Higher-rank Bohr sets and multiplicative diophantine approximation
Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied...
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| Format: | Journal article |
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Foundation Compositio Mathematica
2019
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| _version_ | 1826280279402086400 |
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| author | Chow, S Technau, N |
| author_facet | Chow, S Technau, N |
| author_sort | Chow, S |
| collection | OXFORD |
| description | Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem. |
| first_indexed | 2024-03-07T00:11:19Z |
| format | Journal article |
| id | oxford-uuid:79560e36-bf8e-49dd-8caa-c8528a97a383 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:11:19Z |
| publishDate | 2019 |
| publisher | Foundation Compositio Mathematica |
| record_format | dspace |
| spelling | oxford-uuid:79560e36-bf8e-49dd-8caa-c8528a97a3832022-03-26T20:36:42ZHigher-rank Bohr sets and multiplicative diophantine approximationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:79560e36-bf8e-49dd-8caa-c8528a97a383Symplectic Elements at OxfordFoundation Compositio Mathematica2019Chow, STechnau, NGallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem. |
| spellingShingle | Chow, S Technau, N Higher-rank Bohr sets and multiplicative diophantine approximation |
| title | Higher-rank Bohr sets and multiplicative diophantine approximation |
| title_full | Higher-rank Bohr sets and multiplicative diophantine approximation |
| title_fullStr | Higher-rank Bohr sets and multiplicative diophantine approximation |
| title_full_unstemmed | Higher-rank Bohr sets and multiplicative diophantine approximation |
| title_short | Higher-rank Bohr sets and multiplicative diophantine approximation |
| title_sort | higher rank bohr sets and multiplicative diophantine approximation |
| work_keys_str_mv | AT chows higherrankbohrsetsandmultiplicativediophantineapproximation AT technaun higherrankbohrsetsandmultiplicativediophantineapproximation |