On the representation of friable integers by linear forms

TL;DR

This paper investigates the distribution of integers with small prime factors when represented by systems of affine-linear forms, extending previous results and providing improved bounds for the count of such integers within convex regions.

Contribution

It advances the understanding of friable integers represented by linear forms, improving bounds over prior work specifically for products of linear forms.

Findings

Provides new bounds for the count of friable integers in affine-linear form systems.

Extends previous results to more general systems of forms.

Improves upon earlier work by Balog, Blomer, Dartyge, and Tenenbaum.

Abstract

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}\Psi\_{F\_1\cdots F\_t}\left(\mathcal{K}\cap[-N,N]^d,N^{1/u}\right):=\\#\left\{\mathcal{K}\in {\mathbf{N}}\cap[-N,N]^d:\vphantom{P^+(F\_1(\boldsymbol{n})\cdots F\_t(\boldsymbol{n}))\leq N^{1/u}}\right.\left.P^+(F\_1(\boldsymbol{n})\cdots F\_t(\boldsymbol{n}))\leq N^{1/u}\right\}\end{align*} where $(F\_1,\ldots,F\_t)$ is a system of affine-linear forms of $\mathbf{Z}[X\_1,\ldots,X\_d]$ no two of which are affinely related and $\mathcal{K}$ is a convex body. This improves upon Balog, Blomer, Dartyge and Tenenbaum's work~\cite{BBDT12} in the case of product of linear forms.