Let 
for all 
For 
In recent years, it has been realised that one can make more progress on this conjecture if one works instead with the logarithmically averaged version
of the conjecture, where we use the logarithmic averaging notation
Using the summation by parts (or telescoping series) identity
it is not difficult to show that the Chowla conjecture (1) for a given 
is equivalent to the prime number theorem; but the logarithmically averaged analogue
is significantly easier to show (a proof with the Liouville function 
“>this previous blog post). And indeed, significantly more is now known for the logarithmically averaged Chowla conjecture; in this paper of mine I had proven (2) for 
In view of this emerging consensus that the logarithmically averaged Chowla conjecture was easier than the ordinary Chowla conjecture, it was thus somewhat of a surprise for me to read a recent paper of Gomilko, Kwietniak, and Lemanczyk who (among other things) established the following statement:
Theorem 1 Assume that the logarithmically averaged Chowla conjecture (2) is true for all
going to infinity such that the Chowla conjecture (1) is true for all
for all
This implication does not use any special properties of the Liouville function (other than that they are bounded), and in fact proceeds by ergodic theoretic methods, focusing in particular on the ergodic decomposition of invariant measures of a shift into ergodic measures. Ergodic methods have proven remarkably fruitful in understanding these sorts of number theoretic and combinatorial problems, as could already be seen by the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recently by the work of Frantzikinakis and Host on Sarnak’s conjecture. (My first paper with Teravainen also uses ergodic theory tools.) Indeed, many other results in the subject were first discovered using ergodic theory methods.
On the other hand, many results in this subject that were first proven ergodic theoretically have since been reproven by more combinatorial means; my second paper with Teravainen is an instance of this. As it turns out, one can also prove Theorem 1 by a standard combinatorial (or probabilistic) technique known as the second moment method. In fact, one can prove slightly more:
Theorem 2 Let
. Then there exists a set
of natural numbers of logarithmic density
(that is,
) such that
for any distinct
It is not difficult to deduce Theorem 1 from Theorem 2 using a diagonalisation argument. Unfortunately, the known cases of the logarithmically averaged Chowla conjecture (
We now sketch the proof of Theorem 2. For any distinct 
We can expand this as
If all the 
By Markov’s inequality (and (3)), we conclude that for any fixed 
such that
By deleting at most finitely many elements, we may assume that 
For any 
for some 
interchanging the sums and using
We conclude on taking
as required.
be a monic polynomial of degree 
as 
(possibly with repetition).
of two variables with given initial data 
for which 
is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series 
, then the polynomial evolves by the formula 
evolves in time, the zeroes 
evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution?
are real and the discriminant 
is initially positive, we see that we start with two real zeroes centred around 
, which then approach each other until time 
, at which point the roots collide and then move off from each other in an imaginary direction.
) leading to the equations of motion 
are real, each term 
represents a (first-order) attraction in the dynamics between 
, but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of 
instead of 
. The evolution 
is the “energy” 
, one expects 
) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time 
.
, or else by using the equations of motion 
level, which is unfortunate since the first non-trivial odd case 
requires quantitative control on the 
level. (But it may be possible to make the Gowers uniformity bounds for 
of Schur polynomials, namely that such ratios are monotone non-decreasing in each coordinate of 
if 
(using the convention that the skew-Schur polynomial 
vanishes if 
and 
denotes the pointwise min and max of 
respectively). It is fairly easy to derive the monotonicity of 
from this, by using the expansion
be any 
of increasing elements of 
, one has
denotes the 
minor formed from the rows in 
and columns in 
.
, one has
), or
, 
, 
, 
). It turns out that this claim can be proven relatively easy by an induction argument, relying on the Dodgson and Karlin identities from 
that is expressed in block-matrix form as 
matrix, 
is an 
matrix, 
is an 
matrix, and 
is a 
. If 
of course. Indeed, to solve the equation 
are 
column vectors and 
are 
column vectors, we can expand this out as a system 
is invertible) 
of 
of the minor 
denote the 
matrix 
the 
be its transpose: 
(which we identify with 
, we conclude that 
is the 
basis column vector.
minor 
of the identity from 
of 
of 
; for instance, 
for some real diagonal matrix 
will be the same as the law of 
, where 
and eigenvectors 
, we have 
). Thus the coefficients 
with respect to the orthonormal basis 
can be thought of as independent (complex) Gaussian vectors, divided by that magnitude. Using this and the Chernoff inequality again, we see (for 
away from the real axis at least) that one has the concentration of measure 
are roughly constant). Similarly we have 
, we thus conclude the approximate relationship 
for some 
, one expects the top left 
of 
solves the Burgers-type equation 
.
of the identity, then 
. One checks that 
is a steady state solution to 
times the identity. 
, then by the semi-circular law (see 
(using an appropriate branch of the square root). One can then verify the self-similar solution 
when 
, we consider the characteristic flow 
formed by solving the ODE 
, ignoring for this discussion the problems of existence and uniqueness. Then from the chain rule, the equation 
. Inserting this back into 
crosses the real axis, as 
necessarily has negative or zero imaginary part. 
as in Example 
, which after making the change of variables 
becomes 
as the weak limit of 
as 
; we write this informally as 
converge weakly to the Dirac mass at 
.
be the spectral measure associated to 
, then the curve 
from ![{(0,1]}](https://s0.wp.com/latex.php?latex=%7B%280%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{(0,1]}")
to the space of measures is the high-dimensional limit 
of a Gelfand-Tsetlin pattern (discussed in 
as 
, then the curve is 
, corresponding to a pattern that is entirely filled with 
is a semicircular distribution, then the pattern is 
from the top, the pattern is semicircular on the interval ![{[-2\sigma \sqrt{\alpha}, 2\sigma \sqrt{\alpha}]}](https://s0.wp.com/latex.php?latex=%7B%5B-2%5Csigma+%5Csqrt%7B%5Calpha%7D%2C+2%5Csigma+%5Csqrt%7B%5Calpha%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[-2\sigma \sqrt{\alpha}, 2\sigma \sqrt{\alpha}]}")
. The interlacing property of Gelfand-Tsetlin patterns translates to the claim that 
(resp. 
) is non-decreasing (resp. non-increasing) in 
, which corresponds to 
, where 
-dimensional subspace of 
. Here we have 
, there are simple poles at 
, and the associated measure is 
of the 
eigenvalues of the 
minor to be equal to 
, the poles disappear and one just has 
, one has an inverse semicircle distribution 
-transform and 
-transform from free probability. Formally, letlet 
be the inverse of 
, and then define the 
. 
is the reciprocal of a natural number 
, then 
is the free arithmetic mean of 
of 
. In terms of random matrices, this is asserting that the top 
minor of a random matrix 
of 
(or 
), the 
with 
, then the 
is defined by 
we obtain 
to be the 
; from the link between 
is the free product of 
and 
, where 
is asymptotically the free product of that of 
in which all three elements lie in a single set. This will be deduced from an easier variant of the theorem in which one locates (a family of) length three progressions in which just the first two elements 
of the progression lie in a good set (and some other properties of the family are also required). This is in turn derived from an even easier variant in which now just the first element of the progression is required to be in the good set.
be a natural number, and let 
. Then, whenever 
of 3-term arithmetic progressions with the following properties:
, 
and 
lie in 
lie in 
are in arithmetic progression.
are in blue and elements of 
are in grey:
of non-increasing non-negative integers 
. (Here our partitions are “weak” in the sense that we allow some parts of the partition to be zero. Henceforth we will omit the modifier “weak”, as we will not need to consider the more usual notion of “strong” partitions.) To each such partition 
row containing 
boxes. A 
of shape 
that is weakly increasing along rows (moving rightwards) and strictly increasing along columns (moving downwards). The collection of such tableaux will be denoted 
. The weight 
of a tableau 
, where 
is the number of occurrences of the integer 
in the tableau. For instance, if 
, an example of a Young tableau of shape 
.
on 
, which we will define as 
to the Schur polynomial 
. In the case of partitions of the form 
, the Schur polynomial 
is just the 
of degree 
, the general linear group 
, or the unitary group 
. For instance, one can view 
as the character of an irreducible polynomial representation of 
associated with the partition 
and 
of some shape 
by removing all the appearances of 
row). For instance, with 
in this case is 
. As Young tableaux are required to be strictly increasing down columns, we can see that the reduced partition 
; we denote this interspersion relation as 
(though we caution that this is not intended to be a partial ordering). In the converse direction, if 
, one can form a Young tableau 
, and the size 
of a partition 
.
, where 
denotes the 
if 
is negative. Thus for instance 
, where each 
is a partition of length 
that intersperses the next partition 
, with 
set equal to 
, whenever 
, thus the Schur polynomial 
with 
. (One could relabel the variables here to be something like 
instead, but this labeling seems slightly more natural, particularly in view of identities such as 
, and 
are partitions of lengths 
respectively. This gives another recursive way to understand Schur polynomials and skew Schur polynomials. For instance, one can use it to establish the generalised Jacobi-Trudi identity 
for 
are required to be integer-valued, and their definition similarly involves summation over a discrete set. It turns out that there are “continuous” (or “classical”) analogues of these functions, in which the parameters 
now take real values rather than integers, and are defined via integration rather than summation. One can view these continuous analogues as a “semiclassical limit” of their discrete counterparts, in a manner that can be made precise using the machinery of 
real partition 
precisely as before, by requiring that the inequalities 
for 
recursively by defining 
and the integral is with respect to 
as before. Thus for instance 
for 
, and 
recursively by defining 
. Thus for instance 
in the first integral real Gelfand-Tsetlin patterns based at 
![\displaystyle N^{-k(k-1)/2} s_{\lfloor N \lambda \rfloor}( \exp[ x/N ] ) \rightarrow S_\lambda(x) \ \ \ \ \ (11)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++N%5E%7B-k%28k-1%29%2F2%7D+s_%7B%5Clfloor+N+%5Clambda+%5Crfloor%7D%28+%5Cexp%5B+x%2FN+%5D+%29+%5Crightarrow+S_%5Clambda%28x%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0 "\displaystyle N^{-k(k-1)/2} s_{\lfloor N \lambda \rfloor}( \exp[ x/N ] ) \rightarrow S_\lambda(x) \ \ \ \ \ (11)")
![\displaystyle \exp[x/N] := (\exp(x_1/N), \dots, \exp(x_k/N)).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cexp%5Bx%2FN%5D+%3A%3D+%28%5Cexp%28x_1%2FN%29%2C+%5Cdots%2C+%5Cexp%28x_k%2FN%29%29.&bg=ffffff&fg=000000&s=0 "\displaystyle \exp[x/N] := (\exp(x_1/N), \dots, \exp(x_k/N)).")
![\displaystyle N^{j(j-1)/2-k(k-1)/2} s_{\lfloor N \lambda \rfloor / \lfloor N \mu \rfloor}( \exp[ x/N ] ) \rightarrow S_{\lambda/\mu}(x)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++N%5E%7Bj%28j-1%29%2F2-k%28k-1%29%2F2%7D+s_%7B%5Clfloor+N+%5Clambda+%5Crfloor+%2F+%5Clfloor+N+%5Cmu+%5Crfloor%7D%28+%5Cexp%5B+x%2FN+%5D+%29+%5Crightarrow+S_%7B%5Clambda%2F%5Cmu%7D%28x%29&bg=ffffff&fg=000000&s=0 "\displaystyle N^{j(j-1)/2-k(k-1)/2} s_{\lfloor N \lambda \rfloor / \lfloor N \mu \rfloor}( \exp[ x/N ] ) \rightarrow S_{\lambda/\mu}(x)")
, and any 
of an 
for 
. This multiplicativity can in turn be used to establish many further identities; in particular, as shown in 
matrix, and 
and 
denotes the 
column, and similarly 
denotes the 
matrix formed from 
rows and 
columns. Thus for instance when 
we obtain 
. (Charles Dodgson, better known by his pen name 
rows, and similarly for the columns, it is easy to see that the Dodgson condensation identity is equivalent to the variant 
are 
column vectors, 
are 
row vectors, and 
are scalars. If 
of the 
determinant). To remove the requirement that 
, 
denotes the matrix formed from 
. (The Dodgson condensation identity is basically the 
rows of 
of the matrix vanish. Similarly, using elementary column operations instead of row operations we may assume that 
vanish. All matrices involved are now block-diagonal and the identity follows from a routine computation.
, any 
column vectors 
, and any 
matrix 
determinants and we can reduce to the 
case (eliminating the role of 
and 
which forces it to be a scalar multiple of 
, at which point one can test the identity at a single point (e.g. 
and 
for the standard basis 
) to conclude the argument. (One can also derive this identity from the Sylvester determinant identity but I think the calculations are a little messier if one goes by that route. Conversely, one can recover the Dodgson condensation identity from Karlin’s identity by setting 
, 
(for instance) and then permuting some rows and columns.) 
in some space 
(in many cases 
and 
are some functionals of 
norm), and 
of symmetries 
acting on the underlying space. For instance, if 
for some shift 
), or perhaps dilations with some normalisation (e.g. 
for some dilation factor 
and some normalisation exponent 
, which can be thought of as the dimensionality of length one is assigning to 
and all 
, we say that 
in 
has a limit 
, in which case we obtain the amplification 
is itself 
has a limit 
as 
and 
exist, the limiting functionals 
, and then bounding 
of this inequality to conclude that
of the inequality to conclude that
existed) which would likely also be easier to prove (one can take whatever proofs one had in mind of the inequalities in 
. Both sides of this inequality are invariant with respect to interchanging 
with 
, 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
to conclude the proof. Here, the intermediate quantity 
is not invariant with respect to interchange of 
at most), and disappears completely in the most extremal case 
, 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
. One could try to leverage the triangle inequality 
for real numbers by using the crude estimate
, 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 
; this reduces the loss from 
to 
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 
, 
, and was tasked to conclude the corresponding inequality 
. The quickest way to do this is of course to exploit a bijection between the natural numbers 
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 
, 
by some shift 
, 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 
, whereas the inequality 
of 
for some 
(that is to say, 
). Then, if one wishes to prove an inequality 
of this inequality are 
. By restricting to those 
, for if they did not, then the arguments would work in the more general setting 
or 
. It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of 
being a nonnegative real; this is of course possible since any complex number 
. Once 
and 
are two 
Hermitian matrices that are positive semi-definite, then their 
and 
are rank one positive semi-definite matrices, since in this case 
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.
with non-negative coefficients 
is entrywise positivity preserving on the space 
of ![\displaystyle P[A] := (P(a_{ij}))_{1 \leq i,j \leq N}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++P%5BA%5D+%3A%3D+%28P%28a_%7Bij%7D%29%29_%7B1+%5Cleq+i%2Cj+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0 "\displaystyle P[A] := (P(a_{ij}))_{1 \leq i,j \leq N}")
![{P[A]}](https://s0.wp.com/latex.php?latex=%7BP%5BA%5D%7D&bg=ffffff&fg=000000&s=0 "{P[A]}")
is not the application 
of ![\displaystyle P[A] = c_0 A^{\circ 0} + c_1 A^{\circ 1} + \dots + c_d A^{\circ d},](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++P%5BA%5D+%3D+c_0+A%5E%7B%5Ccirc+0%7D+%2B+c_1+A%5E%7B%5Ccirc+1%7D+%2B+%5Cdots+%2B+c_d+A%5E%7B%5Ccirc+d%7D%2C&bg=ffffff&fg=000000&s=0 "\displaystyle P[A] = c_0 A^{\circ 0} + c_1 A^{\circ 1} + \dots + c_d A^{\circ d},")
is the Hadamard product of 
and 
of positive definite matrices with entries in ![{I = [0,\rho]}](https://s0.wp.com/latex.php?latex=%7BI+%3D+%5B0%2C%5Crho%5D%7D&bg=ffffff&fg=000000&s=0 "{I = [0,\rho]}")
, such functions are precisely the 
is a function that is entrywise positivity preserving on 
for all 
replaced by other domains, appear in the work of 
, a function would be entrywise positivity preserving on 
if and only if 
order must be non-negative in order for 
for some 
. In particular, if 
must be non-negative. In fact, a stronger assertion can be made, namely that the first 
, one can furthermore show that any negative term in 
be a sign. 
is preceded by at least 
with 
). Then, for any 
, with each 
having the sign of 
, which is entrywise positivity preserving on 
. 
with 
). Then there exists a convergent power series 
, with each 
.
, or the complex disk 
, 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.
![{\mathrm{det} P[ {\bf u} {\bf u}^* ]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdet%7D+P%5B+%7B%5Cbf+u%7D+%7B%5Cbf+u%7D%5E%2A+%5D%7D&bg=ffffff&fg=000000&s=0 "{\mathrm{det} P[ {\bf u} {\bf u}^* ]}")
of polynomials 
; the positivity of these determinants can be used (together with a continuity argument) to establish the positive definiteness of ![{P[uu^*]}](https://s0.wp.com/latex.php?latex=%7BP%5Buu%5E%2A%5D%7D&bg=ffffff&fg=000000&s=0 "{P[uu^*]}")
for various ranges of 
and the signs of the coefficients in the linear combination are determined by the signs of the coefficients of 
, where the weight 
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 
and another positive semi-definite matrix 
matrix). Using the fundamental theorem of calculus to continuously deform the rank one matrix ![{P[{\bf u} {\bf u}^*]}](https://s0.wp.com/latex.php?latex=%7BP%5B%7B%5Cbf+u%7D+%7B%5Cbf+u%7D%5E%2A%5D%7D&bg=ffffff&fg=000000&s=0 "{P[{\bf u} {\bf u}^*]}")
combined with an induction hypothesis on 
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