{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T11:14:47Z","timestamp":1686222887255},"reference-count":18,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2014,12]]},"abstract":"<jats:p>We examine the sequences A that are low for dimension, i.e. those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-L\u00f6f random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of K<jats:sup>A<\/jats:sup>(n)\/K(n) is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order n<jats:sup>\u03b5<\/jats:sup>, for any \u03b5 &gt; 0.<\/jats:p>","DOI":"10.1142\/s0219061314500111","type":"journal-article","created":{"date-parts":[[2014,12,15]],"date-time":"2014-12-15T10:36:29Z","timestamp":1418639789000},"page":"1450011","source":"Crossref","is-referenced-by-count":2,"title":["Lowness for effective Hausdorff dimension"],"prefix":"10.1142","volume":"14","author":[{"given":"Steffen","family":"Lempp","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph S.","family":"Miller","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Keng Meng","family":"Ng","sequence":"additional","affiliation":[{"name":"Division of Mathematical Sciences, School of Physical and Mathematical Sciences, College of Science, Nanyang Technological University, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel D.","family":"Turetsky","sequence":"additional","affiliation":[{"name":"Kurt G\u00f6del Research Center, University of Vienna, 1090 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rebecca","family":"Weber","sequence":"additional","affiliation":[{"name":"White River Junction, VT 05001, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2015,1,28]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-012-0012-5"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(76)90005-0"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1147\/rd.214.0350"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-011-0217-z"},{"key":"rf7","volume-title":"Theory and Applications of Computability","author":"Downey R. 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