Non-planarity and metric Diophantine approximation for systems of linear forms
In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of ‘weak non-planarity’ of manifolds and more generally measures on the spaceMm,n ofm × nmatrices over ℝ is introduced and studied. This notion generalizes the one of non-planarity...
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| Veröffentlicht in: | Journal de theorie des nombres de bordeaux 2015-01, Vol.27 (1), p.1-31 |
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| container_title | Journal de theorie des nombres de bordeaux |
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| creator | BERESNEVICH, Victor KLEINBOCK, Dmitry MARGULIS, Gregory |
| description | In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of ‘weak non-planarity’ of manifolds and more generally measures on the spaceMm,n
ofm × nmatrices over ℝ is introduced and studied. This notion generalizes the one of non-planarity in ℝⁿand is used to establish strong (Diophantine) extremality of manifolds and measures inMm,n
. Thus our results contribute to resolving a problem stated in [20, §9.1] regarding the strong extremality of manifolds inMm,n
. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results of the first named author and Velani [11] and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart. |
| doi_str_mv | 10.5802/jtnb.890 |
| format | Article |
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ofm × nmatrices over ℝ is introduced and studied. This notion generalizes the one of non-planarity in ℝⁿand is used to establish strong (Diophantine) extremality of manifolds and measures inMm,n
. Thus our results contribute to resolving a problem stated in [20, §9.1] regarding the strong extremality of manifolds inMm,n
. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results of the first named author and Velani [11] and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.</description><identifier>ISSN: 1246-7405</identifier><identifier>EISSN: 2118-8572</identifier><identifier>DOI: 10.5802/jtnb.890</identifier><language>eng</language><publisher>cedram</publisher><subject>Approximation ; Diophantine sets ; Euclidean space ; Hausdorff dimensions ; Hyperplanes ; Lebesgue measures ; Mathematical manifolds ; Mathematical theorems ; Mathematics ; Transference</subject><ispartof>Journal de theorie des nombres de bordeaux, 2015-01, Vol.27 (1), p.1-31</ispartof><rights>Société Arithmétique de Bordeaux, 2015</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c283t-e1e513b5b3a8b3da11a5df108591356f520af8d9c53616c5d56d9bff8fdab8c63</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/26273980$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/26273980$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,776,780,799,828,4010,27900,27901,27902,57992,57996,58225,58229</link.rule.ids></links><search><creatorcontrib>BERESNEVICH, Victor</creatorcontrib><creatorcontrib>KLEINBOCK, Dmitry</creatorcontrib><creatorcontrib>MARGULIS, Gregory</creatorcontrib><title>Non-planarity and metric Diophantine approximation for systems of linear forms</title><title>Journal de theorie des nombres de bordeaux</title><description>In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of ‘weak non-planarity’ of manifolds and more generally measures on the spaceMm,n
ofm × nmatrices over ℝ is introduced and studied. This notion generalizes the one of non-planarity in ℝⁿand is used to establish strong (Diophantine) extremality of manifolds and measures inMm,n
. Thus our results contribute to resolving a problem stated in [20, §9.1] regarding the strong extremality of manifolds inMm,n
. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results of the first named author and Velani [11] and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.</description><subject>Approximation</subject><subject>Diophantine sets</subject><subject>Euclidean space</subject><subject>Hausdorff dimensions</subject><subject>Hyperplanes</subject><subject>Lebesgue measures</subject><subject>Mathematical manifolds</subject><subject>Mathematical theorems</subject><subject>Mathematics</subject><subject>Transference</subject><issn>1246-7405</issn><issn>2118-8572</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNo9kEtLxDAYRYMoWEfBPyBk6aZjHpM0Wcr4hGHc6Lp8bRJsaZOSZGH_vS0jri5cDpfLQeiWkq1QhD302TdbpckZKhilqlSiYueooGwny2pHxCW6SqknhHGpVYGOx-DLaQAPscszBm_waHPsWvzUhekbfO68xTBNMfx0I-QueOxCxGlO2Y4JB4eHhYC4tmO6RhcOhmRv_nKDvl6eP_dv5eHj9X3_eChbpnguLbWC8kY0HFTDDVAKwjhKlNCUC-kEI-CU0a3gkspWGCGNbpxTzkCjWsk36P6028aQUrSunuJyL841JfXqoV491IuHBb07oX3KIf5zTLKKa0X4LxHKXFU</recordid><startdate>20150101</startdate><enddate>20150101</enddate><creator>BERESNEVICH, Victor</creator><creator>KLEINBOCK, Dmitry</creator><creator>MARGULIS, Gregory</creator><general>cedram</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20150101</creationdate><title>Non-planarity and metric Diophantine approximation for systems of linear forms</title><author>BERESNEVICH, Victor ; KLEINBOCK, Dmitry ; MARGULIS, Gregory</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c283t-e1e513b5b3a8b3da11a5df108591356f520af8d9c53616c5d56d9bff8fdab8c63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Approximation</topic><topic>Diophantine sets</topic><topic>Euclidean space</topic><topic>Hausdorff dimensions</topic><topic>Hyperplanes</topic><topic>Lebesgue measures</topic><topic>Mathematical manifolds</topic><topic>Mathematical theorems</topic><topic>Mathematics</topic><topic>Transference</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>BERESNEVICH, Victor</creatorcontrib><creatorcontrib>KLEINBOCK, Dmitry</creatorcontrib><creatorcontrib>MARGULIS, Gregory</creatorcontrib><collection>CrossRef</collection><jtitle>Journal de theorie des nombres de bordeaux</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>BERESNEVICH, Victor</au><au>KLEINBOCK, Dmitry</au><au>MARGULIS, Gregory</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Non-planarity and metric Diophantine approximation for systems of linear forms</atitle><jtitle>Journal de theorie des nombres de bordeaux</jtitle><date>2015-01-01</date><risdate>2015</risdate><volume>27</volume><issue>1</issue><spage>1</spage><epage>31</epage><pages>1-31</pages><issn>1246-7405</issn><eissn>2118-8572</eissn><abstract>In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of ‘weak non-planarity’ of manifolds and more generally measures on the spaceMm,n
ofm × nmatrices over ℝ is introduced and studied. This notion generalizes the one of non-planarity in ℝⁿand is used to establish strong (Diophantine) extremality of manifolds and measures inMm,n
. Thus our results contribute to resolving a problem stated in [20, §9.1] regarding the strong extremality of manifolds inMm,n
. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results of the first named author and Velani [11] and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.</abstract><pub>cedram</pub><doi>10.5802/jtnb.890</doi><tpages>31</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Journal de theorie des nombres de bordeaux, 2015-01, Vol.27 (1), p.1-31 |
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| source | Jstor Complete Legacy; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; NUMDAM; JSTOR Mathematics & Statistics |
| subjects | Approximation Diophantine sets Euclidean space Hausdorff dimensions Hyperplanes Lebesgue measures Mathematical manifolds Mathematical theorems Mathematics Transference |
| title | Non-planarity and metric Diophantine approximation for systems of linear forms |
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