{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9347","name":"Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension","abstract":"We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways.\r\n\r\n1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2.\r\n\r\n2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y) and Mdim(x:y).","keywords":["algorithmic randomness","conditional dimension","geometric measure theory","Kakeya sets","Kolmogorov complexity"],"author":[{"@type":"Person","name":"Lutz, Jack H.","givenName":"Jack H.","familyName":"Lutz"},{"@type":"Person","name":"Lutz, Neil","givenName":"Neil","familyName":"Lutz"}],"position":53,"pageStart":"53:1","pageEnd":"53:13","dateCreated":"2017-03-06","datePublished":"2017-03-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Lutz, Jack H.","givenName":"Jack H.","familyName":"Lutz"},{"@type":"Person","name":"Lutz, Neil","givenName":"Neil","familyName":"Lutz"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2017.53","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1137\/S0097539703446912","http:\/\/dx.doi.org\/10.1109\/FOCS.2006.63","http:\/\/dx.doi.org\/10.1137\/070684689"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6269","volumeNumber":66,"name":"34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)","dateCreated":"2017-03-06","datePublished":"2017-03-06","editor":[{"@type":"Person","name":"Vollmer, Heribert","givenName":"Heribert","familyName":"Vollmer"},{"@type":"Person","name":"Vall\u00e9e, Brigitte","givenName":"Brigitte","familyName":"Vall\u00e9e"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9347","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6269"}}}