{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T08:16:23Z","timestamp":1770884183038,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2017,4,27]],"date-time":"2017-04-27T00:00:00Z","timestamp":1493251200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Theory Comput Syst"],"published-print":{"date-parts":[[2018,10]]},"DOI":"10.1007\/s00224-017-9767-8","type":"journal-article","created":{"date-parts":[[2017,4,27]],"date-time":"2017-04-27T05:49:04Z","timestamp":1493272144000},"page":"1573-1585","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Liouville, Computable, Borel Normal and Martin-L\u00f6f Random Numbers"],"prefix":"10.1007","volume":"62","author":[{"given":"Cristian S.","family":"Calude","sequence":"first","affiliation":[]},{"given":"Ludwig","family":"Staiger","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,4,27]]},"reference":[{"key":"9767_CR1","unstructured":"Abbott, A.: Inapplicability of certain correlations in Borel-normal sequences, manuscript, 2 January 2014, 3 pages"},{"issue":"4","key":"9767_CR2","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1080\/10586458.2002.10504704","volume":"11","author":"DH Bailey","year":"2002","unstructured":"Bailey, D.H., Crandall, R.E.: Random generators and normal numbers. Experiment. Math. 11(4), 527\u2013546 (2002)","journal-title":"Experiment. Math."},{"key":"9767_CR3","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1090\/proc\/12841","volume":"144","author":"V Becher","year":"2016","unstructured":"Becher, V., Bugeaud, Y., Slaman, T.A.: The irrationality exponents of computable numbers. Proceedings of the American Mathematical Society 144, 1509\u20131521 (2016)","journal-title":"Proceedings of the American Mathematical Society"},{"key":"9767_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ic.2013.08.013","volume":"232","author":"V Becher","year":"2013","unstructured":"Becher, V., Heiber, P., Slaman, T.A.: A polynomial-time algorithm for computing absolutely normal numbers. Information and Computation 232, 1\u20139 (2013)","journal-title":"Information and Computation"},{"key":"9767_CR5","doi-asserted-by":"crossref","first-page":"2939","DOI":"10.1090\/mcom\/2964","volume":"86","author":"V Becher","year":"2015","unstructured":"Becher, V., Heiber, P., Slaman, T.A.: A computable absolutely normal Liouville number. Math. Comput. 86, 2939\u20132952 (2015)","journal-title":"Math. Comput."},{"key":"9767_CR6","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF03019651","volume":"27","author":"\u00c9 Borel","year":"1909","unstructured":"Borel, \u00c9. : Les probabilit\u00e9s d\u00e9nombrables et leurs applications arithm\u00e9tiques. Rendiconti del Circolo Matematico di Palermo 27, 247\u2013271 (1909)","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"9767_CR7","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511902659","volume-title":"Neverending Fractions. an Introduction to Continued Fractions","author":"J Borwein","year":"2014","unstructured":"Borwein, J., van der Poorten, A., Shallit, J., Zudilin, W.: Neverending Fractions. an Introduction to Continued Fractions. Cambridge University Press, Cambridge (2014)"},{"key":"9767_CR8","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/S1631-073X(02)02456-1","volume":"335","author":"Y Bugeaud","year":"2002","unstructured":"Bugeaud, Y.: Nombres de Liouville et nombres normaux. C. R. Acad. Sci. Paris, Ser. I 335, 117\u2013120 (2002)","journal-title":"C. R. Acad. Sci. Paris, Ser. I"},{"key":"9767_CR9","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-04978-5","volume-title":"Information and Randomness\u2014An Algorithmic Perspective","author":"CS Calude","year":"2002","unstructured":"Calude, C.S.: Information and Randomness\u2014An Algorithmic Perspective, 2nd ed. Springer, Berlin (2002)","edition":"2nd ed."},{"key":"9767_CR10","first-page":"27","volume":"2","author":"C Calude","year":"1988","unstructured":"Calude, C., Chi\u0163escu, I.: Random sequences: some topological and measure-theoretical properties. An. Univ. Bucure\u015fti Mat.-Inf. 2, 27\u201332 (1988)","journal-title":"An. Univ. Bucure\u015fti Mat.-Inf."},{"key":"9767_CR11","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/3-540-58131-6_37","volume-title":"Results and Trends in Theoretical Computer Science, Lecture Notes in Comput. Sci., Vol. 812","author":"C Calude","year":"1994","unstructured":"Calude, C., J\u00fcrgensen, H., Rozenberg, G.: Randomness as an invariant for number representations. In: Maurer, H., Karhum\u00e4ki, J. (eds.) Results and Trends in Theoretical Computer Science, Lecture Notes in Comput. Sci., Vol. 812, pp. 44\u201366. Springer-Verlag, Berlin (1994)"},{"key":"9767_CR12","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.ipl.2003.07.002","volume":"88","author":"CS Calude","year":"2003","unstructured":"Calude, C.S., Marcus, S., Staiger, L.: A topological characterization of random sequences, Inform. Process. Lett. 88, 245\u2013250 (2003)","journal-title":"Process. Lett."},{"key":"9767_CR13","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.apal.2005.06.004","volume":"138","author":"CS Calude","year":"2006","unstructured":"Calude, C.S., Staiger, L., Terwijn, S.A.: On partial randomness. Ann. Appl. Pur. Log. 138, 20\u201330 (2006)","journal-title":"Ann. Appl. Pur. Log."},{"key":"9767_CR14","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1112\/jlms\/s1-8.4.254","volume":"8","author":"DG Champernowne","year":"1933","unstructured":"Champernowne, D.G.: The construction of decimals normal in the scale of ten. J. Lond. Math. Soc. 8, 254\u2013260 (1933)","journal-title":"J. Lond. Math. Soc."},{"issue":"1","key":"9767_CR15","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s11139-015-9685-9","volume":"37","author":"M Coons","year":"2015","unstructured":"Coons, M.: Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers. Ramanujan J. 37(1), 109\u2013111 (2015)","journal-title":"Ramanujan J."},{"key":"9767_CR16","first-page":"758","volume":"49","author":"N de Bruijn","year":"1946","unstructured":"de Bruijn, N.: A combinatorial problem. Proc. Kon. Nederl. Akad. Wetensch. 49, 758\u2013764 (1946)","journal-title":"Proc. Kon. Nederl. Akad. Wetensch."},{"key":"9767_CR17","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-68441-3","volume-title":"Algorithmic Randomness and Complexity","author":"R Downey","year":"2010","unstructured":"Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Heidelberg (2010)"},{"key":"9767_CR18","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-74749-1","volume-title":"Measure, Topology, and Fractal Geometry","author":"G Edgar","year":"2008","unstructured":"Edgar, G.: Measure, Topology, and Fractal Geometry. Springer, New York (2008)"},{"key":"9767_CR19","volume-title":"Fractal Geometry","author":"K Falconer","year":"1990","unstructured":"Falconer, K.: Fractal Geometry. Wiley, New York (1990)"},{"key":"9767_CR20","volume-title":"Combinatorial Theory","author":"MJr Hall","year":"1967","unstructured":"Hall, M. Jr.: Combinatorial Theory. Blaisdell, Waltham (1967)"},{"issue":"1","key":"9767_CR21","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/S0890-5401(02)00034-2","volume":"181","author":"P Hertling","year":"2003","unstructured":"Hertling, P., Weihrauch, K.: Random elements in effective topological spaces with measure. Information and Computation 181(1), 32\u201356 (2003)","journal-title":"Information and Computation"},{"key":"9767_CR22","first-page":"371","volume":"36","author":"V Jarn\u00edk","year":"1929","unstructured":"Jarn\u00edk, V.: Diophantische Approximationen und Hausdorffsches Ma\u00df. Rec. Math. Moscou 36, 371\u2013382 (1929)","journal-title":"Rec. Math. Moscou"},{"key":"9767_CR23","unstructured":"J\u00fcrgensen, H., Thierrin, G.: Some structural properties of \u00fa-languages 13th Nat. School with Internat. Participation \u201cApplications of Mathematics in Technology\u201d, pp. 56\u201363, Sofia (1988)"},{"key":"9767_CR24","doi-asserted-by":"crossref","first-page":"265","DOI":"10.4064\/aa-39-3-265-267","volume":"39","author":"R Kaufman","year":"1981","unstructured":"Kaufman, R.: On the theorem of Jarn\u00edk and Besicovitch. Acta Arithmetica 39, 265\u2013267 (1981)","journal-title":"Acta Arithmetica"},{"issue":"10","key":"9767_CR25","first-page":"947","volume":"98","author":"MJ Knight","year":"1991","unstructured":"Knight, M.J.: An \u201cOceans of Zeros\u201d proof that a certain Non-Liouville number is transcendental. American Mathematical Monthly 98(10), 947\u2013949 (1991)","journal-title":"American Mathematical Monthly"},{"key":"9767_CR26","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1109\/TIT.1968.1054210","volume":"14","author":"AN Kolmogorov","year":"1968","unstructured":"Kolmogorov, A.N.: Logical basis for information theory and probability theory. IEEE Trans. Inform. Theory 14, 662\u2013664 (1968)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9767_CR27","first-page":"883","volume":"18","author":"J Liouville","year":"1844","unstructured":"Liouville, J.: M\u00e9moires et Communications des Membres et des Correspondants de l\u2019Acad\u00e9mie. C. R. Acad. Sci. 18, 883\u2013885 (1844)","journal-title":"C. R. Acad. Sci."},{"key":"9767_CR28","first-page":"410","volume":"138","author":"E Maillet","year":"1904","unstructured":"Maillet, E.: Sur les nombres quasi-rationnels et les fractions arithm\u00e9tiques ordinaires ou continues quasi-p\u00e9riodiques. C. R. Acad. Sci. Paris 138, 410\u2013411 (1904)","journal-title":"C. R. Acad. Sci. Paris"},{"issue":"8","key":"9767_CR29","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1080\/00029890.2001.11919806","volume":"108","author":"G Martin","year":"2001","unstructured":"Martin, G.: Absolutely abnormal numbers. American Mathematical Monthly 108(8), 746\u2013754 (2001)","journal-title":"American Mathematical Monthly"},{"key":"9767_CR30","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/S0019-9958(66)80018-9","volume":"9","author":"P Martin-L\u00f6f","year":"1966","unstructured":"Martin-L\u00f6f, P.: The definition of random sequences. Inf. Control. 9, 602\u2013619 (1966)","journal-title":"Inf. Control."},{"key":"9767_CR31","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.entcs.2008.12.016","volume":"221","author":"J Mielke","year":"2008","unstructured":"Mielke, J.: Refined bounds on Kolmogorov complexity for \u03c9-languages. Electr. Notes Theor. Comput. Sci. 221, 181\u2013189 (2008)","journal-title":"Electr. Notes Theor. Comput. Sci."},{"key":"9767_CR32","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1007\/s00224-014-9554-8","volume":"58","author":"S Nandakumar","year":"2016","unstructured":"Nandakumar, S., Vangapelli, S.K.: Normality and finite-state dimension of Liouville numbers. Theory of Computing Systems 58, 392\u2013402 (2016)","journal-title":"Theory of Computing Systems"},{"key":"9767_CR33","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9339-9","volume-title":"Measure and Category","author":"JC Oxtoby","year":"1980","unstructured":"Oxtoby, J.C.: Measure and Category, 2nd ed. Springer-Verlag, Berlin (1980)","edition":"2nd ed."},{"issue":"3","key":"9767_CR34","first-page":"16","volume":"22","author":"B Ya","year":"1986","unstructured":"Ya, B.: Ryabko. Noiseless coding of combinatorial sources, Hausdorff dimension and Kolmogorov complexity. Problemy Peredachi Informatsii 22(3), 16\u201326 (1986)","journal-title":"Problemy Peredachi Informatsii"},{"key":"9767_CR35","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-94694-3","volume-title":"Einf\u00fchrung in Die Transzendenten Zahlen","author":"T Schneider","year":"1957","unstructured":"Schneider, T.: Einf\u00fchrung in Die Transzendenten Zahlen. Springer-Verlag, Berlin (1957)"},{"key":"9767_CR36","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1006\/inco.1993.1017","volume":"103","author":"L Staiger","year":"1993","unstructured":"Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Inf. Comput. 103, 159\u2013194 (1993)","journal-title":"Inf. Comput."},{"key":"9767_CR37","first-page":"455","volume":"284","author":"L Staiger","year":"2002","unstructured":"Staiger, L.: The Kolmogorov complexity of real numbers, Theoret. Comput. Sci. 284, 455\u2013466 (2002)","journal-title":"Comput. Sci."},{"key":"9767_CR38","volume-title":"A Course in Combinatorics","author":"JH van Lint","year":"1993","unstructured":"van Lint, J.H., Wilson, R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (1993)"}],"container-title":["Theory of Computing Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00224-017-9767-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00224-017-9767-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00224-017-9767-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,7,24]],"date-time":"2018-07-24T05:24:09Z","timestamp":1532409849000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00224-017-9767-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,27]]},"references-count":38,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2018,10]]}},"alternative-id":["9767"],"URL":"https:\/\/doi.org\/10.1007\/s00224-017-9767-8","relation":{},"ISSN":["1432-4350","1433-0490"],"issn-type":[{"value":"1432-4350","type":"print"},{"value":"1433-0490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,4,27]]}}}