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On the density version of quadratic Waring's problem and the quadratic Waring--Goldbach problem

Zi Li Lim

TL;DR

This work establishes sharp density-type results for quadratic Waring and quadratic Waring–Goldbach problems in modular and integer settings. It develops a transference framework, anchored by restriction estimates for squares in cyclic groups and detailed sumset analyses, to prove that a dense subset of squares in $(\mathbb{Z}/W)^{(2)}$ densely generates all residues via sums of $s$ terms when $s\ge 5$ and $W$ avoids small primes. For integers, it provides improvements on density thresholds in the Waring–Goldbach context, including an explicit local-to-global scheme with thresholds $D_s$ and $d_s$ and a congruence condition; the results improve prior density bounds in several regimes and illuminate the role of local moduli. The paper combines analytic tools (restriction theory, Gauss sums), additive combinatorics (sumsets, Green–Ruzsa), and a combinatorial lemma with computer-assisted base cases to bridge local and global results, including extensions to small moduli via a refined $W$-trick. Overall, it advances the understanding of density phenomena in quadratic representations and connects modular and integer Waring-type problems through a flexible transference paradigm.

Abstract

We prove a sharp density theorem for quadratic Waring's problem over cyclic groups, when the number of variables is at least $5$. Also, we obtain some new improvements on the density version of the quadratic Waring--Goldbach problem over integers.

On the density version of quadratic Waring's problem and the quadratic Waring--Goldbach problem

TL;DR

This work establishes sharp density-type results for quadratic Waring and quadratic Waring–Goldbach problems in modular and integer settings. It develops a transference framework, anchored by restriction estimates for squares in cyclic groups and detailed sumset analyses, to prove that a dense subset of squares in densely generates all residues via sums of terms when and avoids small primes. For integers, it provides improvements on density thresholds in the Waring–Goldbach context, including an explicit local-to-global scheme with thresholds and and a congruence condition; the results improve prior density bounds in several regimes and illuminate the role of local moduli. The paper combines analytic tools (restriction theory, Gauss sums), additive combinatorics (sumsets, Green–Ruzsa), and a combinatorial lemma with computer-assisted base cases to bridge local and global results, including extensions to small moduli via a refined -trick. Overall, it advances the understanding of density phenomena in quadratic representations and connects modular and integer Waring-type problems through a flexible transference paradigm.

Abstract

We prove a sharp density theorem for quadratic Waring's problem over cyclic groups, when the number of variables is at least . Also, we obtain some new improvements on the density version of the quadratic Waring--Goldbach problem over integers.

Paper Structure

This paper contains 17 sections, 22 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Sketch of the proofs
  4. Future directions
  5. Road map
  6. Acknowledgments
  7. Preliminaries
  8. Squares in $\mathbb{Z}/W$
  9. Other notation
  10. Restriction Estimates
  11. Sumsets of Dense Subsets of $\mathbb{Z}/W$
  12. Transference Principle
  13. Counterexamples
  14. A Combinatorial Lemma
  15. Extensions to Small Moduli
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $s\geq 5$ be an integer and $\theta\in (1/s,1]$. There exists a constant $M(s,\theta)$ depending only on $s$ and $\theta$ such that the following holds. Suppose $W=\prod_{i} p_i^{n_i}$ is an integer with $p_i\geq M(s,\theta)$ for all $i$, and $A$ is a subset of the squares in $\mathbb{Z}/W$ with for some constant $c(s,\theta)>0$ depending only on $s$ and $\theta$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 37 more