A Short Character Sum in \mathbbF_p³

2025/05/26 by Chattopadhyay, Aishik
#11L26 #11L40 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.19654

Abstract

We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in \mathbbF_p³ and higher-dimensional lattices in \mathbbF_p^d, extending an earlier work of Mei-Chu Chang on Burgess inequality in \mathbbF_p². In particular, we show that for intervals of size p^3/8+ε, the sum ∑_{x, y} χ(x + ωy), with ω∈ \mathbbF_p³ ∖ \mathbbF_p, exhibits nontrivial cancellation uniformly in ω. This is further generalized to codimension-one sublattices in \mathbbF_p^d, and applied to obtain an alternative estimate for character sums on binary cubic forms.

Citations

• Small solutions of generic ternary quadratic congruences to general moduli
• Burgess bounds for short character sums evaluated at forms II: the mixed case
• Estimates for character sums in finite fields of order p² and p³
• Large character sums: Burgess's theorem and zeros of L-functions
• Large character sums

Related

• Estimates for short character sums evaluated at homogeneous polynomials
• Burgess bounds for short character sums evaluated at forms II: the mixed case
• Estimates for character sums in finite fields of order p² and p³
• Character sums over sparse elements of finite fields
• Sums of inverses in thin sets of finite fields