
Mathematicians Catch a Pattern by Figuring Out How to Avoid It - otoburb
https://www.quantamagazine.org/mathematicians-catch-a-pattern-by-figuring-out-how-to-avoid-it-20191125/
======
DavidSJ
I was confused by this article. It says that for every k, then all
sufficiently large sets have an arithmetic progression of length k.

But suppose we set k = 3, and take the set {1, 2, 4, 8, ..., 2^n}. Then no
matter how big n is, we have no arithmetic progression of length 3.

Looking into this, it appears that the article misstated the result, called
Szemerédi's theorem: “if the positive integers are partitioned into finitely
many classes, then at least one of these classes contains arbitrarily long
arithmetic progressions”. [https://www.semanticscholar.org/paper/A-new-proof-
of-Szemeré...](https://www.semanticscholar.org/paper/A-new-proof-of-
Szemerédi's-theorem-Gowers/3e47e427a64759a41e54dd60c27e5663068886ca)

~~~
mauricioc
The theorem you quoted is Van der Waerden's theorem [0], not Szemerédi's. It
is significantly easier to prove than Szemerédi's result.

The article correctly states Szemerédi's theorem, if you interpret
"sufficiently large" to mean "positive density": For every density delta > 0
and length k, there exists an n0 such that every subset of {1, ..., n} (n >
n0) of density at least delta (that is, at least delta * n elements) contains
an arithmetic progression of length k.

For n large, your set has density roughly (log N)/N (where N = 2^n), which is
not bounded from below by any constant delta > 0; this is why Szemerédi's
theorem does not apply. Speaking of your example, you might be interested in
Behrend's construction [1], a way of constructing "large" sets (for another
definition of large) with no 3-term arithmetic progression.

Szemerédi's theorem easily implies Van der Waerden's theorem because if you
colour the integers with finitely many colours, you can apply Szemerédi's
theorem to the largest colour class.

[0]
[https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem](https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem)
[1]
[https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078964/pdf/pna...](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078964/pdf/pnas01693-0039.pdf)

------
mikorym
> catch a pattern by figuring how to avoid it

In my own research, the main focus is to take a (usually known) property and
to rephrase it so that the new statement is self-dual. The analogy would be
that you "understand a pattern only if you know how to phrase it in a self-
dual context".

None of these different approaches necessarily try to presuppose or subsume
each other, but they hint towards your outlook and interests. That is why
independent discoveries are usually "clearly" independent, in the sense that
the whole framework is usually different.

This happened for example with Lawvere and the eventual standard definition of
a topos; so too with Grothendieck toposes and the eventual standard definition
of a topos. Newton and Leibniz would be the more accessible example.

------
notelonmusk
Bounds for sets with no polynomial progressions

Abstract:

Let P_1,..., P_m ∈ Z[y] be polynomials with distinct degrees, each having zero
constant term. We show that any subset A of {1,..., N} with no nontrivial
progressions of the form x, x + P_1(y),..., x + P_m(y) has size |A| ≪ N/(log
log N)^(cP1,...,Pm) . Along the way, we prove a general result controlling
weighted counts of polynomial progressions by Gowers norms.

[-1]
[https://arxiv.org/pdf/1909.00309.pdf](https://arxiv.org/pdf/1909.00309.pdf)

------
kazinator
These observations have a bit of the flavor of the birthday paradox and
pigeonhole principle: if the size of a certain object is at least so and so,
then a certain property is guaranteed. If we have more than 365 people, then a
birthday has to repeat.

Also, the Mean Value Theorem comes to mind for some reason.

~~~
mikorym
The mean value theorem also guarantees existence (of a parallel line tangent).

------
rini17
Did not read the paper but it seems to me bit weird definition: that adding a
constant to geometric progression makes it polynomial progression.

~~~
pure-awesome
When you say "weird definition", do you mean it's a strange concept, or a
strange name for that concept?

Edit: Or do you mean that "polynomial progression" has an existing definition,
and this definition is equivalent, but is a strange way of stating it? Or do
you mean the definition as stated in the article is wrong / differs from the
standard definition?

~~~
Chris2048
Possibly that it doesn't seem to warrant an entirely new name? that "shifted
geometric progression" or something might as well do.

~~~
pure-awesome
See my other response to rini17 for more info.

But yeah, seems to me these "shifted geometric progressions" are actually just
a special case of a more general concept of a "polynomial progression".

------
ksd482
I am a Math student and I am curious about the topics in this article.

I am curious as to what all branches of Mathematics is the paper related to?
It's a result about polynomials, so Algebra and Analysis?

~~~
impendia
This area is usually known as "additive combinatorics".

[https://en.wikipedia.org/wiki/Arithmetic_combinatorics](https://en.wikipedia.org/wiki/Arithmetic_combinatorics)

~~~
ksd482
Thanks! The Math in that paper is mind boggling! I hope to get to a point to
understand it someday.

Very naive question: the paper uses integrals (perhaps Lebesgue --- can you
tell how naive I am, yet?) for proving some lemmas etc. As a beginner it's
hard for me to understand how integrals come into picture in number theory.

Not that it is my plan to do so, can you tell me what all fields of Math
(specifically textbooks) and techniques I need to be familiar with to
understand the paper?

For e.g., Abstract Algebra (D&F), Functional Analysis (graduate),
Combinatorics etc.?

This will give me a much clearer picture on how various fields of Mathematics
come into play.

Thanks again!

~~~
impendia
If you want to understand Peluse's paper, I think the most directly relevant
background reading would be Tao and Vu's _Additive Combinatorics_.

Dummit and Foote's _Abstract Algebra_ is an excellent book -- even if not very
directly relevant to additive combinatorics. Analysis background would be
useful, such as to be found in Stein and Shakarchi's series. Some introductory
combinatorics would also be good (e.g. Stanley's _Enumerative Combinatorics_ ,
or Van Lint and Wilson).

Finally, analytic number theory would also be good to learn. For example, you
might read Davenport's _Multiplicative Number Theory_. (Especially if you want
to see integrals come into the picture.)

~~~
ksd482
Thank you! This is exactly what I was looking for. This gives me a sense as to
where I stand relative to understanding this paper.

------
nwallin
> Peluse answered that question in a counterintuitive way — by thinking about
> exactly what it would take for a set of numbers not to contain the pattern
> you’re looking for.

Proof by contradiction isn't that counter intuitive.

And it's utterly bizarre that this article doesn't explain that this technique
is proof by contradiction, or give an example of what it is. The proof that
the square root of two can't be expressed as a ratio of integers (hence is
not-ratio-able, or irrational) is approachable by highschool freshmen, and
would add so much to the article.

~~~
willis936
When I was coming up with and solving problems with a math friend I learned
that proof by contradiction was perhaps the most common way to prove things.
It’s strange to people who don’t work with math everyday, to everyone else
it’s secondhand.

~~~
pacaro
It's not just math, it's a pretty basic concept in philosophy, or any area
where critical thinking is required

