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In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. (The principle can apply to other mathematical statements than inequalities, with the “hypothesis” and “conclusion” of that statement generally playing the role of the “right-hand side” and “left-hand side” of an inequality, but for sake of discussion I will restrict attention here to inequalities.) One can formalise this principle as follows. Many inequalities in analysis can be expressed in the form

\displaystyle A(f) \leq B(f) \ \ \ \ \ (1)

for all {f} in some space {X} (in many cases {X} will be a function space, and {f} a function in that space), where {A(f)} and {B(f)} are some functionals of {f} (that is to say, real-valued functions of {f}). For instance, {B(f)} might be some function space norm of {f} (e.g. an {L^p} norm), and {A(f)} might be some function space norm of some transform of {f}. In addition, we assume we have some group {G} of symmetries {T: X \rightarrow X} acting on the underlying space. For instance, if {X} is a space of functions on some spatial domain, the group might consist of translations (e.g. {Tf(x) = f(x-h)} for some shift {h}), or perhaps dilations with some normalisation (e.g. {Tf(x) = \frac{1}{\lambda^\alpha} f(\frac{x}{\lambda})} for some dilation factor {\lambda > 0} and some normalisation exponent {\alpha \in {\bf R}}, which can be thought of as the dimensionality of length one is assigning to {f}). If we have

\displaystyle A(Tf) = A(f)

for all symmetries {T \in G} and all {f \in X}, we say that {A} is invariant with respect to the symmetries in {G}; otherwise, it is not.

Suppose we know that the inequality (1) holds for all {f \in X}, but that there is an imbalance of symmetry: either {A} is {G}-invariant and {B} is not, or vice versa. Suppose first that {A} is {G}-invariant and {B} is not. Substituting {f} by {Tf} in (1) and taking infima, we can then amplify (1) to the stronger inequality

\displaystyle A(f) \leq \inf_{T \in G} B(Tf).

In particular, it is often the case that there is a way to send {T} off to infinity in such a way that the functional {B(Tf)} has a limit {B_\infty(f)}, in which case we obtain the amplification

\displaystyle A(f) \leq B_\infty(f) \ \ \ \ \ (2)

of (1). Note that these amplified inequalities will now be {G}-invariant on both sides (assuming that the way in which we take limits as {T \rightarrow \infty} is itself {G}-invariant, which it often is in practice). Similarly, if {B} is {G}-invariant but {A} is not, we may instead amplify (1) to

\displaystyle \sup_{T \in G} A(Tf) \leq B(f)

and in particular (if {A(Tf)} has a limit {A_\infty(f)} as {T \rightarrow \infty})

\displaystyle A_\infty(f) \leq B(f). \ \ \ \ \ (3)

If neither {A(f)} nor {B(f)} has a {G}-symmetry, one can still use the {G}-symmetry by replacing {f} by {Tf} and taking a limit to conclude that

\displaystyle A_\infty(f) \leq B_\infty(f),

though now this inequality is not obviously stronger than the original inequality (1) (for instance it could well be trivial). In some cases one can also average over {G} instead of taking a limit as {T \rightarrow \infty}, thus averaging a non-invariant inequality into an invariant one.

As discussed in the previous post, this use of amplification gives rise to a general principle about inequalities: the most efficient inequalities are those in which the left-hand side and right-hand side enjoy the same symmetries. It is certainly possible to have true inequalities that have an imbalance of symmetry, but as shown above, such inequalities can always be amplified to more efficient and more symmetric inequalities. In the case when limits such as {A_\infty} and {B_\infty} exist, the limiting functionals {A_\infty(f)} and {B_\infty(f)} are often simpler in form, or more tractable analytically, than their non-limiting counterparts {A(f)} and {B(f)} (this is one of the main reasons why we take limits at infinity in the first place!), and so in many applications there is really no reason to use the weaker and more complicated inequality (1), when stronger, simpler, and more symmetric inequalities such as (2), (3) are available. Among other things, this explains why many of the most useful and natural inequalities one sees in analysis are dimensionally consistent.

One often tries to prove inequalities (1) by directly chaining together simpler inequalities. For instance, one might attempt to prove (1) by by first bounding {A(f)} by some auxiliary quantity {C(f)}, and then bounding {C(f)} by {B(f)}, thus obtaining (1) by chaining together two inequalities

\displaystyle A(f) \leq C(f) \leq B(f). \ \ \ \ \ (4)

A variant of the above principle then asserts that when proving inequalities by such direct methods, one should, whenever possible, try to maintain the symmetries that are present in both sides of the inequality. Why? Well, suppose that we ignored this principle and tried to prove (1) by establishing (4) for some {C} that is not {G}-invariant. Assuming for sake of argument that (4) were actually true, we could amplify the first half {A(f) \leq C(f)} of this inequality to conclude that

\displaystyle A(f) \leq \inf_{T \in G} C(Tf)

and also amplify the second half {C(f) \leq B(f)} of the inequality to conclude that

\displaystyle \sup_{T \in G} C(Tf) \leq B(f)

and hence (4) amplifies to

\displaystyle A(f) \leq \inf_{T \in G} C(Tf) \leq \sup_{T \in G} C(Tf) \leq B(f). \ \ \ \ \ (5)

Let’s say for sake of argument that all the quantities involved here are positive numbers (which is often the case in analysis). Then we see in particular that

\displaystyle \frac{\sup_{T \in G} C(Tf)}{\inf_{T \in G} C(Tf)} \leq \frac{B(f)}{A(f)}. \ \ \ \ \ (6)

Informally, (6) asserts that in order for the strategy (4) used to prove (1) to work, the extent to which {C} fails to be {G}-invariant cannot exceed the amount of “room” present in (1). In particular, when dealing with those “extremal” {f} for which the left and right-hand sides of (1) are comparable to each other, one can only have a bounded amount of non-{G}-invariance in the functional {C}. If {C} fails so badly to be {G}-invariant that one does not expect the left-hand side of (6) to be at all bounded in such extremal situations, then the strategy of proving (1) using the intermediate quantity {C} is doomed to failure – even if one has already produced some clever proof of one of the two inequalities {A(f) \leq C(f)} or {C(f) \leq B(f)} needed to make this strategy work. And even if it did work, one could amplify (4) to a simpler inequality

\displaystyle A(f) \leq C_\infty(f) \leq B(f) \ \ \ \ \ (7)

(assuming that the appropriate limit {C_\infty(f) = \lim_{T \rightarrow \infty} C(Tf)} existed) which would likely also be easier to prove (one can take whatever proofs one had in mind of the inequalities in (4), conjugate them by {T}, and take a limit as {T \rightarrow \infty} to extract a proof of (7)).

Here are some simple (but somewhat contrived) examples to illustrate these points. Suppose one wishes to prove the inequality

\displaystyle xy \leq x^2 + y^2 \ \ \ \ \ (8)

for all {x,y>0}. Both sides of this inequality are invariant with respect to interchanging {x} with {y}, so the principle suggests that when proving this inequality directly, one should only use sub-inequalities that are also invariant with respect to this interchange. However, in this particular case there is enough “room” in the inequality that it is possible (though somewhat unnatural) to violate this principle. For instance, one could decide (for whatever reason) to start with the inequality

\displaystyle 0 \leq (x - y/2)^2 = x^2 - xy + y^2/4

to conclude that

\displaystyle xy \leq x^2 + y^2/4

and then use the obvious inequality {x^2 + y^2/4 \leq x^2+y^2} to conclude the proof. Here, the intermediate quantity {x^2 + y^2/4} is not invariant with respect to interchange of {x} and {y}, but the failure is fairly mild (changing {x} and {y} only modifies the quantity {x^2 + y^2/4} by a multiplicative factor of {4} at most), and disappears completely in the most extremal case {x=y}, which helps explain why one could get away with using this quantity in the proof here. But it would be significantly harder (though still not impossible) to use non-symmetric intermediaries to prove the sharp version

\displaystyle xy \leq \frac{x^2 + y^2}{2}

of (8) (that is to say, the arithmetic mean-geometric mean inequality). Try it!

Similarly, consider the task of proving the triangle inequality

\displaystyle |z+w| \leq |z| + |w| \ \ \ \ \ (9)

for complex numbers {z, w}. One could try to leverage the triangle inequality {|x+y| \leq |x| + |y|} for real numbers by using the crude estimate

\displaystyle |z+w| \leq |\hbox{Re}(z+w)| + |\hbox{Im}(z+w)|

and then use the real triangle inequality to obtain

\displaystyle |\hbox{Re}(z+w)| \leq |\hbox{Re}(z)| + |\hbox{Re}(w)|

and

\displaystyle |\hbox{Im}(z+w)| \leq |\hbox{Im}(z)| + |\hbox{Im}(w)|

and then finally use the inequalities

\displaystyle |\hbox{Re}(z)|, |\hbox{Im}(z)| \leq |z| \ \ \ \ \ (10)

and

\displaystyle |\hbox{Re}(w)|, |\hbox{Im}(w)| \leq |w| \ \ \ \ \ (11)

but when one puts this all together at the end of the day, one loses a factor of two:

\displaystyle |z+w| \leq 2(|z| + |w|).

One can “blame” this loss on the fact that while the original inequality (9) was invariant with respect to phase rotation {(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}, the intermediate expressions we tried to use when proving it were not, leading to inefficient estimates. One can try to be smarter than this by using Pythagoras’ theorem {|z|^2 = |\hbox{Re}(z)|^2 + |\hbox{Im}(z)|^2}; this reduces the loss from {2} to {\sqrt{2}} but does not eliminate it completely, which is to be expected as one is still using non-invariant estimates in the proof. But one can remove the loss completely by using amplification; see the previous blog post for details (we also give a reformulation of this amplification below).

Here is a slight variant of the above example. Suppose that you had just learned in class to prove the triangle inequality

\displaystyle (\sum_{n=1}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=1}^\infty |a_n|^2)^{1/2} + (\sum_{n=1}^\infty |b_n|^2)^{1/2} \ \ \ \ \ (12)

for (say) real square-summable sequences {(a_n)_{n=1}^\infty}, {(b_n)_{n=1}^\infty}, and was tasked to conclude the corresponding inequality

\displaystyle (\sum_{n \in {\bf Z}} |a_n+b_n|^2)^{1/2} \leq (\sum_{n \in {\bf Z}} |a_n|^2)^{1/2} + (\sum_{n \in {\bf Z}} |b_n|^2)^{1/2} \ \ \ \ \ (13)

for doubly infinite square-summable sequences {(a_n)_{n \in {\bf Z}}, (b_n)_{n \in {\bf Z}}}. The quickest way to do this is of course to exploit a bijection between the natural numbers {1,2,\dots} and the integers, but let us say for sake of argument that one was unaware of such a bijection. One could then proceed instead by splitting the integers into the positive integers and the non-positive integers, and use (12) on each component separately; this is very similar to the strategy of proving (9) by splitting a complex number into real and imaginary parts, and will similarly lose a factor of {2} or {\sqrt{2}}. In this case, one can “blame” this loss on the abandonment of translation invariance: both sides of the inequality (13) are invariant with respect to shifting the sequences {(a_n)_{n \in {\bf Z}}}, {(b_n)_{n \in {\bf Z}}} by some shift {h} to arrive at {(a_{n-h})_{n \in {\bf Z}}, (b_{n-h})_{n \in {\bf Z}}}, but the intermediate quantities caused by splitting the integers into two subsets are not invariant. Another way of thinking about this is that the splitting of the integers gives a privileged role to the origin {n=0}, whereas the inequality (13) treats all values of {n} equally thanks to the translation invariance, and so using such a splitting is unnatural and not likely to lead to optimal estimates. On the other hand, one can deduce (13) from (12) by sending this symmetry to infinity; indeed, after applying a shift to (12) we see that

\displaystyle (\sum_{n=-N}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=-N}^\infty |a_n|^2)^{1/2} + (\sum_{n=-N}^\infty |b_n|^2)^{1/2}

for any {N}, and on sending {N \rightarrow \infty} we obtain (13) (one could invoke the monotone convergence theorem here to justify the limit, though in this case it is simple enough that one can just use first principles).

Note that the principle of preserving symmetry only applies to direct approaches to proving inequalities such as (1). There is a complementary approach, discussed for instance in this previous post, which is to spend the symmetry to place the variable {f} “without loss of generality” in a “normal form”, “convenient coordinate system”, or a “good gauge”. Abstractly: suppose that there is some subset {Y} of {X} with the property that every {f \in X} can be expressed in the form {f = Tg} for some {T \in G} and {g \in Y} (that is to say, {X = GY}). Then, if one wishes to prove an inequality (1) for all {f \in X}, and one knows that both sides {A(f), B(f)} of this inequality are {G}-invariant, then it suffices to check (1) just for those {f} in {Y}, as this together with the {G}-invariance will imply the same inequality (1) for all {f} in {GY=X}. By restricting to those {f} in {Y}, one has given up (or spent) the {G}-invariance, as the set {Y} will in typical not be preserved by the group action {G}. But by the same token, by eliminating the invariance, one also eliminates the prohibition on using non-invariant proof techniques, and one is now free to use a wider range of inequalities in order to try to establish (1). Of course, such inequalities should make crucial use of the restriction {f \in Y}, for if they did not, then the arguments would work in the more general setting {f \in X}, and then the previous principle would again kick in and warn us that the use of non-invariant inequalities would be inefficient. Thus one should “spend” the symmetry wisely to “buy” a restriction {f \in Y} that will be of maximal utility in calculations (for instance by setting as many annoying factors and terms in one’s analysis to be {0} or {1} as possible).

As a simple example of this, let us revisit the complex triangle inequality (9). As already noted, both sides of this inequality are invariant with respect to the phase rotation symmetry {(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}. This seems to limit one to using phase-rotation-invariant techniques to establish the inequality, in particular ruling out the use of real and imaginary parts as discussed previously. However, we can instead spend the phase rotation symmetry to restrict to a special class of {z} and {w}. It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of {z+w} being a nonnegative real; this is of course possible since any complex number {z+w} can be turned into a nonnegative real by multiplying by an appropriate phase {e^{i\theta}}. Once {z+w} is a nonnegative real, the imaginary part disappears and we have

\displaystyle |z+w| = \hbox{Re}(z+w) = \hbox{Re}(z) + \hbox{Re}(w),

and the triangle inequality (9) is now an immediate consequence of (10), (11). (But note that if one had unwisely spent the symmetry to normalise, say, {z} to be a non-negative real, then one is no closer to establishing (9) than before one had spent the symmetry.)

Apoorva Khare and I have just uploaded to the arXiv our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“. This paper explores the relationship between positive definiteness of Hermitian matrices, and entrywise operations on these matrices. The starting point for this theory is the Schur product theorem, which asserts that if {A = (a_{ij})_{1 \leq i,j \leq N}} and {B = (b_{ij})_{1 \leq i,j \leq N}} are two {N \times N} Hermitian matrices that are positive semi-definite, then their Hadamard product

\displaystyle  A \circ B := (a_{ij} b_{ij})_{1 \leq i,j \leq N}

is also positive semi-definite. (One should caution that the Hadamard product is not the same as the usual matrix product.) To prove this theorem, first observe that the claim is easy when {A = {\bf u} {\bf u}^*} and {B = {\bf v} {\bf v}^*} are rank one positive semi-definite matrices, since in this case {A \circ B = ({\bf u} \circ {\bf v}) ({\bf u} \circ {\bf v})^*} is also a rank one positive semi-definite matrix. The general case then follows by noting from the spectral theorem that a general positive semi-definite matrix can be expressed as a non-negative linear combination of rank one positive semi-definite matrices, and using the bilinearity of the Hadamard product and the fact that the set of positive semi-definite matrices form a convex cone. A modification of this argument also lets one replace “positive semi-definite” by “positive definite” in the statement of the Schur product theorem.

One corollary of the Schur product theorem is that any polynomial {P(z) = c_0 + c_1 z + \dots + c_d z^d} with non-negative coefficients {c_n \geq 0} is entrywise positivity preserving on the space {{\mathbb P}_N({\bf C})} of {N \times N} positive semi-definite Hermitian matrices, in the sense that for any matrix {A = (a_{ij})_{1 \leq i,j \leq N}} in {{\mathbb P}_N({\bf C})}, the entrywise application

\displaystyle  P[A] := (P(a_{ij}))_{1 \leq i,j \leq N}

of {P} to {A} is also positive semi-definite. (As before, one should caution that {P[A]} is not the application {P(A)} of {P} to {A} by the usual functional calculus.) Indeed, one can expand

\displaystyle  P[A] = c_0 A^{\circ 0} + c_1 A^{\circ 1} + \dots + c_d A^{\circ d},

where {A^{\circ i}} is the Hadamard product of {i} copies of {A}, and the claim now follows from the Schur product theorem and the fact that {{\mathbb P}_N({\bf C})} is a convex cone.

A slight variant of this argument, already observed by Pólya and Szegö in 1925, shows that if {I} is any subset of {{\bf C}} and

\displaystyle  f(z) = \sum_{n=0}^\infty c_n z^n \ \ \ \ \ (1)

is a power series with non-negative coefficients {c_n \geq 0} that is absolutely and uniformly convergent on {I}, then {f} will be entrywise positivity preserving on the set {{\mathbb P}_N(I)} of positive definite matrices with entries in {I}. (In the case that {I} is of the form {I = [0,\rho]}, such functions are precisely the absolutely monotonic functions on {I}.)

In the work of Schoenberg and of Rudin, we have a converse: if {f: (-1,1) \rightarrow {\bf C}} is a function that is entrywise positivity preserving on {{\mathbb P}_N((-1,1))} for all {N}, then it must be of the form (1) with {c_n \geq 0}. Variants of this result, with {(-1,1)} replaced by other domains, appear in the work of Horn, Vasudeva, and Guillot-Khare-Rajaratnam.

This gives a satisfactory classification of functions {f} that are entrywise positivity preservers in all dimensions {N} simultaneously. However, the question remains as to what happens if one fixes the dimension {N}, in which case one may have a larger class of entrywise positivity preservers. For instance, in the trivial case {N=1}, a function would be entrywise positivity preserving on {{\mathbb P}_1((0,\rho))} if and only if {f} is non-negative on {I}. For higher {N}, there is a necessary condition of Horn (refined slightly by Guillot-Khare-Rajaratnam) which asserts (at least in the case of smooth {f}) that all derivatives of {f} at zero up to {(N-1)^{th}} order must be non-negative in order for {f} to be entrywise positivity preserving on {{\mathbb P}_N((0,\rho))} for some {0 < \rho < \infty}. In particular, if {f} is of the form (1), then {c_0,\dots,c_{N-1}} must be non-negative. In fact, a stronger assertion can be made, namely that the first {N} non-zero coefficients in (1) (if they exist) must be positive, or equivalently any negative term in (1) must be preceded (though not necessarily immediately) by at least {N} positive terms. If {f} is of the form (1) is entrywise positivity preserving on the larger set {{\mathbb P}_N((0,+\infty))}, one can furthermore show that any negative term in (1) must also be followed (though not necessarily immediately) by at least {N} positive terms.

The main result of this paper is that these sign conditions are the only constraints for entrywise positivity preserving power series. More precisely:

Theorem 1 For each {n}, let {\epsilon_n \in \{-1,0,+1\}} be a sign.

  • Suppose that any negative sign {\epsilon_M = -1} is preceded by at least {N} positive signs (thus there exists {0 \leq n_0 < \dots < n_{N-1}< M} with {\epsilon_{n_0} = \dots = \epsilon_{n_{N-1}} = +1}). Then, for any {0 < \rho < \infty}, there exists a convergent power series (1) on {(0,\rho)}, with each {c_n} having the sign of {\epsilon_n}, which is entrywise positivity preserving on {{\bf P}_N((0,\rho))}.
  • Suppose in addition that any negative sign {\epsilon_M = -1} is followed by at least {N} positive signs (thus there exists {M < n_N < \dots < n_{2N-1}} with {\epsilon_{n_N} = \dots = \epsilon_{n_{2N-1}} = +1}). Then there exists a convergent power series (1) on {(0,+\infty)}, with each {c_n} having the sign of {\epsilon_n}, which is entrywise positivity preserving on {{\bf P}_N((0,+\infty))}.

One can ask the same question with {(0,\rho)} or {(0,+\infty)} replaced by other domains such as {(-\rho,\rho)}, or the complex disk {D(0,\rho)}, but it turns out that there are far fewer entrywise positivity preserving functions in those cases basically because of the non-trivial zeroes of Schur polynomials in these ranges; see the paper for further discussion. We also have some quantitative bounds on how negative some of the coefficients can be compared to the positive coefficients, but they are a bit technical to state here.

The heart of the proofs of these results is an analysis of the determinants {\mathrm{det} P[ {\bf u} {\bf u}^* ]} of polynomials {P} applied entrywise to rank one matrices {uu^*}; the positivity of these determinants can be used (together with a continuity argument) to establish the positive definiteness of {P[uu^*]} for various ranges of {P} and {u}. Using the Cauchy-Binet formula, one can rewrite such determinants as linear combinations of squares of magnitudes of generalised Vandermonde determinants

\displaystyle  \mathrm{det}( u_i^{n_j} )_{1 \leq i,j \leq N},

where {{\bf u} = (u_1,\dots,u_N)} and the signs of the coefficients in the linear combination are determined by the signs of the coefficients of {P}. The task is then to find upper and lower bounds for the magnitudes of such generalised Vandermonde determinants. These determinants oscillate in sign, which makes the problem look difficult; however, an algebraic miracle intervenes, namely the factorisation

\displaystyle  \mathrm{det}( u_i^{n_j} )_{1 \leq i,j \leq N} = \pm V({\bf u}) s_\lambda({\bf u})

of the generalised Vandermonde determinant into the ordinary Vandermonde determinant

\displaystyle  V({\bf u}) = \prod_{1 \leq i < j \leq N} (u_j - u_i)

and a Schur polynomial {s_\lambda} applied to {{\bf u}}, where the weight {\lambda} of the Schur polynomial is determined by {n_1,\dots,n_N} in a simple fashion. The problem then boils down to obtaining upper and lower bounds for these Schur polynomials. Because we are restricting attention to matrices taking values in {(0,\rho)} or {(0,+\infty)}, the entries of {{\bf u}} can be taken to be non-negative. One can then take advantage of the total positivity of the Schur polynomials to compare these polynomials with a monomial, at which point one can obtain good criteria for {P[A]} to be positive definite when {A} is a rank one positive definite matrix {A = {\bf u} {\bf u}^*}.

If we allow the exponents {n_1,\dots,n_N} to be real numbers rather than integers (thus replacing polynomials or power series by Pusieux series or Hahn series), then we lose the above algebraic miracle, but we can replace it with a geometric miracle, namely the Harish-Chandra-Itzykson-Zuber identity, which I discussed in this previous blog post. This factors the above generalised Vandermonde determinant as the product of the ordinary Vandermonde determinant and an integral of a positive quantity over the orthogonal group, which one can again compare with a monomial after some fairly elementary estimates.

It remains to understand what happens for more general positive semi-definite matrices {A}. Here we use a trick of FitzGerald and Horn to amplify the rank one case to the general case, by expressing a general positive semi-definite matrix {A} as a linear combination of a rank one matrix {{\bf u} {\bf u}^*} and another positive semi-definite matrix {B} that vanishes on the last row and column (and is thus effectively a positive definite {N-1 \times N-1} matrix). Using the fundamental theorem of calculus to continuously deform the rank one matrix {{\bf u} {\bf u}^*} to {A} in the direction {B}, one can then obtain positivity results for {P[A]} from positivity results for {P[{\bf u} {\bf u}^*]} combined with an induction hypothesis on {N}.

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that

\displaystyle  \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+h_0) g_1(n+h_1)}{n} = 0

whenever {1 \leq \omega_m \leq x_m} were sequences going to infinity, {h_0,h_1} were distinct integers, and {g_0,g_1: {\bf N} \rightarrow {\bf C}} were {1}-bounded multiplicative functions which were non-pretentious in the sense that

\displaystyle  \liminf_{X \rightarrow \infty} \inf_{|t_j| \leq X} \sum_{p \leq X} \frac{1-\mathrm{Re}( g_j(p) \overline{\chi_j}(p) p^{it_j})}{p} = \infty \ \ \ \ \ (1)

for all Dirichlet characters {\chi_j} and for {j=0,1}. Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+h)}{n} = o(\log x)

for fixed any non-zero {h}, where {\lambda} was the Liouville function.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

\displaystyle  \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log x) \ \ \ \ \ (2)

for all odd {k} and all integers {h_1,\dots,h_k} (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument {n}).

For the more general Elliott conjecture, we can show that

\displaystyle  \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+h_1) \dots g_k(n+h_k)}{n} = 0

for any {k}, any integers {h_1,\dots,h_k} and any bounded multiplicative functions {g_1,\dots,g_k}, unless the product {g_1 \dots g_k} weakly pretends to be a Dirichlet character {\chi} in the sense that

\displaystyle  \sum_{p \leq X} \frac{1 - \hbox{Re}( g_1 \dots g_k(p) \overline{\chi}(p)}{p} = o(\log\log X).

This can be seen to imply (2) as a special case. Even when {g_1,\dots,g_k} does pretend to be a Dirichlet character {\chi}, we can still say something: if the limits

\displaystyle  f(a) := \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+ah_1) \dots g_k(n+ah_k)}{n}

exist for each {a \in {\bf Z}} (which can be guaranteed if we pass to a suitable subsequence), then {f} is the uniform limit of periodic functions {f_i}, each of which is {\chi}isotypic in the sense that {f_i(ab) = f_i(a) \chi(b)} whenever {a,b} are integers with {b} coprime to the periods of {\chi} and {f_i}. This does not pin down the value of any single correlation {f(a)}, but does put significant constraints on how these correlations may vary with {a}.

Among other things, this allows us to show that all {16} possible length four sign patterns {(\lambda(n+1),\dots,\lambda(n+4)) \in \{-1,+1\}^4} of the Liouville function occur with positive density, and all {65} possible length four sign patterns {(\mu(n+1),\dots,\mu(n+4)) \in \{-1,0,+1\}^4 \backslash \{-1,+1\}^4} occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

\displaystyle  f(a) := \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+a) \dots \lambda(n+(k-1)a)}{n}, \ \ \ \ \ (3)

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function {f}. The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime {p} and observe that {\lambda(pn)=-\lambda(n)} for any {n}, which allows us to rewrite (3) as

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle  \sum_{n \leq X} \frac{\lambda(pn) \lambda(pn+ap) \dots \lambda(pn+(k-1)ap)}{n}.

Making the change of variables {n' = pn}, we obtain

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle \sum_{n' \leq pX} \frac{\lambda(n') \lambda(n'+ap) \dots \lambda(n'+(k-1)ap)}{n'} p 1_{p|n'}.

The difference between {n' \leq pX} and {n' \leq X} is negligible in the limit (here is where we crucially rely on the log-averaging), hence

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} p 1_{p|n}

and thus by (3) we have

\displaystyle  (-1)^k f(a) = f(ap) + \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} (p 1_{p|n}-1).

The entropy decrement argument can be used to show that the latter limit is small for most {p} (roughly speaking, this is because the factors {p 1_{p|n}-1} behave like independent random variables as {p} varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the {\lambda} factors). We thus obtain the approximate isotopy property

\displaystyle  (-1)^k f(a) \approx f(ap) \ \ \ \ \ (4)

for most {a} and {p}.

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express {f(a)} as a multiple correlation

\displaystyle  f(a) = \int_X g(x) g(T^a x) \dots g(T^{(k-1)a} x)\ d\mu(x)

for some probability space {(X,\mu)} equipped with a measure-preserving invertible map {T: X \rightarrow X}. Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

\displaystyle  f(a) = f_1(a) + f_2(a) \ \ \ \ \ (5)

where {f_1} is a nilsequence, and {f_2} goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on {X}, which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on {f_1} so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error {f_2(a)}, we can now combine (5) to conclude that

\displaystyle  f(a) \approx (-1)^k f_1(ap).

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up {f_1} further into a periodic piece {f_0} and an “irrational” or “minor arc” piece {f_3}. The contribution of the minor arc piece {f_3} can be shown to mostly cancel itself out after dilating by primes {p} and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

\displaystyle  f(a) \approx (-1)^k f_0(ap),

which already shows (heuristically, at least) the claim that {f} can be approximated by periodic functions {f_0} which are isotopic in the sense that

\displaystyle  f_0(a) \approx (-1)^k f_0(ap).

But if {k} is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes {p} that are {1} modulo the period of {f_0}, and conclude now that {f_0} vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in {p} using the “{W}-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

\displaystyle  (-1)^k f(a) \approx {\bf E}_{b: (b,W)=1} f(ab)

where {b} ranges over a large range of integers coprime to some primorial {W = \prod_{p \leq w} p}. On the other hand, by iterating (4) we have

\displaystyle  f(a) \approx f(apq)

for most semiprimes {pq}, and by again averaging over semiprimes one can obtain an approximation of the form

\displaystyle  f(a) \approx {\bf E}_{b: (b,W)=1} f(ab).

For {k} odd, one can combine the two approximations to conclude that {f(a)=0}. (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

The complete homogeneous symmetric polynomial {h_d(x_1,\dots,x_n)} of {n} variables {x_1,\dots,x_n} and degree {d} can be defined as

\displaystyle h_d(x_1,\dots,x_n) := \sum_{1 \leq i_1 \leq \dots \leq i_d \leq n} x_{i_1} \dots x_{i_d},

thus for instance

\displaystyle h_0(x_1,\dots,x_n) = 0,

\displaystyle h_1(x_1,\dots,x_n) = x_1 + \dots + x_n,

and

\displaystyle h_2(x_1,\dots,x_n) = x_1^2 + \dots + x_n^2 + \sum_{1 \leq i < j \leq n} x_i x_j.

One can also define all the complete homogeneous symmetric polynomials of {n} variables simultaneously by means of the generating function

\displaystyle \sum_{d=0}^\infty h_d(x_1,\dots,x_n) t^d = \frac{1}{(1-t x_1) \dots (1-tx_n)}. \ \ \ \ \ (1)

We will think of the variables {x_1,\dots,x_n} as taking values in the real numbers. When one does so, one might observe that the degree two polynomial {h_2} is a positive definite quadratic form, since it has the sum of squares representation

\displaystyle h_2(x_1,\dots,x_n) = \frac{1}{2} \sum_{i=1}^n x_i^2 + \frac{1}{2} (x_1+\dots+x_n)^2.

In particular, {h_2(x_1,\dots,x_n) > 0} unless {x_1=\dots=x_n=0}. This can be compared against the superficially similar quadratic form

\displaystyle x_1^2 + \dots + x_n^2 + \sum_{1 \leq i < j \leq n} \epsilon_{ij} x_i x_j

where {\epsilon_{ij} = \pm 1} are independent randomly chosen signs. The Wigner semicircle law says that for large {n}, the eigenvalues of this form will be mostly distributed in the interval {[-\sqrt{n}, \sqrt{n}]} using the semicircle distribution, so in particular the form is quite far from being positive definite despite the presence of the first {n} positive terms. Thus the positive definiteness is coming from the finer algebraic structure of {h_2}, and not just from the magnitudes of its coefficients.

One could ask whether the same positivity holds for other degrees {d} than two. For odd degrees, the answer is clearly no, since {h_d(-x_1,\dots,-x_n) = -h_d(x_1,\dots,x_n)} in that case. But one could hope for instance that

\displaystyle h_4(x_1,\dots,x_n) = \sum_{1 \leq i \leq j \leq k \leq l \leq n} x_i x_j x_k x_l

also has a sum of squares representation that demonstrates positive definiteness. This turns out to be true, but is remarkably tedious to establish directly. Nevertheless, we have a nice result of Hunter that gives positive definiteness for all even degrees {d}. In fact, a modification of his argument gives a little bit more:

Theorem 1 Let {n \geq 1}, let {d \geq 0} be even, and let {x_1,\dots,x_n} be reals.

  • (i) (Positive definiteness) One has {h_d(x_1,\dots,x_n) \geq 0}, with strict inequality unless {x_1=\dots=x_n=0}.
  • (ii) (Schur convexity) One has {h_d(x_1,\dots,x_n) \geq h_d(y_1,\dots,y_n)} whenever {(x_1,\dots,x_n)} majorises {(y_1,\dots,y_n)}, with equality if and only if {(y_1,\dots,y_n)} is a permutation of {(x_1,\dots,x_n)}.
  • (iii) (Schur-Ostrowski criterion for Schur convexity) For any {1 \leq i < j \leq n}, one has {(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j}) h_d(x_1,\dots,x_n) \geq 0}, with strict inequality unless {x_i=x_j}.

Proof: We induct on {d} (allowing {n} to be arbitrary). The claim is trivially true for {d=0}, and easily verified for {d=2}, so suppose that {d \geq 4} and the claims (i), (ii), (iii) have already been proven for {d-2} (and for arbitrary {n}).

If we apply the differential operator {(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j})} to {\frac{1}{(1-t x_1) \dots (1-tx_n)}} using the product rule, one obtains after a brief calculation

\displaystyle \frac{(x_i-x_j)^2 t^2}{(1-t x_1) \dots (1-tx_n) (1-t x_i) (1-t x_j)}.

Using (1) and extracting the {t^d} coefficient, we obtain the identity

\displaystyle (x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j}) h_d(x_1,\dots,x_n)

\displaystyle = (x_i-x_j)^2 h_{d-2}( x_1,\dots,x_n,x_i,x_j). \ \ \ \ \ (2)

The claim (iii) then follows from (i) and the induction hypothesis.

To obtain (ii), we use the more general statement (known as the Schur-Ostrowski criterion) that (ii) is implied from (iii) if we replace {h_d} by an arbitrary symmetric, continuously differentiable function. To establish this criterion, we induct on {n} (this argument can be made independently of the existing induction on {d}). If {(y_1,\dots,y_n)} is majorised by {(x_1,\dots,x_n)}, it lies in the permutahedron of {(x_1,\dots,x_n)}. If {(y_1,\dots,y_n)} lies on a face of this permutahedron, then after permuting both the {x_i} and {y_j} we may assume that {(y_1,\dots,y_m)} is majorised by {(x_1,\dots,x_m)}, and {(y_{m+1},\dots,y_n)} is majorised by {(x_{m+1},\dots,x_n)} for some {1 \leq m < n}, and the claim then follows from two applications of the induction hypothesis. If instead {(y_1,\dots,y_n)} lies in the interior of the permutahedron, one can follow it to the boundary by using one of the vector fields {(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j})}, and the claim follows from the boundary case.

Finally, to obtain (i), we observe that {(x_1,\dots,x_n)} majorises {(x,\dots,x)}, where {x} is the arithmetic mean of {x_1,\dots,x_n}. But {h_d(x,\dots,x)} is clearly a positive multiple of {x^d}, and the claim now follows from (ii). \Box

If the variables {x_1,\dots,x_n} are restricted to be nonnegative, the same argument gives Schur convexity for odd degrees also.

The proof in Hunter of positive definiteness is arranged a little differently than the one above, but still relies ultimately on the identity (2). I wonder if there is a genuinely different way to establish positive definiteness that does not go through this identity.

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