Linear truncation for conditioned prime-factor fibres

TL;DR

This paper improves the truncation bounds in the conditional effective Erdos-Wintner theorem for prime-factor fibers by establishing a linear truncation lemma under certain ratio estimates, enhancing previous results.

Contribution

It introduces an effective linear truncation lemma that sharpens bounds in the Erdos-Wintner theorem for prime-factor fibers, extending to various fiber types.

Findings

Improved truncation bounds from $ ext{eta}_f(R)^{r/(r+1)}$ to $r ext{eta}_f(R)$ in the central window.

Applicable to prime-set restrictions, $ ext{Omega}$-fibres, and weighted fibres with ratio estimates.

Provides a more precise truncation step in the analysis of additive functions on fibers.

Abstract

In previous joint work with Tenenbaum, the truncation step $f \mapsto f_R$ in the conditional effective Erdos-Wintner theorem on the fibre $\omega(n)=k$ yields, in the continuous case for real strongly additive $f$, a remainder of size $\eta_f(R)^{r/(r+1)}$, where $R$ is the truncation level and $r=k/\log\log x$. We prove an effective linear truncation lemma showing that, in the central window $\kappa \le r \le 1/\kappa$, this bound improves to the natural linear scale $r\eta_f(R)$ under an effective Sathe-Selberg-type ratio estimate for the fibre. This yields a direct effective sharpening of the truncation step in the previous joint work. The same truncation upgrade also applies to prime-set restrictions, $\Omega$-fibres, and weighted fibres whenever the corresponding ratio estimate is available.