Bounds on the exceptional set in the $abc$ conjecture
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915996378333184 |
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| author | Bernert, Christian Browning, Tim Lichtman, Jared Duker Teräväinen, Joni |
| author_facet | Bernert, Christian Browning, Tim Lichtman, Jared Duker Teräväinen, Joni |
| contents | We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $ε>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-ε}$ with finitely many exceptions. In this article we obtain a power-saving bound on the size of the exceptional set of triples. The proof is based on a combination of upper bounds for the density of integer points on certain high-dimensional varieties, coming from the geometry of numbers and from Fourier analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12234 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on the exceptional set in the $abc$ conjecture Bernert, Christian Browning, Tim Lichtman, Jared Duker Teräväinen, Joni Number Theory Combinatorics 11D45 (11D41, 11D75) We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $ε>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-ε}$ with finitely many exceptions. In this article we obtain a power-saving bound on the size of the exceptional set of triples. The proof is based on a combination of upper bounds for the density of integer points on certain high-dimensional varieties, coming from the geometry of numbers and from Fourier analysis. |
| title | Bounds on the exceptional set in the $abc$ conjecture |
| topic | Number Theory Combinatorics 11D45 (11D41, 11D75) |
| url | https://arxiv.org/abs/2410.12234 |