Roth's Theorem in Super Smooth Numbers

TL;DR

This paper proves Roth's theorem for super smooth numbers, a set characterized by very smooth prime factorizations, extending previous results to a broader class of numbers.

Contribution

It establishes Roth's theorem in the context of super smooth numbers, significantly broadening the scope of arithmetic progression results in number theory.

Findings

Roth's theorem holds for super smooth numbers

Extends Harper's previous weaker hypothesis result

Demonstrates the abundance of 3-term arithmetic progressions in super smooth numbers

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.