Roth's Theorem in Super Smooth Numbers

Summary · qwen2.5:32b

The article demonstrates that Roth's theorem on arithmetic progressions holds for sets of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$, defined as super smooth when $y=\log^KN$ for a large constant $K$. This extends Harper’s previous work, which established the theorem under less stringent conditions.

Suggested post angle

Exploring Roth's Theorem in Super Smooth Numbers: A New Advancement in Number Theory

Excerpt

arXiv:2510.18024v2 Announce Type: replace Abstract: We say that the set of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$ is super smooth if $y=\log^KN$ for a large fixed constant $K$. We show that the Roth's theorem on arithmetic progressions is true in super smooth numbers case. This extends the result of Harper where he showed the statement is true under a weaker hypothesis.