Burgess-type character sum estimates over generalized arithmetic progressions of rank $2$

TL;DR

This work extends classical Burgess-type character-sum estimates to proper rank-2 generalized arithmetic progressions in $\mathbb{F}_p$, establishing a nontrivial bound for short sums when $|A|\ge p^{1/4+\varepsilon}$. The authors prove a sharp bound for the multiplicative energy $E_{\times}(A)$ of rank-2 GAPs, $E_{\times}(A) \ll (|A|^2+|A|^4/p)\log p$, by adapting Konyagin's method and leveraging geometry-of-numbers techniques, together with new upper bounds on Bohr-set sizes. This energy bound directly yields a Burgess-type estimate for sums over rank-2 GAPs, improving our understanding of character sums on structured sets in finite fields and highlighting the role of sum-product phenomena in this context. The work also introduces Bohr-set bounds that may be of independent interest and discusses obstacles to generalizing the approach to higher-rank GAPs, outlining future directions in extending Burgess-type results beyond rank 2.

Abstract

We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank $2$ in prime fields $\mathbb{F}_p$. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. A key step in our argument involves establishing new upper bounds for the sizes of Bohr sets, which may be of independent interest.

Theorems & Definitions (12)

• Theorem : Pólya-Vinogadov
• Theorem : Burgess
• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Minkowksi's Second Theorem
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 3.1
• proof : Proof of Proposition \ref{['prop:bohr']}