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On additive decompositions of primitive elements in finite fields

Hai-Liang Wu, Yue-Feng She

TL;DR

This work addresses additive decompositions of primitive elements in finite fields by bounding when a primitive-set $P_p$ can be expressed as $A+B$. It employs multiplicative character sums, Weil bounds, and Shkredov-type additive-combinatorics techniques to derive new constraints: (i) a nonexistence result for certain small-set decompositions (Theorem A), (ii) asymptotically sharp size bounds for $A$ and $B$ in decompositions $A+B=P_p$ (Theorem B), and (iii) precise size-impossibility and size-bound dualities when $|A|=|B|$ (Theorem C). These results refine earlier bounds of Dartyge–Sárközy and Shparlinski and advance understanding of the additive structure of primitive elements, with implications for additive combinatorics in finite fields and related conjectures. The methods combine explicit representations of $P_p$, Weil-type estimates, and Shkredov’s energy-based framework to tighten the known landscape of decompositions.

Abstract

In this paper, we study several topics on additive decompositions of primitive elemements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy and Shparlinski.

On additive decompositions of primitive elements in finite fields

TL;DR

This work addresses additive decompositions of primitive elements in finite fields by bounding when a primitive-set can be expressed as . It employs multiplicative character sums, Weil bounds, and Shkredov-type additive-combinatorics techniques to derive new constraints: (i) a nonexistence result for certain small-set decompositions (Theorem A), (ii) asymptotically sharp size bounds for and in decompositions (Theorem B), and (iii) precise size-impossibility and size-bound dualities when (Theorem C). These results refine earlier bounds of Dartyge–Sárközy and Shparlinski and advance understanding of the additive structure of primitive elements, with implications for additive combinatorics in finite fields and related conjectures. The methods combine explicit representations of , Weil-type estimates, and Shkredov’s energy-based framework to tighten the known landscape of decompositions.

Abstract

In this paper, we study several topics on additive decompositions of primitive elemements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy and Shparlinski.
Paper Structure (6 sections, 10 theorems, 69 equations)

This paper contains 6 sections, 10 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Some related results
  3. Main Results
  4. Proof of Theorem \ref{['Thm. A']}
  5. Proof of Theorem \ref{['Thm. B']}.
  6. Proof of Theorem \ref{['Thm. C']}.

Key Result

Theorem 1.1

Let $p>3$ be an odd prime. For an integer $k\ge2$, if then $A+B\neq P_p$ for any $A,B\subseteq\mathbb{F}_p$ with $|A|,|B|\ge2$ and $|B|=k$.

Theorems & Definitions (13)

  • Conjecture 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 3 more