On the representation of friable integers by linear forms (1609.08872v2)

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.