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

TL;DR

This paper proves that every sufficiently large integer can be expressed as the sum of two squares minus a square, with all squares bounded by the integer, solving a longstanding problem in number theory.

Contribution

It introduces a novel approach using quadratic forms and hyperboloid point-counting, applying Duke's theorem to settle Erdős Problem 1148.

Findings

Every large integer n can be written as x^2 + y^2 - z^2 with max(x^2,y^2,z^2) ≤ n

The proof employs measure-theoretic duality and point-counting on hyperboloids

The result confirms a conjecture in additive number theory

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 Erd\H{o}s Problem 1148.