{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T17:09:21Z","timestamp":1760202561997},"reference-count":18,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2008,8,27]],"date-time":"2008-08-27T00:00:00Z","timestamp":1219795200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2008,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper initiates the study of sets in Euclidean spaces \u211d<jats:sup><jats:italic>n<\/jats:italic> <\/jats:sup> (<jats:italic>n<\/jats:italic> \u2265 2) that are defined in terms of the dimensions of their elements. Specifically, given an interval <jats:italic>I<\/jats:italic> \u2286 [0, <jats:italic>n<\/jats:italic> ], we are interested in the connectivity properties of the set DIM<jats:sup><jats:italic>I<\/jats:italic> <\/jats:sup>, consisting of all points in \u211d<jats:sup><jats:italic>n<\/jats:italic> <\/jats:sup> whose (constructive Hausdorff) dimensions lie in <jats:italic>I<\/jats:italic>, and of its dual DIM<jats:sup><jats:italic>I<\/jats:italic> <\/jats:sup><jats:sub>str<\/jats:sub>, consisting of all points whose strong (constructive packing) dimensions lie in <jats:italic>I<\/jats:italic>. If <jats:italic>I<\/jats:italic> is [0, 1) or (<jats:italic>n<\/jats:italic> \u2013 1, <jats:italic>n<\/jats:italic> ], it is easy to see that the sets DIM<jats:sup><jats:italic>I<\/jats:italic> <\/jats:sup> and DIM<jats:sup><jats:italic>I<\/jats:italic> <\/jats:sup><jats:sub>str<\/jats:sub> are totally disconnected. In contrast, we show that if <jats:italic>I<\/jats:italic> is [0, 1] or [<jats:italic>n<\/jats:italic> \u2013 1, <jats:italic>n<\/jats:italic> ], then the sets DIM<jats:sup><jats:italic>I<\/jats:italic> <\/jats:sup> and DIM<jats:sup><jats:italic>I<\/jats:italic> <\/jats:sup><jats:sub>str<\/jats:sub> are path\u2010connected. Our proof of this fact uses geometric properties of Kolmogorov complexity in Euclidean spaces. (\u00a9 2008 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200710060","type":"journal-article","created":{"date-parts":[[2008,8,27]],"date-time":"2008-08-27T09:57:26Z","timestamp":1219831046000},"page":"483-491","source":"Crossref","is-referenced-by-count":9,"title":["Connectivity properties of dimension level sets"],"prefix":"10.1002","volume":"54","author":[{"given":"Jack H.","family":"Lutz","sequence":"first","affiliation":[]},{"given":"Klaus","family":"Weihrauch","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2008,8,27]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539703446912"},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1215\/ijm\/1255455863","article-title":"Hausdorff dimension in probability theory","volume":"4","author":"Billingsley P.","year":"1960","journal-title":"Illinois J. 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Available athttp:\/\/www.cs.uwyo.edu\/\u223cjhitchco\/bib\/dim.shtml."},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00224-004-1122-1"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1145\/1227839.1227845"},{"key":"e_1_2_1_12_2","doi-asserted-by":"crossref","unstructured":"M.Li andP. M. B.Vit\u00e1nyi An Introduction to Kolmogorov Complexity and its Applications second edition (Springer 1997).","DOI":"10.1007\/978-1-4757-2606-0"},{"key":"e_1_2_1_13_2","doi-asserted-by":"crossref","unstructured":"J. H.Lutz Gales and the constructive dimension of individual sequences. In: Proceedings of the 27th International Colloquium on Automata Languages and Programming. Lecture Notes in Computer Science 1853 pp. 902\u2013913 (Springer 2000). Revised as [13].","DOI":"10.1007\/3-540-45022-X_76"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0890-5401(03)00187-1"},{"key":"e_1_2_1_15_2","unstructured":"J. H.Lutz andE.Mayordomo Dimensions of points in self\u2010similar fractals. To appear in the SIAM Journal on Computing. Preliminary version available athttp:\/\/www.cs.iastate.edu\/\u223clutz\/=PAPERS\/dpssf.pdf."},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0190(02)00343-5"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100022684"},{"key":"e_1_2_1_18_2","unstructured":"M.van Lambalgen Random sequences. Ph. 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