Roth's Theorem in Super Smooth Numbers

TL;DR

This work proves a Roth-type result for 3-term arithmetic progressions inside the set of $y$-smooth numbers in the super smooth regime $y=\log^K N$ with fixed large $K$. Building on Harper's findings, the authors deploy a Green--Tao–style transference principle together with a $W$-trick to transfer a dense Roth theorem to the sparse, structured setting of $\mathcal{S}(N,y)$. The analysis combines distribution results for smooth numbers in progressions, exponential-sum estimates over smooth numbers on major/minor arcs, and a carefully crafted weighted majorant to enable a density increment argument in a relative setting. The main result shows that any $A\subseteq \mathcal{S}(N,y)$ with $|A|\ge\delta\Psi(N,y)$ contains a length-$3$ arithmetic progression for large $N$, with the construction Yukawa mapping back to $A$ via the $W$-trick, and relies on a transference framework that controls the relevant trilinear averages.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Proposition 3.6
• Theorem 3.7
• Proposition 4.1
• Proposition 4.2