Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers

Let $x⩾3$, for $1⩽n⩽x$ an integer, let $\omega (n)$ be its number of distinct prime factors. We show that, among the values $n⩽x$ with $\omega (n)=k$ where $1⩽k\ll {\log }_{2}x$, $\omega (n-1)$ satisfies an Erdős-Kac type theorem around $2{\log }_{2}x$, so in large deviation regime, when weighted by $2^{\omega (n-1)}$. This sharpens a result of Gorodetsky and Grimmelt with a quantitative and quasi-optimal error term. The proof of the main theorem is based on the characteristic function method and uses recent progress on Titchmarsh’s divisor problem.