Several years ago, I developed a public lecture on the cosmic distance ladder in astronomy from a historical perspective (and emphasising the role of mathematics in building the ladder). I previously blogged about the lecture here; the most recent version of the slides can be found here. Recently, I have begun working with Tanya Klowden (a long time friend with a background in popular writing on a variety of topics, including astronomy) to expand the lecture into a popular science book, with the tentative format being non-technical chapters interspersed with some more mathematical sections to give some technical details. We are still in the middle of the writing process, but we have produced a sample chapter (which deals with what we call the “fourth rung” of the distance ladder – the distances and orbits of the planets – and how the work of Copernicus, Brahe, Kepler and others led to accurate measurements of these orbits, as well as Kepler’s famous laws of planetary motion). As always, any feedback on the chapter is welcome. (Due to various pandemic-related uncertainties, we do not have a definite target deadline for when the book will be completed, but presumably this will occur sometime in the next year.)
The book is currently under contract with Yale University Press. My coauthor Tanya Klowden can be reached at tklowden@gmail.com.
of a finite tile 
in a standard lattice 
(which we call tiling sets) such that every element of 
of 
for a natural number 
is periodic but allowed to be non-constant).
will be periodic (by which we mean translation invariant with respect to some lattice (a finite index subgroup) of 
is the unit square 
and 
. However one can modify some tilings to make them less periodic. For instance, keeping 
is an arbitrary function. This tiling set is periodic in a single direction 
, but is not doubly periodic. For the slightly modified tile 
, the set 
can be verified to be a tiling set, which in general will not exhibit any periodicity whatsoever; however, it is weakly periodic in the sense that it is the disjoint union of finitely many sets, each of which is periodic in one direction.
has at least one tiling set, then it has a tiling set which is periodic. 
there is a simple pigeonhole principle argument 
case was settled more recently by 
remain open in general.
of the tile is bounded:
) 
-periodic for some 
. 
. (Even the decidability of this problem was not known until the result of Bhattacharya.)
disjoint sets, each of which is periodic in a direction of magnitude 
. 
:
be finite, and normalized so that 
. Then every tiling set of 
-periodic, where 
is the least common multiple of all primes up to 
, and 
is the least common multiple of the magnitudes 
of all 
. 
, due 
tiling 
such that 
.
for any 
coprime to 
. We need to show that 
. It suffices to show the claim 
, since both sides take values in 
. The convolution algebra ![{{\bf F}_p[{\bf Z}^d]}](https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5B%7B%5Cbf+Z%7D%5Ed%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{{\bf F}_p[{\bf Z}^d]}")
(or group algebra) of finitely supported functions from 
is a commutative algebra of characteristic 
, so we have the Frobenius identity 
for any 
. As a consequence we see that 
. The claim now follows by convolving the identity 
by 
further copies of 
.
![{\varphi_f: {\bf Z}^d \rightarrow [0,1]}](https://s0.wp.com/latex.php?latex=%7B%5Cvarphi_f%3A+%7B%5Cbf+Z%7D%5Ed+%5Crightarrow+%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\varphi_f: {\bf Z}^d \rightarrow [0,1]}")
is a function that is periodic in the direction 
, where 
for any 
, where 
is the Kronecker delta at 
. Now average over 
, particularly in one and two dimensions. On one hand, one can work modulo 
to eliminate the 
and 
behave in a polynomial fashion, after collecting commensurable terms). On the other hand we have the inequality 
-algebra
be a measure-preserving system for some group 
, thus 
is a (concrete) probability space and 
is a group homomorphism from 
of the probability space. (Here we are abusing notation by using 
to refer both to the measure-preserving system and to the underlying set. In the notation of the paper we would instead distinguish these two objects as 
and 
respectively, reflecting two of the (many) categories one might wish to view 
is a compact group, we define a (concrete) cocycle to be a collection of measurable functions 
for 
that obey the cocycle equation 
and all 
. (One could weaken this requirement by only demanding the cocycle equation to hold for almost all 
, rather than all 
of 
where
is the Cartesian product of 
is the product measure of 
and Haar probability measure on 
is given by the formula 
is a homomorphism, and the (left) invariance of Haar measure and Fubini’s theorem guarantees that the 
remain measure preserving. There is also the more general notion of a homogeneous skew-product 
in which the group 
for some closed subgroup of 
, noting that 
which is a relatively tractable object to manipulate. (This is not to say that the cohomology of measure-preserving systems is trivial, but at least there are many tools one can use to study them, such as the 
comes with a factor map 
and a coordinate map 
, which by 
is measure-preserving (using the product measure on 
), indeed the way we have constructed things this map is just the identity map.
, and weakening the requirement that 
be measure-preserving. Namely, define a group extension of 
equipped with a measure-preserving map 
. There is also a more general notion of a homogeneous extension in which 
takes values in 
is a group extension of 
is a closed subgroup of 
can be viewed as a group extension of 
to be the vertical coordinate 
(viewing 
now as an element of 
pushes forward to the wrong measure on 
rather than 
. 
for some measurable “gauge function” 
, then 
. 
for some 
outside of 
but now use the measure 
.
is 
of 
is equal modulo null sets to 
(or 
) is invariant but has measure 
. We then have the following fundamental result of Mackey and Zimmer:
given by the formula 
-systems, where the 
to 
are closely tied the group 
for some measurable 
, while in example (ii), the invariant functions on 
take the form 
. In either case, the invariant factor is isomorphic to 
, and can be viewed as a factor of the invariant factor of 
, which is isomorphic to 
of 
in this process. The main remaining task is to lift the quotient 
back up to a map 
, formed by pushing forward 
. This measure in turn has several lifts up to a probability measure on 
via the Riesz representation theorem by demanding 
. This measure does not come from a graph of any single lift 
, and this can be used as a substitute for a measurable selection theorem. A variant of this Krein-Milman argument can also be used to express any homogeneous extension as a quotient of a group extension, giving the second part of the Mackey-Zimmer theorem.
of three infinite sets of reals has the property that there exists a positive measure set 
of 
) is basically proven by a sequential application of these tools (except for dyadic pigeonholing, which turns out not to be needed here).
, for instance, might appear in the middle of such a string, taking in an input 
on the right and returning an output 
on its left that might then be fed into some other operation. There are a few mathematical notations which are two-dimensional, such as the commutative diagrams in homological algebra, the tree expansions of solutions to nonlinear PDE (particularly stochastic nonlinear PDE), or the Feynman diagrams and 
, but by using the basic fact that two points determine a line (or two lines intersect in at most one point), a simple application of Cauchy-Schwarz improves this bound to 
. In graph theoretic terms, the point is that the bipartite incidence graph between points and lines does not contain a copy of 
(there does not exist two points and two lines that are all incident to each other). Without any other further hypotheses, this bound is basically sharp: consider for instance the collection of 
points and 
lines in a finite plane 
, that has 
incidences (one can make the situation more symmetric by working with a projective plane rather than an affine plane). If however one considers lines in the real plane 
, the famous 
. Thus the incidence graph between real points and lines contains more structure than merely the absence of 
(or 
in the hypergraph case) is known as Zarankiewicz’s problem. We have results for all 
case.
. 
. 
to implement a “divide and conquer” strategy in which one can efficiently control incidences between 
points and boxes. For (ii) there is additional order-theoretic structure one can work with: first there is an easy pruning device to reduce to the case when no rectangle is completely contained inside another, and then one can impose the “tile partial order” in which one dyadic rectangle 
is less than another 
if 
and 
. The point is that this order is “locally linear” in the sense that for any two dyadic rectangles 
, the set ![{[R_-,R_+] := \{ R: R_- \leq R \leq R_+\}}](https://s0.wp.com/latex.php?latex=%7B%5BR_-%2CR_%2B%5D+%3A%3D+%5C%7B+R%3A+R_-+%5Cleq+R+%5Cleq+R_%2B%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[R_-,R_+] := \{ R: R_- \leq R \leq R_+\}}")
is linearly ordered, and this can be exploited by elementary double counting arguments to obtain a bound which eventually becomes 
after optimising certain parameters in the argument. The proof also suggests how to construct the counterexample in (iii), which is achieved by an elementary iterative construction.
matrix 
, in which one takes the successive upper left 
minors 
of 
in non-decreasing order. These eigenvalues are related to each other by the 
, and are often arranged in a triangular array known as a Gelfand-Tsetlin pattern, as discussed in 
is large and the matrix 
on the real line, then under suitable hypotheses (e.g., unitary conjugation invariance of the random matrix ensemble 
for any fixed 
converging to a specific measure 
that depends only on 
of 
. For instance, if 
, then 
. At the random matrix level, this reflects the fact that the minor of a GUE matrix is again a GUE matrix (up to a renormalizing constant).
(see 
-transform 
of 
, in which case one can show that 
, the 
.)
with distribution 
, then the “minor” ![{[pAp]}](https://s0.wp.com/latex.php?latex=%7B%5BpAp%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[pAp]}")
of 
whose elements are minors ![{[pXp]}](https://s0.wp.com/latex.php?latex=%7B%5BpXp%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[pXp]}")
, 
with trace ![{\tau_p([pXp]) := k \tau(pXp)}](https://s0.wp.com/latex.php?latex=%7B%5Ctau_p%28%5BpXp%5D%29+%3A%3D+k+%5Ctau%28pXp%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\tau_p([pXp]) := k \tau(pXp)}")
) has the law of 
is monotone non-decreasing for integer 
is monotone non-increasing for integer 
of a noncommutative random variable, which is the analogue of the classical 
is a polynomial and 
is a semicircular element free of 
for a sufficiently regular 
, where 
is the 
relative to some subalgebra 
of noncommutative random variables.
![{X \mapsto [pXp]}](https://s0.wp.com/latex.php?latex=%7BX+%5Cmapsto+%5BpXp%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{X \mapsto [pXp]}")
, in particular by standard calculations one can establish the identity ![\displaystyle J( [pXp] : [pBp] ) = k {\bf E}( [p J(X:B) p] | [pXp], [pBp] )](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++J%28+%5BpXp%5D+%3A+%5BpBp%5D+%29+%3D+k+%7B%5Cbf+E%7D%28+%5Bp+J%28X%3AB%29+p%5D+%7C+%5BpXp%5D%2C+%5BpBp%5D+%29&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle J( [pXp] : [pBp] ) = k {\bf E}( [p J(X:B) p] | [pXp], [pBp] )")
). The monotonicity of free entropy then follows from an integral representation of free entropy as an integral of free Fisher information along the free Ornstein-Uhlenbeck process (or equivalently, free Fisher information is essentially the rate of change of free entropy with respect to perturbation by semicircular noise). The argument also shows when equality holds in the monotonicity inequalities; this occurs precisely when 
for sufficiently regular 
. It is not difficult to show that 
is a positive semi-definite kernel, which gives the required monotonicity. It would be interesting to obtain some more insightful interpretation of the kernel ![{A \mapsto [pAp]}](https://s0.wp.com/latex.php?latex=%7BA+%5Cmapsto+%5BpAp%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{A \mapsto [pAp]}")
being associated to some sort of “gradient flow” in the 
on the “Gelfand-Tsetlin pyramid” 
![\displaystyle \mu^{\boxplus 1/s}((-\infty,\lambda(s,y)/s])=y/s,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmu%5E%7B%5Cboxplus+1%2Fs%7D%28%28-%5Cinfty%2C%5Clambda%28s%2Cy%29%2Fs%5D%29%3Dy%2Fs%2C&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \mu^{\boxplus 1/s}((-\infty,\lambda(s,y)/s])=y/s,")
eigenvalue of the 
upper left minor of a random 
is in some sense a continuum limit of Gelfand-Tsetlin patterns. Thus for instance the Cauchy interlacing laws in this asymptotic limit regime become 
. The quantity 
measures the mean eigenvalue spacing at some location of the Gelfand-Tsetlin pyramid, and the ratio 
measures mean eigenvalue drift in the minor process. This suggests that this Lagrangian density is some sort of measure of entropy of the asymptotic microscale point process emerging from the minor process at this spacing and drift. There is 
(which were also used in 
then defines a discrete Fourier multiplier operator 
for any 
by the formula 
is the Fourier transform on 
defines a continuous Fourier multiplier operator 
by the formula 
. In both cases we refer to 
as the symbol of the multiplier operator 
.
, we define the discrete Fourier multiplier operator 
for any frequency shift 
, we can form a multifrequency projection operator 
on 
by the formula 
![{[-\varepsilon,\varepsilon]}](https://s0.wp.com/latex.php?latex=%7B%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[-\varepsilon,\varepsilon]}")
, then ![{\bigcup_{\alpha \in \Sigma} \alpha + [-\varepsilon,\varepsilon]}](https://s0.wp.com/latex.php?latex=%7B%5Cbigcup_%7B%5Calpha+%5Cin+%5CSigma%7D+%5Calpha+%2B+%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\bigcup_{\alpha \in \Sigma} \alpha + [-\varepsilon,\varepsilon]}")
.
or 
theory of their continuous counterparts. For instance we have the basic result of ![{[-1/2,1/2]}](https://s0.wp.com/latex.php?latex=%7B%5B-1%2F2%2C1%2F2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[-1/2,1/2]}")
, then 
, where 
denotes the operator norm of an operator 
. ![{[-1/2Q,1/2Q]}](https://s0.wp.com/latex.php?latex=%7B%5B-1%2F2Q%2C1%2F2Q%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[-1/2Q,1/2Q]}")
for some natural number 
, then 
.
the implied constant in these bounds can be set to equal 
, but I do not know the full answer to this question.
; for instance it shows that a discrete Fourier multiplier with symbol 
for a fixed test function 
is bounded on ![\displaystyle \bigcup_{q=1}^N \bigcup_{\alpha \in \frac{1}{q}{\bf Z}/{\bf Z}} \alpha + [-\varepsilon,\varepsilon] \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbigcup_%7Bq%3D1%7D%5EN+%5Cbigcup_%7B%5Calpha+%5Cin+%5Cfrac%7B1%7D%7Bq%7D%7B%5Cbf+Z%7D%2F%7B%5Cbf+Z%7D%7D+%5Calpha+%2B+%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \bigcup_{q=1}^N \bigcup_{\alpha \in \frac{1}{q}{\bf Z}/{\bf Z}} \alpha + [-\varepsilon,\varepsilon] \ \ \ \ \ (1)")
![\displaystyle \bigcup_{\alpha \in \Sigma_{\leq k}} \alpha + [-\varepsilon,\varepsilon] \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbigcup_%7B%5Calpha+%5Cin+%5CSigma_%7B%5Cleq+k%7D%7D+%5Calpha+%2B+%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \bigcup_{\alpha \in \Sigma_{\leq k}} \alpha + [-\varepsilon,\varepsilon] \ \ \ \ \ (2)")
consists of all rational points in the unit circle of the form 
where 
is an integer. For suitable choices of 
and 
, together with something like the product of all the primes up to 
is small, there is a good theory:
, then 
![{\alpha + [-\varepsilon,\varepsilon]}](https://s0.wp.com/latex.php?latex=%7B%5Calpha+%2B+%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\alpha + [-\varepsilon,\varepsilon]}")
in 
was reduced to 
or 
for some integer 
. We use the following standard sequence of steps:
into various pieces depending on the denominator 
, and exploits “
norm by the 
power of the square function and estimates it by a “nonconcentrated” version in which various factors that arise in the expansion are “disjoint”. 
. 
is the set of 
are various complex numbers, as an average 
is a random partition of 
of being completely shattered by such a random partition. This “decouples” the index set 
which is more convenient for application of the superorthogonality theory. For (ii), the point is to efficiently obtain estimates of the form 
are various non-negative quantities, and a 
that consist of a common “core” 
and disjoint “petals” 
. The other parts of the argument are relatively routine; see for instance 
.
has its Fourier transform supported on 
. In fact, as can be shown from a little bit of routine Fourier analysis, if we narrow the support of the Fourier transform slightly to ![{[-c,c]}](https://s0.wp.com/latex.php?latex=%7B%5B-c%2Cc%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[-c,c]}")
for some 
, then the restriction 
has the same 
behaviour as the original function, in the sense that 
; see Theorem 4.18 of 
.
, and the standard embedding of 
of 
is Schwartz and has Fourier transform 
supported in ![{[-c/Q,c/Q] \times \frac{1}{Q}{\bf Z}/{\bf Z}}](https://s0.wp.com/latex.php?latex=%7B%5B-c%2FQ%2Cc%2FQ%5D+%5Ctimes+%5Cfrac%7B1%7D%7BQ%7D%7B%5Cbf+Z%7D%2F%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[-c/Q,c/Q] \times \frac{1}{Q}{\bf Z}/{\bf Z}}")
, and 
is endowed with probability Haar measure.
(the product of the reals and the 
). By using the Ionescu-Wainger multiplier theorem, we are able to obtain an adelic version of the quantitative sampling estimate 
, 
is 
supported on ![{[-\varepsilon,\varepsilon] \times \Sigma_{\leq k}}](https://s0.wp.com/latex.php?latex=%7B%5B-%5Cvarepsilon%2C%5Cvarepsilon%5D+%5Ctimes+%5CSigma_%7B%5Cleq+k%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{[-\varepsilon,\varepsilon] \times \Sigma_{\leq k}}")
for some sufficiently small 
in a fashion not detailed here). This allows one obtain an “adelic” extension of the Ionescu-Wainger multiplier theorem, in which the 
operator norm of an adelic counterpart to that multiplier operator; in principle this reduces “major arc” harmonic analysis on the integers 
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