partial_t u \rangle} of the Gram-type matrix {G[u,u]}, but fortunately this component is not used in any of the conservation laws and so the loss of this component does not cause any difficulty.

After applying the above-mentioned Nash-embedding theorem, the task is now to locate a matrix {G} obeying all the hypotheses of that theorem, as well as the conservation laws for mass, momentum, and energy (after defining the potential energy field {V} in terms of {G}). This is quite a lot of fields and constraints, but one can cut down significantly on the degrees of freedom by requiring that {G} is spherically symmetric (in a tensorial sense) and also continuously self-similar (not just discretely self-similar). Note that this hypothesis is weaker than the assertion that the original field {u} is spherically symmetric and continuously self-similar; indeed we do not know if non-trivial solutions of this type actually exist. These symmetry hypotheses reduce the number of independent components of the {(2d+4) \times (2d+4)} matrix {G} to just six: {g_{1,1}, g_{1,i\partial_t}, g_{1,i\partial_r}, g_{\partial_r, \partial_r}, g_{\partial_\omega, \partial_\omega}, g_{\partial_r, \partial_t}}, which now take as their domain the {1+1}-dimensional space

\displaystyle  H_1 := ([0,+\infty) \times {\bf R}) \backslash \{(0,0)\}.

One now has to construct these six fields, together with a potential energy field {v}, that obey a number of constraints, notably some positive definiteness constraints as well as the aforementioned conservation laws for mass, momentum, and energy.

The field {g_{1,i\partial_t}} only arises in the equation for the potential {v} (coming from Euler’s identity) and can easily be eliminated. Similarly, the field {g_{\partial_r,\partial_t}} only makes an appearance in the current of the energy conservation law, and so can also be easily eliminated so long as the total energy is conserved. But in the energy-supercritical case, the total energy is infinite, and so it is relatively easy to eliminate the field {g_{\partial_r, \partial_t}} from the problem also. This leaves us with the task of constructing just five fields {g_{1,1}, g_{1,i\partial_r}, g_{\partial_r,\partial_r}, g_{\partial_\omega,\partial_\omega}, v} obeying a number of positivity conditions, symmetry conditions, regularity conditions, and conservation laws for mass and momentum.

The potential field {v} can effectively be absorbed into the angular stress field {g_{\partial_\omega,\partial_\omega}} (after placing an appropriate counterbalancing term in the radial stress field {g_{\partial_r, \partial_r}} so as not to disrupt the conservation laws), so we can also eliminate this field. The angular stress field {g_{\partial_\omega, \partial_\omega}} is then only constrained through the momentum conservation law and a requirement of positivity; one can then eliminate this field by converting the momentum conservation law from an equality to an inequality. Finally, the radial stress field {g_{\partial_r, \partial_r}} is also only constrained through a positive definiteness constraint and the momentum conservation inequality, so it can also be eliminated from the problem after some further modification of the momentum conservation inequality.

The task then reduces to locating just two fields {g_{1,1}, g_{1,i\partial_r}} that obey a mass conservation law

\displaystyle  \partial_t g_{1,1} = 2 \left(\partial_r + \frac{d-1}{r} \right) g_{1,i\partial r}

together with an additional inequality that is the remnant of the momentum conservation law. One can solve for the mass conservation law in terms of a single scalar field {W} using the ansatz

\displaystyle g_{1,1} = 2 r^{1-d} \partial_r (r^d W)

\displaystyle g_{1,i\partial_r} = r^{1-d} \partial_t (r^d W)

so the problem has finally been simplified to the task of locating a single scalar field {W} with some scaling and homogeneity properties that obeys a certain differential inequality relating to momentum conservation. This turns out to be possible by explicitly writing down a specific scalar field {W} using some asymptotic parameters and cutoff functions.

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