Estimates for trilinear and quadrilinear character sums

TL;DR

The paper develops power-saving bounds for two families of multilinear character sums, the trilinear sum $\mathfrak{T}$ and the quadrilinear sum $\mathfrak{Q}$, with arbitrary weights modulo a large prime $p$. It introduces two main results (Type I and Type II) that treat cases with $\boldsymbol{\alpha}$ either constant or general, respectively, and proves a quadrilinear analogue bound; the methods combine amplification, smoothing, and Weil-type estimates, supported by moment and gcd-sum preliminaries. These bounds yield nontrivial consequences, including a Farey-fraction oscillation bound, a modular analogue of Iwaniec–Sárközy type problems on distances to squares, and a range of corollaries for sums involving primes, divisor functions, and polynomial-driven twists. The findings extend Burgess-type phenomena to multilinear settings and provide versatile tools for studying oscillations of characters on structured sets, with potential applications to prime-type equations under various constraints.

Abstract

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to an apparently new problem on oscillations of characters on differences between Farey fractions is given. Other applications include a modular analogue of a multiplicative hybrid problem of Iwaniec and Sárközy (1987) and the solvability of some prime type equations with constraints.

Theorems & Definitions (28)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Corollary 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• Corollary 2.8
• Corollary 2.9
• Remark 2.10