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Sums of squares of integers from residue classes

Daejun Kim

Abstract

A subset $\mathcal{A}\subseteq\mathbb{Z}$ is called $s$-almost square universal if every sufficiently large positive integer can be written as a sum of at most $s$ squares of integers from $\mathcal{A}$. In this article, we study the minimal number $\mathrm{ASU}(\mathcal{A}_{d,m})$ with this property, where $\mathcal{A}_{d,m}$ denotes the residue class of $d$ modulo $m$, with $m\in\mathbb{N}$ and $d\in\mathbb{Z}$. We further prove that $\mathcal{A}_{d,m}$ is $s$-square universal for some $s\in\mathbb{N}$ if and only if $d \equiv \pm 1 \pmod{m}$, and determine the minimal such number $\mathrm{SU}(\mathcal{A}_{d,m})$ in these cases.

Sums of squares of integers from residue classes

Abstract

A subset is called -almost square universal if every sufficiently large positive integer can be written as a sum of at most squares of integers from . In this article, we study the minimal number with this property, where denotes the residue class of modulo , with and . We further prove that is -square universal for some if and only if , and determine the minimal such number in these cases.

Paper Structure

This paper contains 9 sections, 12 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic spaces and lattices
  4. Quadratic lattice cosets
  5. Algebraic structures of lattice cosets
  6. Modular forms
  7. Theta series for cosets
  8. Almost square universality of ${\mathcal{A}}_{d,m}$
  9. Square universality of ${\mathcal{A}}_{d,m}$

Key Result

Theorem 1.1

For $m\in{\mathbb N}$ and $d\in{\mathbb Z}$ with $(m,d)=1$, we have

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • ...and 12 more