Gauss sums over some subsets

Xiaoying Liu; Zhefeng Xu; Xiaoying Liu; Zhefeng Xu

doi:10.3934/math.2026081

AIMS Mathematics

2026, Volume 11, Issue 1: 1954-1967. doi: 10.3934/math.2026081

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Research article

Gauss sums over some subsets

Xiaoying Liu ,
Zhefeng Xu ^,

Research Center for Number Theory and Its Applications, Northwest University, Xi'an 710127, China

Received: 01 December 2025 Revised: 06 January 2026 Accepted: 13 January 2026 Published: 21 January 2026
MSC : 11L05, 11L40

Let $ q \ge 3 $ be an integer, $ \chi $ a non-principal character modulo $ q $, and $ A, B, H\leq q $ with $ (u, q) = 1 $ and $ (v, q) = 1 $. In this paper, by combining estimates for general Kloosterman sums and Gauss sums with properties of trigonometric sums, we derive nontrivial bounds for Gauss sums over the sets $ \aleph(n, q), $ $ \hbar_{u, v}(A, B, H) $, and $ \aleph(n, q)\cap\hbar_{u, v}(A, B, H), $ where
$ \aleph(n,q) = \big\{ a \in\mathbb{Z} \mid (a, q) = 1,\ n \nmid a + \overline{a} \big\}, $
and
$ \hbar_{u,v}(A, B, H) = \{a\in\mathbb{Z}|(a,q) = 1, ab\equiv 1\bmod q, 1\le a\le A, 1\le b\le B, |ua-vb|\le H\}. $
- Gauss sum,
- Lehmer number,
- upper bound
Citation: Xiaoying Liu, Zhefeng Xu. Gauss sums over some subsets[J]. AIMS Mathematics, 2026, 11(1): 1954-1967. doi: 10.3934/math.2026081

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Abstract

Let $ q \ge 3 $ be an integer, $ \chi $ a non-principal character modulo $ q $, and $ A, B, H\leq q $ with $ (u, q) = 1 $ and $ (v, q) = 1 $. In this paper, by combining estimates for general Kloosterman sums and Gauss sums with properties of trigonometric sums, we derive nontrivial bounds for Gauss sums over the sets $ \aleph(n, q), $ $ \hbar_{u, v}(A, B, H) $, and $ \aleph(n, q)\cap\hbar_{u, v}(A, B, H), $ where

$ \aleph(n,q) = \big\{ a \in\mathbb{Z} \mid (a, q) = 1,\ n \nmid a + \overline{a} \big\}, $

and

$ \hbar_{u,v}(A, B, H) = \{a\in\mathbb{Z}|(a,q) = 1, ab\equiv 1\bmod q, 1\le a\le A, 1\le b\le B, |ua-vb|\le H\}. $

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