HAUSDORFF DIMENSIONS OF SOME LIMINF SETS IN DIOPHANTINE APPROXIMATION. Issue 1 (January 2015)

• Record Type: Journal Article
• Authors: Wang, Bao-Wei; Wen, Zhi-Ying; Wu, Jun
• Is Part Of: Mathematika. Volume 61:Issue 1(2015)
• Page Start: 101
• Page End: 120
• Publication Date: 2015-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579314000205
• Languages: English
• ISSNs: 0025-5793

Abstract

<abstract> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9cj5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$Q$]]></tex-math></alternatives></inline-formula> be an infinite subset of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p999k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbb{N}$]]></tex-math></alternatives></inline-formula>. For any <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p99wc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\it\tau}>2$]]></tex-math></alternatives></inline-formula>, denote <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p98sb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$W_{{\it\tau}}(Q)$]]></tex-math></alternatives></inline-formula> (respectively <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9cfm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$W_{{\it\tau}}$]]></tex-math></alternatives></inline-formula>) to be the set of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p98n8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\it\tau}$]]></tex-math></alternatives></inline-formula> well-approximable points by rationals with denominators in <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p99zd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$Q$]]></tex-math></alternatives></inline-formula> (respectively in <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9b3z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbb{N}$]]></tex-math></alternatives></inline-formula>). We consider the Hausdorff dimension of the liminf set <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p98zx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$]]></tex-math></alternatives></inline-formula> after Adiceam. By using the tools of continued fractions, it is shown that if <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p98ps" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$Q$]]></tex-math></alternatives></inline-formula> is a so-called <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9bpr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbb{N}\setminus Q$]]></tex-math></alternatives></inline-formula>-free set, the Hausdorff dimension of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p98dn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$]]></tex-math></alternatives></inline-formula> is the same as that of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9ckp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$W_{{\it\tau}}$]]></tex-math></alternatives></inline-formula>, i.e. <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9b0d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$2/{\it\tau}$]]></tex-math></alternatives></inline-formula>.</p> </abstract>