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Irrationality exponent, Hausdorff dimension and effectivization
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a , we show t...
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| Published in: | Monatshefte für Mathematik 2018-02, Vol.185 (2), p.167-188 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c359t-d1957724c9b877c0668915c2dc501be7c8c42224747eb585d50bf3d8c80485713 |
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| cites | cdi_FETCH-LOGICAL-c359t-d1957724c9b877c0668915c2dc501be7c8c42224747eb585d50bf3d8c80485713 |
| container_end_page | 188 |
| container_issue | 2 |
| container_start_page | 167 |
| container_title | Monatshefte für Mathematik |
| container_volume | 185 |
| creator | Becher, Verónica Reimann, Jan Slaman, Theodore A. |
| description | We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number
a
greater than or equal to 2 and any non-negative real
b
be less than or equal to 2 /
a
, we show that there is a Cantor-like set with Hausdorff dimension equal to
b
such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to
a
. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to
b
and irrationality exponent equal to
a
. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets. |
| doi_str_mv | 10.1007/s00605-017-1094-2 |
| format | article |
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a
greater than or equal to 2 and any non-negative real
b
be less than or equal to 2 /
a
, we show that there is a Cantor-like set with Hausdorff dimension equal to
b
such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to
a
. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to
b
and irrationality exponent equal to
a
. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.</description><identifier>ISSN: 0026-9255</identifier><identifier>EISSN: 1436-5081</identifier><identifier>DOI: 10.1007/s00605-017-1094-2</identifier><language>eng</language><publisher>Vienna: Springer Vienna</publisher><subject>Irrationality ; Mathematics ; Mathematics and Statistics ; Real numbers</subject><ispartof>Monatshefte für Mathematik, 2018-02, Vol.185 (2), p.167-188</ispartof><rights>Springer-Verlag GmbH Austria 2017</rights><rights>Copyright Springer Science & Business Media 2018</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c359t-d1957724c9b877c0668915c2dc501be7c8c42224747eb585d50bf3d8c80485713</citedby><cites>FETCH-LOGICAL-c359t-d1957724c9b877c0668915c2dc501be7c8c42224747eb585d50bf3d8c80485713</cites><orcidid>0000-0003-1156-8390</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Becher, Verónica</creatorcontrib><creatorcontrib>Reimann, Jan</creatorcontrib><creatorcontrib>Slaman, Theodore A.</creatorcontrib><title>Irrationality exponent, Hausdorff dimension and effectivization</title><title>Monatshefte für Mathematik</title><addtitle>Monatsh Math</addtitle><description>We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number
a
greater than or equal to 2 and any non-negative real
b
be less than or equal to 2 /
a
, we show that there is a Cantor-like set with Hausdorff dimension equal to
b
such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to
a
. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to
b
and irrationality exponent equal to
a
. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.</description><subject>Irrationality</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Real numbers</subject><issn>0026-9255</issn><issn>1436-5081</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp1kL1OwzAURi0EEqXwAGyRWDHc69ixPSFUAa1UiQVmK_EPStUmwU4Q5elJCQML013O-XR1CLlEuEEAeZsAChAUUFIEzSk7IjPkeUEFKDwmMwBWUM2EOCVnKW0AAPNCz8jdKsayr9um3Nb9PvOfXdv4pr_OluWQXBtDyFy9800akaxsXOZD8LavP-qvH-2cnIRym_zF752T18eHl8WSrp-fVov7NbW50D11qIWUjFtdKSktFIXSKCxzVgBWXlplOWOMSy59JZRwAqqQO2UVcCUk5nNyNe12sX0ffOrNph3i-HUyqJVmOWqpRwonysY2peiD6WK9K-PeIJhDJzN1MmMnc-hk2OiwyUkj27z5-Gf5X-kbU4Npvw</recordid><startdate>20180201</startdate><enddate>20180201</enddate><creator>Becher, Verónica</creator><creator>Reimann, Jan</creator><creator>Slaman, Theodore A.</creator><general>Springer Vienna</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-1156-8390</orcidid></search><sort><creationdate>20180201</creationdate><title>Irrationality exponent, Hausdorff dimension and effectivization</title><author>Becher, Verónica ; Reimann, Jan ; Slaman, Theodore A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c359t-d1957724c9b877c0668915c2dc501be7c8c42224747eb585d50bf3d8c80485713</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Irrationality</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Real numbers</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Becher, Verónica</creatorcontrib><creatorcontrib>Reimann, Jan</creatorcontrib><creatorcontrib>Slaman, Theodore A.</creatorcontrib><collection>CrossRef</collection><jtitle>Monatshefte für Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Becher, Verónica</au><au>Reimann, Jan</au><au>Slaman, Theodore A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Irrationality exponent, Hausdorff dimension and effectivization</atitle><jtitle>Monatshefte für Mathematik</jtitle><stitle>Monatsh Math</stitle><date>2018-02-01</date><risdate>2018</risdate><volume>185</volume><issue>2</issue><spage>167</spage><epage>188</epage><pages>167-188</pages><issn>0026-9255</issn><eissn>1436-5081</eissn><abstract>We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number
a
greater than or equal to 2 and any non-negative real
b
be less than or equal to 2 /
a
, we show that there is a Cantor-like set with Hausdorff dimension equal to
b
such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to
a
. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to
b
and irrationality exponent equal to
a
. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.</abstract><cop>Vienna</cop><pub>Springer Vienna</pub><doi>10.1007/s00605-017-1094-2</doi><tpages>22</tpages><orcidid>https://orcid.org/0000-0003-1156-8390</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0026-9255 |
| ispartof | Monatshefte für Mathematik, 2018-02, Vol.185 (2), p.167-188 |
| issn | 0026-9255 1436-5081 |
| language | eng |
| recordid | cdi_proquest_journals_1989231979 |
| source | Springer Nature:Jisc Collections:Springer Nature Read and Publish 2023-2025: Springer Reading List |
| subjects | Irrationality Mathematics Mathematics and Statistics Real numbers |
| title | Irrationality exponent, Hausdorff dimension and effectivization |
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