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Bounded Representations by $x^2+y^2-z^2$

Przemyslaw Chojecki

Abstract

We prove that every sufficiently large integer $n$ can be written in the form $n=x^2+y^2-z^2$ with $\textrm{max}(x^2,y^2,z^2)\le n$. The proof converts the problem into finding a primitive binary quadratic form of positive discriminant $4n$ inside a fixed relatively compact open patch of the real hyperboloid $b^2-4ac=4n$. This is then supplied by Duke's theorem in the precise point-counting form deduced from the measure-theoretic duality of Einsiedler-Lindenstrauss-Michel-Venkatesh. A finite parity correction returns to the original ternary variables. This settles Erdős Problem 1148.

Bounded Representations by $x^2+y^2-z^2$

Abstract

We prove that every sufficiently large integer can be written in the form with . The proof converts the problem into finding a primitive binary quadratic form of positive discriminant inside a fixed relatively compact open patch of the real hyperboloid . This is then supplied by Duke's theorem in the precise point-counting form deduced from the measure-theoretic duality of Einsiedler-Lindenstrauss-Michel-Venkatesh. A finite parity correction returns to the original ternary variables. This settles Erdős Problem 1148.
Paper Structure (5 sections, 5 theorems, 52 equations)

This paper contains 5 sections, 5 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. From ternary representations to discriminant points
  3. Three elementary coefficient moves
  4. Duke's theorem in the needed point-counting form
  5. Proof of the main theorem

Key Result

Theorem 1.1

There exists $N$ such that every integer $n\ge N$ admits integers $x,y,z$ with

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 4.1: Point-counting consequence of Duke--ELMV
  • proof
  • Corollary 4.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 1 more