On the representation of friable integers by linear forms

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.
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Armand Lachand. On the representation of friable integers by linear forms. Acta Arithmetica, Instytut Matematyczny PAN, 2017, 181, pp.97-109. ⟨10.4064/aa8153-9-2017⟩. ⟨hal-01081277v3⟩

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