Roth's Theorem in Super Smooth Numbers

Laurence P. Wijaya

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arXiv · 2510.18024

Roth's Theorem in Super Smooth Numbers

Laurence P. Wijaya

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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.

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Laurence P. Wijaya. 2025-10-20. Roth's Theorem in Super Smooth Numbers. https://arxiv.org/abs/2510.18024

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