Exponential sums over integers without large prime divisors

TL;DR

The paper advances exponential-sum bounds over $y$-smooth integers by sharpening the classical Fouvry–Tenenbaum estimates and extending them to sums with $\nu$-th powers. The authors blend a refined factorisation of smooth numbers with Type I/II bilinear techniques, leveraging trace-function bounds (non-exceptional) for general $\nu$ and employing Vinogradov’s method for $\nu=1$ alongside Vaughan–Fouvry–Kowalski–Michel machinery for $\nu\neq 1$. They establish new bounds for $S_{\nu,a,q}(x,y)=\sum_{n\in{\mathcal S}(x,y)}{\mathbf{e}}_q(a n^{\nu})$, including a FM-type bound with piecewise behaviour in $y$ relative to $q$ and $x$, and a FMK-type bound that yields power savings in broader ranges. The results improve the range of nontriviality, refine the dependence on $y$ and $q$, and open avenues for further extensions to twisted sums and trace-function contexts, with potential impact on Waring-type problems and circle-method analyses for smooth numbers.

Abstract

We obtain a new bound on exponential sums over integers without large prime divisors, improving that of Fouvry and Tenenbaum (1991). For a fixed integer $ν\ne 0$, we also obtain new bounds on exponential sums with $ν$-th powers of such integers. The improvement is based on exploiting more precisely the factorisation of integers without large prime divisors, along with existing Type~I and Type~II bounds. For $ν=1$ we use the classical bounds of Vinogradov (1937), while for $ν\neq 1$ we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).

Figures (2)

• Figure 1.1: Ranges where the bounds from Theorem \ref{['thm: Sum Snu']} are relevant. Here $y=x^\alpha$ and $q=x^\beta$.
• Figure 4.1: Three different regimes of $\alpha$ and $\beta$.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4