Bounded Representations by $x^2+y^2-z^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917352322367488 |
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| author | Chojecki, Przemyslaw |
| author_facet | Chojecki, Przemyslaw |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18087 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounded Representations by $x^2+y^2-z^2$ Chojecki, Przemyslaw Number Theory 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. |
| title | Bounded Representations by $x^2+y^2-z^2$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.18087 |