Problem Statement
If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$, must $A$ contain arbitrarily long arithmetic progressions?
Categories:
Number Theory Additive Combinatorics Arithmetic Progressions
Progress
Recent Breakthroughs
For k=3: Bloom and Sisask established the base case. Kelley and Meka proved improved bounds.
General Case
- Gowers: $r_k(N) \ll \frac{N}{(\log\log N)^{c_k}}$
- Green and Tao: $r_4(N) \ll N/(\log N)^c$
- Leng, Sah, and Sawhney (current best): $r_k(N) \ll \frac{N}{\exp((\log\log N)^{c_k})}$
Related
- Connected to Green-Tao theorem on primes in arithmetic progressions
Source: erdosproblems.com/3 | Last verified: January 13, 2026