Mass transference principle from rectangles to rectangles in Diophantine approximation
Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the lat...
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| Autor*in: |
Wang, Baowei [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Mathematische Annalen - Springer Berlin Heidelberg, 1869, 381(2021), 1-2 vom: 26. Apr., Seite 243-317 |
|---|---|
| Übergeordnetes Werk: |
volume:381 ; year:2021 ; number:1-2 ; day:26 ; month:04 ; pages:243-317 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00208-021-02187-6 |
|---|
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OLC2077129425 |
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| 520 | |a Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. | ||
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10.1007/s00208-021-02187-6 doi (DE-627)OLC2077129425 (DE-He213)s00208-021-02187-6-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn 31.00 bkl Wang, Baowei verfasserin aut Mass transference principle from rectangles to rectangles in Diophantine approximation 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. Wu, Jun aut Enthalten in Mathematische Annalen Springer Berlin Heidelberg, 1869 381(2021), 1-2 vom: 26. Apr., Seite 243-317 (DE-627)129060151 (DE-600)285-9 (DE-576)014390825 0025-5831 nnns volume:381 year:2021 number:1-2 day:26 month:04 pages:243-317 https://doi.org/10.1007/s00208-021-02187-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-FOR GBV_ILN_40 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 381 2021 1-2 26 04 243-317 |
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10.1007/s00208-021-02187-6 doi (DE-627)OLC2077129425 (DE-He213)s00208-021-02187-6-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn 31.00 bkl Wang, Baowei verfasserin aut Mass transference principle from rectangles to rectangles in Diophantine approximation 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. Wu, Jun aut Enthalten in Mathematische Annalen Springer Berlin Heidelberg, 1869 381(2021), 1-2 vom: 26. Apr., Seite 243-317 (DE-627)129060151 (DE-600)285-9 (DE-576)014390825 0025-5831 nnns volume:381 year:2021 number:1-2 day:26 month:04 pages:243-317 https://doi.org/10.1007/s00208-021-02187-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-FOR GBV_ILN_40 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 381 2021 1-2 26 04 243-317 |
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10.1007/s00208-021-02187-6 doi (DE-627)OLC2077129425 (DE-He213)s00208-021-02187-6-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn 31.00 bkl Wang, Baowei verfasserin aut Mass transference principle from rectangles to rectangles in Diophantine approximation 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. Wu, Jun aut Enthalten in Mathematische Annalen Springer Berlin Heidelberg, 1869 381(2021), 1-2 vom: 26. Apr., Seite 243-317 (DE-627)129060151 (DE-600)285-9 (DE-576)014390825 0025-5831 nnns volume:381 year:2021 number:1-2 day:26 month:04 pages:243-317 https://doi.org/10.1007/s00208-021-02187-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-FOR GBV_ILN_40 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 381 2021 1-2 26 04 243-317 |
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Mass transference principle from rectangles to rectangles in Diophantine approximation |
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Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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Abstract The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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