In logic, there is a subtle but important distinction between the concept of mutual knowledge – information that everyone (or almost everyone) knows – and common knowledge, which is not only knowledge that (almost) everyone knows, but something that (almost) everyone knows that everyone else knows (and that everyone knows that everyone else knows that everyone else knows, and so forth). A classic example arises from Hans Christian Andersens’ fable of the Emperor’s New Clothes: the fact that the emperor in fact has no clothes is mutual knowledge, but not common knowledge, because everyone (save, eventually, for a small child) is refusing to acknowledge the emperor’s nakedness, thus perpetuating the charade that the emperor is actually wearing some incredibly expensive and special clothing that is only visible to a select few. My own personal favourite example of the distinction comes from the blue-eyed islander puzzle, discussed previously here, here and here on the blog. (By the way, I would ask that any commentary about that puzzle be directed to those blog posts, rather than to the current one.)
I believe that there is now a real-life instance of this situation in the US presidential election, regarding the following
Proposition 1. The presumptive nominee of the Republican Party, Donald Trump, is not even remotely qualified to carry out the duties of the presidency of the United States of America.
Proposition 1 is a statement which I think is approaching the level of mutual knowledge amongst the US population (and probably a large proportion of people following US politics overseas): even many of Trump’s nominal supporters secretly suspect that this proposition is true, even if they are hesitant to say it out loud. And there have been many prominent people, from both major parties, that have made the case for Proposition 1: for instance Mitt Romney, the Republican presidential nominee in 2012, did so back in March, and just a few days ago Hillary Clinton, the likely Democratic presidential nominee this year, did so in this speech:
I highly recommend watching the entirety of the (35 mins or so) speech, followed by the entirety of Trump’s rebuttal.
However, even if Proposition 1 is approaching the status of “mutual knowledge”, it does not yet seem to be close to the status of “common knowledge”: one may secretly believe that Trump cannot be considered as a serious candidate for the US presidency, but must continue to entertain this possibility, because they feel that others around them, or in politics or the media, appear to be doing so. To reconcile these views can require taking on some implausible hypotheses that are not otherwise supported by any evidence, such as the hypothesis that Trump’s displays of policy ignorance, pettiness, and other clearly unpresidential behaviour are merely “for show”, and that behind this facade there is actually a competent and qualified presidential candidate; much like the emperor’s new clothes, this alleged competence is supposedly only visible to a select few. And so the charade continues.
I feel that it is time for the charade to end: Trump is unfit to be president, and everybody knows it. But more people need to say so, openly.
Important note: I anticipate there will be any number of “tu quoque” responses, asserting for instance that Hillary Clinton is also unfit to be the US president. I personally do not believe that to be the case (and certainly not to the extent that Trump exhibits), but in any event such an assertion has no logical bearing on the qualification of Trump for the presidency. As such, any comments that are purely of this “tu quoque” nature, and which do not directly address the validity or epistemological status of Proposition 1, will be deleted as off-topic. However, there is a legitimate case to be made that there is a fundamental weakness in the current mechanics of the US presidential election, particularly with the “first-past-the-post” voting system, in that (once the presidential primaries are concluded) a voter in the presidential election is effectively limited to choosing between just two viable choices, one from each of the two major parties, or else refusing to vote or making a largely symbolic protest vote. This weakness is particularly evident when at least one of these two major choices is demonstrably unfit for office, as per Proposition 1. I think there is a serious case for debating the possibility of major electoral reform in the US (I am particularly partial to the Instant Runoff Voting system, used for instance in my home country of Australia, which allows for meaningful votes to third parties), and I would consider such a debate to be on-topic for this post. But this is very much a longer term issue, as there is absolutely no chance that any such reform would be implemented by the time of the US elections in November (particularly given that any significant reform would almost certainly require, at minimum, a constitutional amendment).
true-false questions on an examination, with the answers randomised so that each question is equally likely to have “true” as the correct answer as “false”, with no correlation between different questions. Suppose that the students taking the examination must answer each question with exactly one of “true” or “false” (they are not allowed to skip any question). Then it is easy to see how to grade the exam: one can simply count how many questions each student answered correctly (i.e. each correct answer scores one point, and each incorrect answer scores zero points), and give that number 
as the final grade of the examination. More generally, one could assign some score of 
points to each correct answer and some score (possibly negative) of 
points to each incorrect answer, giving a total grade of 
points. As long as 
, this grade is simply an affine rescaling of the simple grading scheme 
and a given question 
, the student 
and false with probability 
, where 
is some quantity that can be viewed as a measure of confidence 
, and confident that the answer is false if 
); for simplicity let us assume that in 
and false if 
. (If 
, the student 
out of 
questions correctly, was it because he or she actually knew the right answer for seven of the questions, or was it because he or she was making educated guesses for the ten questions that turned out to be slightly better than random chance? There seems to be no way to discern this if the only input the student is allowed to provide for each question is the single binary choice of true/false.
confident that the answer is true” (and hence 
confidence the answer is false). Such answers would give more insight as to how well the student actually knew the material; in particular, we would theoretically be able to actually see the student’s subjective probabilities 
confidence is awarded one point. How many points should one award a correct answer given in ![{f: [0,1] \rightarrow {\bf R}}](https://s0.wp.com/latex.php?latex=%7Bf%3A+%5B0%2C1%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0 "{f: [0,1] \rightarrow {\bf R}}")
and then awarding a student 
points whenever they indicate the correct answer with a confidence of 
. For instance, if the student was 
points if the correct answer actually was “true”, and 
points if the correct answer actually was “false”. One can then ask the question of what functions 
would be “best” for this scheme?
– thus for instance a correct answer given with 
points. But is this the “best” option?
for a student 
), then student would expect a grade of
, thus we should have
. This suggests that the function 
should be constant. (Strictly speaking, it only gives the weaker constraint that 
; but if one generalised the problem to allow for multiple-choice questions with more than two possible answers, with a grading scheme that depended only on the confidence assigned to the correct answer, the same analysis would in fact force 
to be constant in 
for some 
; by monotonicity we expect 
(so that no points are awarded for a 
split in confidence between true and false) and 
, one arrives at the grading scheme
, he or she will be awarded 
points when the correct answer is “true”, and 
points if the correct answer is “false”. The following table gives some illustrative values for this scheme:
over the finite field 
of three elements is a subset 
, where 
and 
. A basic problem in additive combinatorics (discussed in one of the 
. Using Fourier methods (and the density increment argument of Roth), the bound of 
was obtained 
for some absolute constant 
, using a version of the polynomial method recently introduced 
.) Given the success of the polynomial method in superficially similar problems such as the finite field Kakeya problem (discussed in 
). The basic starting point is this: if 
, where 
is the Kronecker delta function, which we view as taking values in 
has solutions precisely when 
are either all equal, or form a line, and the latter is ruled out precisely when 
taking values in a field 
is of rank one if it is non-zero and of the form 
for some 
, and that the 
as a linear combination. More generally, if 
, we define the rank of a function 
to be the least number of “rank one” functions of the form
and some functions 
, 
, that are needed to generate 
, the rank one functions take the form 
, 
, 
, and linear combinations of 
such rank one functions will give a function of rank at most 
, let 
be a coefficient. Then the rank of the function 
.
follows from standard linear algebra, so suppose now that 
and the claim has already been proven for 
.
(which cannot increase the rank), we may assume without loss of generality that all the 
, then we obtain a representation
of cardinalities adding up to at most 
and 
.
that are orthogonal to all the 
, 
in the sense that
is at least 
. A basis of this space generates a 
coordinate matrix of full rank, which implies that there is at least one non-singular 
minor. This implies that there exists a function 
of 
.
and sum in 
, we conclude that
, since the summands are rank one functions. On the other hand, from induction hypothesis the left-hand side has rank at least 
, the rank of the function 
is at most 
, where
for 
, we have
in 
with
is at most 
.
. We can regroup this contribution as
ranges over those 
with 
is the monomial
is some explicitly computable function whose exact form will not be of relevance to our argument. The number of such 
and 
similarly have rank at most 
, the rank of 
, 
, 
for some 
summing to 
is the entropy function
is the maximum entropy 
subject to the constraints
, giving rise to the claimed bound of 
.
that has no non-trivial solutions to the equation 
, where 
are non-zero constants that sum to zero. Of course one replaces the function 
in this case.
by finding a more efficient way to decompose 
into rank one functions, however I was not able to do so (though such improvements are reminiscent of the 
progressions, in (say) 
. One can run the above arguments, replacing the function
where
and so this bound is in fact slightly worse than the trivial bound 
! However, there is a slim chance that there is a more efficient way to decompose 
into rank one functions that would give a non-trivial bound on 
as taking values in another field than 
. However, as soon as one works in a field of characteristic other than 
, one can adjoin a cube root 
of unity, and one now has the Fourier decomposition
, and so one cannot improve upon the trivial bound of 
was demonstrated, where 
denoted the Liouville function. The proof proceeded using a method I call the “entropy decrement argument”, which ultimately reduced matters to establishing a bound of the form
was a slowly growing function of 
. This was in turn established in a 
. As it turns out (and as is detailed in the current paper), the entropy decrement argument extends to this setting (after using some known facts about linear equations in primes), and allows one to reduce the above estimate to an estimate of the form
![\displaystyle \sum_{n \leq x} \frac{1}{n} \| \lambda \|_{U^d[n, n+H]} = o( \log x )](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cleq+x%7D+%5Cfrac%7B1%7D%7Bn%7D+%5C%7C+%5Clambda+%5C%7C_%7BU%5Ed%5Bn%2C+n%2BH%5D%7D+%3D+o%28+%5Clog+x+%29&bg=ffffff&fg=000000&s=0 "\displaystyle \sum_{n \leq x} \frac{1}{n} \| \lambda \|_{U^d[n, n+H]} = o( \log x )")
for 
), where 
is the (normalised) local Gowers uniformity norm. (In the case 
, this becomes the Fourier-uniformity conjecture discussed in 
of controlled step and complexity (see the paper for definitions of these terms). 
is a zero entropy (i.e. deterministic) sequence. Morally speaking, this follows from the well-known fact that nilsequences have zero entropy, but the presence of the supremum in 
can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space 
. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an 
from 
which embeds into 
but not 
, or the Klein bottle that embeds into 
but not 
). One can also relax the compactness hypothesis on 
.
be a compact Riemannian manifold. Then there exists a 
, basically because this is the number of degrees of freedom in the Riemannian metric 
.) Nash’s original proof of theorem used what is now known as 
, 
to form a combined map 
that can be closer to an embedding or an isometric embedding than the original maps 
. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of 
and the 
, but there are countless other examples of both. The 
of continuous functions on 
that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm 
), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. 
convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.
be a Hausdorff topological vector space. Then, up to equivalence of norms, there is at most one norm 
one can place on 
so that 
is a Banach space whose topology is at least as strong as 
. In particular, there is at most one topology stronger than 
on 
and 
were both Banach spaces with topologies stronger than 
from the Banach space 
is a sequence in this graph that converged in the product topology to 
, then 
converges to 
norm and hence in 
in 
norm and hence in 
. Applying the closed graph theorem (see also 
is the space of 
) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on 
of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm 
be the standard basis of 
would have to converge in 
is constant in the first variable (thus 
is constant for each 
is constant for each 
such that 
for all 
such that 
for all 
; in fact it must take the form 
. (This can be seen for instance by using the affine linearity in 
from one additive group 
to another, we say that 
and 
, where 
is the difference operator
along one subgroup 
of 
along another subgroup 
of 
. Then 
along the subgroup 
of 
, 
, 
, 
, and 
.
or 
(in which 
and the claim has already been proven for smaller values of 
.
along 
to obtain
is the shift of 
. Then we take a further shift by a direction 
to obtain
has degree less than 
along 
is also of degree less than 
is of degree less than 
, and hence 
involved, and take advantage of the recursive nature of degree (in this case, the fact that a function is of less than degree 
).
of 
on some probability space 
; we call the pair 
(or more precisely 
) a 
is a generalised eigenfunction of degree less than 
if one has
, and some functions 
of degree less than 
along 
on the skew shift 
with 
action given by the generator 
for some irrational 
for every integer 
is a generalised eigenfunction of degree less than 
even along directions 
by 
.)
-system given by the 
with generating shifts
can then be checked to be a generalised eigenfunction of degree less than 
, and also less than 
, and less than 
of a 
. Namely, 
of 
and
denotes the ergodic average along a Følner sequence in 
and 
and with respect to the factor 
for some 
of 
.
(along a subgroup 
, and a combinatorial proof based on a combinatorial analogue of this proposition which is harder to state (but which roughly speaking asserts that a function which is nearly orthogonal to all bounded functions of small 
norm, and also to all bounded functions of small 
norm, is also nearly orthogonal to alll bounded functions of small 
norm). The combinatorial proof parallels the proof of Proposition 
obey a property analogous to being a generalised eigenfunction, namely that
and 
is a “structured function of order 
and 
. This sort of inheritance is quite easy to accomplish in the ergodic setting, as there is a ready-made language of factors to encapsulate the concept of structure, and the shift-invariance and 
-algebra properties of factors make it easy to show that just about any “natural” operation one performs on a function measurable with respect to a given factor, returns a function that is still measurable in that factor. In the finitary combinatorial setting, though, encoding the fact (ii) becomes a remarkably complicated notational nightmare, requiring a huge amount of “epsilon management” and “second-order epsilon management” (in which one manages not only scalar epsilons, but also function-valued epsilons that depend on other parameters). In order to avoid all this we were forced to utilise a nonstandard analysis framework for the combinatorial theorems, which made the arguments greatly resemble the ergodic arguments in many respects (though the two settings are still not equivalent, see 
and a bounded function is small in 
norm for most 
, then it is also small in 
norm for most 
. (Here is a baby version one may wish to warm up on: if a function 
for some large prime 
are replaced by more general coset progressions 
(of bounded rank), so that one has a Bessel inequailty controlling “local” Gowers uniformity norms such as 
by “global” Gowers uniformity norms such as 
. This turns out to be particularly useful when attempting to compute polynomial averages such as 
. After repeated use of the van der Corput lemma, one can control such averages by expressions such as
Gowers uniformity norms of 
for various 
. Using the above Bessel inequality, this can be controlled in turn by an average of various 
Gowers uniformity norms along rank two generalised arithmetic progressions of the form 
for various 
. But for generic 
, this rank two progression is close in a certain technical sense to the “global” interval 
(this is ultimately due to the basic fact that two randomly chosen large integers are likely to be coprime, or at least have a small gcd). As a consequence, one can use the concatenation theorems from our first paper to control expressions such as 
and 
are the Möbius and von Mangoldt functions respectively. This is because we are able to control global Gowers uniformity norms of such functions (thanks to results such as 
be polynomials of degree at most 
is admissible in the sense that for every prime 
such that 
are all coprime to 
, 
(actually for minor technical reasons we reduce the range of 
). In fact one could in principle obtain asymptotics for smaller values of 
differ by a constant, provided one had good enough local uniformity results for the Möbius or von Mangoldt functions. For instance, we can obtain an asymptotic for triplets of the form 
unconditionally for 
, and conditionally on GRH for all 
case of this theorem was obtained in 
(though our results there only give lower bounds on the number of pairs 
, and no asymptotics). Both of these results generalise my 
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