Consider a disk 
Theorem 1 (Holomorphic images of disks) Letbe a disk in the complex plane, and
be a holomorphic function with
.
- (i) (Open mapping theorem)
contains a disk
for some
.
- (ii) (Bloch theorem)
for some absolute constant
and some
. (In fact there is even a holomorphic right inverse of
from
- (iii) (Koebe quarter theorem) If
.
- (iv) If
, then
.
- (v) If one has a bound of the form
for all
and some
, then
for some absolute constant
Parts (i), (ii), (iii) of this theorem are standard, as indicated by the given links. I found part (iv) as (a consequence of) Theorem 2 of this paper of Degot, who remarks that it “seems not already known in spite of its simplicity”. The proof is simple:
Proof: (Proof of (iv)) Let 
). But by the fundamental theorem of algebra we have 
are the roots of the polynomial (counting multiplicity). By the pigeonhole principle, there must therefore exist a root 
of 
such that 
. Thus contains 
, and the claim follows. 
The constant 
. This example also shows that despite parts (ii), (iii) of the theorem, one cannot hope for a general inclusion of the form 
.
Part (v) is implicit in the standard proof of Bloch’s theorem (part (ii)), and is easy to establish:
Proof: (Proof of (v)) From the Cauchy inequalities one has 
Note that part (v) implies part (i). A standard point picking argument also lets one deduce part (ii) from part (v):
Proof: (Proof of (ii)) By shrinking 
such that 
. Iterating this procedure in the obvious fashion we either are done, or obtain a Cauchy sequence 
in such that 
goes to infinity as 
, which contradicts the analytic nature of (and hence continuous nature of 
) on the closure of . This gives the claim.
on an interval 
as an “irrational nilsequence” 
of controlled complexity, plus some “negligible” errors (where one uses the Gowers uniformity norm as the main norm to control the neglibility of the error); and 
for various affine-linear forms 
when the functions ![{\Psi^{[i]}}](https://s0.wp.com/latex.php?latex=%7B%5CPsi%5E%7B%5Bi%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\Psi^{[i]}}")
for various natural numbers 
, defined as the linear span of the vectors 
as 
ranges over the parameter space 
. Roughly speaking, these spaces encode some constraints one would expect to see amongst the forms 
. For instance, in the case of length four arithmetic progressions when 
, 
, and 
, then ![{\Psi^{[1]}}](https://s0.wp.com/latex.php?latex=%7B%5CPsi%5E%7B%5B1%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\Psi^{[1]}}")
is spanned by the vectors 
and 
and can thus be described as the two-dimensional linear space ![\displaystyle \Psi^{[1]} = \{ (a,b,c,d): a-2b+c = b-2c+d = 0\} \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CPsi%5E%7B%5B1%5D%7D+%3D+%5C%7B+%28a%2Cb%2Cc%2Cd%29%3A+a-2b%2Bc+%3D+b-2c%2Bd+%3D+0%5C%7D+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \Psi^{[1]} = \{ (a,b,c,d): a-2b+c = b-2c+d = 0\} \ \ \ \ \ (1)")
![{\Psi^{[2]}}](https://s0.wp.com/latex.php?latex=%7B%5CPsi%5E%7B%5B2%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\Psi^{[2]}}")
is spanned by the vectors 
and can be described as the hyperplane ![\displaystyle \Psi^{[2]} = \{ (a,b,c,d): a-3b+3c-d = 0 \}. \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CPsi%5E%7B%5B2%5D%7D+%3D+%5C%7B+%28a%2Cb%2Cc%2Cd%29%3A+a-3b%2B3c-d+%3D+0+%5C%7D.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \Psi^{[2]} = \{ (a,b,c,d): a-3b+3c-d = 0 \}. \ \ \ \ \ (2)")
for some irrational 
, some arbitrary 
, and some smooth 
, then the limiting value of the average ![\displaystyle {\bf E}_{n, r \in [N]} f(n) f(n+r) f(n+2r) f(n+3r)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+E%7D_%7Bn%2C+r+%5Cin+%5BN%5D%7D+f%28n%29+f%28n%2Br%29+f%28n%2B2r%29+f%28n%2B3r%29&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle {\bf E}_{n, r \in [N]} f(n) f(n+r) f(n+2r) f(n+3r)")
is equal to 
![{\Psi^{[1]} \subset \Psi^{[2]}}](https://s0.wp.com/latex.php?latex=%7B%5CPsi%5E%7B%5B1%5D%7D+%5Csubset+%5CPsi%5E%7B%5B2%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\Psi^{[1]} \subset \Psi^{[2]}}")
to dispense with the lower order term 
(which then plays no further role in the analysis).
![{\Psi^{[i]} \subset \Psi^{[i+1]}}](https://s0.wp.com/latex.php?latex=%7B%5CPsi%5E%7B%5Bi%5D%7D+%5Csubset+%5CPsi%5E%7B%5Bi%2B1%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{\Psi^{[i]} \subset \Psi^{[i+1]}}")
for all 
![\displaystyle \Psi^{[1]} = \{ (a,b,c,d): d-c=3a; b-2a=2d\}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CPsi%5E%7B%5B1%5D%7D+%3D+%5C%7B+%28a%2Cb%2Cc%2Cd%29%3A+d-c%3D3a%3B+b-2a%3D2d%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \Psi^{[1]} = \{ (a,b,c,d): d-c=3a; b-2a=2d\}")
![\displaystyle \Psi^{[2]} = \{ (a,b,c,d): 24a+3b-4c-8d=0 \}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CPsi%5E%7B%5B2%5D%7D+%3D+%5C%7B+%28a%2Cb%2Cc%2Cd%29%3A+24a%2B3b-4c-8d%3D0+%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \Psi^{[2]} = \{ (a,b,c,d): 24a+3b-4c-8d=0 \}")
vanishes, but not when 
, as well as “complexity one” cases. Nevertheless non-flag property systems of affine forms do exist, thus limiting the range of applicability of the techniques in this paper. In particular, the conjecture of Gowers and Wolf (Theorem 1.13 in the paper) is now open again in the non-flag property case.
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 
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 
of all 
. 
, due 
tiling 
such that 
.
for any 
. 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 
, 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.
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