Problem 3 — Visual explainer (undergrad)
========================================

Problem statement
---------------
If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions?

Picture
-------
Number line picture of an arithmetic progression (equal spacing):

---o-----o-----o-----o-----o---
   a    a+d   a+2d  a+3d  a+4d

Question: what to look at
------------------------
- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
-------
Area hint: additive combinatorics / additive number theory.

Benchmarks / known results (from comments)
----------------------------------------
- This is essentially asking for good bounds on $r_k(N)$, the size of the
  largest subset of $\{1,\ldots,N\}$ without a non-trivial $k$-term arithmetic
  progression.
- For example, a bound like\[r_k(N) \ll_k \frac{N}{(\log N)(\log\log
  N)^2}\]would be sufficient.
- Even the case $k=3$ is non-trivial, but was proved by Bloom and Sisask.
- Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka.