{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:34:42Z","timestamp":1774625682821,"version":"3.50.1"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2003,12]]},"abstract":"<jats:p>We introduce a family of multiplicative distributions {\u03bc<jats:sub>s<\/jats:sub>|s\u2208(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R\u2286F is a function \u03bc(R) : s \u2192 \u03bc<jats:sub>s<\/jats:sub>(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions \u03bc(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function \u03bc(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.<\/jats:p>","DOI":"10.1142\/s0218196703001596","type":"journal-article","created":{"date-parts":[[2004,1,27]],"date-time":"2004-01-27T09:43:48Z","timestamp":1075196628000},"page":"705-731","source":"Crossref","is-referenced-by-count":22,"title":["MULTIPLICATIVE MEASURES ON FREE GROUPS"],"prefix":"10.1142","volume":"13","author":[{"given":"ALEXANDRE V.","family":"BOROVIK","sequence":"first","affiliation":[{"name":"Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, United Kingdom"}]},{"given":"ALEXEI G.","family":"MYASNIKOV","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The City College of New York, New York, NY 10031, USA"}]},{"given":"VLADIMIR N.","family":"REMESLENNIKOV","sequence":"additional","affiliation":[{"name":"Omsk Branch of Mathematical Institute SB RAS, 13 Pevtsova Street, Omsk 644099, Russia"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"3205","volume":"12","author":"Arzhantseva G. N.","journal-title":"Proc. Amer. Math. Soc."},{"key":"rf2","first-page":"83","volume":"45","author":"Bartholdi L.","journal-title":"Ensign. Math."},{"key":"rf5","unstructured":"A. V.\u00a0Borovik, A. G.\u00a0Myasnikov and V.\u00a0Shpilrain, Computational and Statistical Group Theory, Contemporary Math\u00a0298 (Amer. Math. Soc., Providence, RI)\u00a0pp. 21\u201342."},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1995.1067"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)72023-8"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(82)90090-8"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388581"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1201\/9781439865699"},{"key":"rf12","volume-title":"Algebraic Combinatorics","author":"Godsil C. D.","year":"1993"},{"key":"rf13","first-page":"217","volume":"32","author":"Grigorchuk R. I.","journal-title":"Russian Math. Surv."},{"key":"rf15","first-page":"490","volume":"31","author":"Grigorchuk R. I.","journal-title":"Ukrainian Math. J."},{"key":"rf16","unstructured":"R. I.\u00a0Grigorchuk, Multicomponent Random Systems, eds. R. L.\u00a0Dobrushin and Ya. G.\u00a0Sinai (Dekker, New York, 1980)\u00a0pp. 285\u2013325."},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698687"},{"key":"rf18","volume-title":"Divergent Series","author":"Hardy G. H.","year":"1991"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.2001.9033"},{"key":"rf20","volume-title":"Denumerable Markov Chains","author":"Kemeny J. G.","year":"1966"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58822-8"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-98-04741-8"},{"key":"rf23","series-title":"Ann. Math. Stud.","volume":"68","author":"Miller C. F.","year":"1971"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(83)90003-X"},{"key":"rf25","doi-asserted-by":"crossref","unstructured":"A. D.\u00a0Myasnikov and A. G.\u00a0Myasnikov, Groups and Computation III, eds. W.\u00a0Kantor and A.\u00a0Seress (de Gruyter, Berlin, 2001)\u00a0pp. 257\u2013264.","DOI":"10.1515\/9783110872743.257"},{"key":"rf26","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/33\/32\/306"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196792000025"},{"key":"rf29","volume-title":"A First Course in Formal Language Theory","author":"Rayward-Smith V. J.","year":"1983"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511609589"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/173\/01830"},{"key":"rf33","volume-title":"Cambridge Tracts in Mathematics","volume":"100","author":"Varopoulos N.","year":"1992"},{"key":"rf34","doi-asserted-by":"publisher","DOI":"10.1007\/BF01371408"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196703001596","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,12]],"date-time":"2024-01-12T01:08:25Z","timestamp":1705021705000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196703001596"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,12]]},"references-count":28,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2003,12]]}},"alternative-id":["10.1142\/S0218196703001596"],"URL":"https:\/\/doi.org\/10.1142\/s0218196703001596","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,12]]}}}