{"volume":11,"date_published":"2001-12-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"953-970","publication_identifier":{"issn":["1016-443X","1420-8970"]},"date_created":"2020-09-18T10:50:11Z","date_updated":"2021-01-12T08:19:52Z","extern":"1","author":[{"last_name":"Kaloshin","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","full_name":"Kaloshin, Vadim","first_name":"Vadim","orcid":"0000-0002-6051-2628"},{"full_name":"Rodnianski, I.","first_name":"I.","last_name":"Rodnianski"}],"quality_controlled":"1","month":"12","publication":"Geometric And Functional Analysis","abstract":[{"lang":"eng","text":"A number α∈R is diophantine if it is not well approximable by rationals, i.e. for some C,ε>0 and any relatively prime p,q∈Z we have |αq−p|>Cq−1−ε. It is well-known and is easy to prove that almost every α in R is diophantine. In this paper we address a noncommutative version of the diophantine properties. Consider a pair A,B∈SO(3) and for each n∈Z+ take all possible words in A, A -1, B, and B - 1 of length n, i.e. for a multiindex I=(i1,i1,…,im,jm) define |I|=∑mk=1(|ik|+|jk|)=n and \\( W_n(A,B ) = \\{W_{\\cal I}(A,B) = A^{i_1} B^{j_1} \\dots A^{i_m} B^{j_m}\\}_{|{\\cal I|}=n \\).¶Gamburd—Jakobson—Sarnak [GJS] raised the problem: prove that for Haar almost every pair A,B∈SO(3) the closest distance of words of length n to the identity, i.e. sA,B(n)=min|I|=n∥WI(A,B)−E∥, is bounded from below by an exponential function in n. This is the analog of the diophantine property for elements of SO(3). In this paper we prove that s A,B (n) is bounded from below by an exponential function in n 2. We also exhibit obstructions to a “simple” proof of the exponential estimate in n."}],"article_type":"original","language":[{"iso":"eng"}],"intvolume":"        11","oa_version":"None","title":"Diophantine properties of elements of SO(3)","article_processing_charge":"No","doi":"10.1007/s00039-001-8222-8","type":"journal_article","citation":{"ama":"Kaloshin V, Rodnianski I. Diophantine properties of elements of SO(3). <i>Geometric And Functional Analysis</i>. 2001;11(5):953-970. doi:<a href=\"https://doi.org/10.1007/s00039-001-8222-8\">10.1007/s00039-001-8222-8</a>","mla":"Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of SO(3).” <i>Geometric And Functional Analysis</i>, vol. 11, no. 5, Springer Nature, 2001, pp. 953–70, doi:<a href=\"https://doi.org/10.1007/s00039-001-8222-8\">10.1007/s00039-001-8222-8</a>.","ista":"Kaloshin V, Rodnianski I. 2001. Diophantine properties of elements of SO(3). Geometric And Functional Analysis. 11(5), 953–970.","apa":"Kaloshin, V., &#38; Rodnianski, I. (2001). Diophantine properties of elements of SO(3). <i>Geometric And Functional Analysis</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00039-001-8222-8\">https://doi.org/10.1007/s00039-001-8222-8</a>","short":"V. Kaloshin, I. Rodnianski, Geometric And Functional Analysis 11 (2001) 953–970.","chicago":"Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of SO(3).” <i>Geometric And Functional Analysis</i>. Springer Nature, 2001. <a href=\"https://doi.org/10.1007/s00039-001-8222-8\">https://doi.org/10.1007/s00039-001-8222-8</a>.","ieee":"V. Kaloshin and I. Rodnianski, “Diophantine properties of elements of SO(3),” <i>Geometric And Functional Analysis</i>, vol. 11, no. 5. Springer Nature, pp. 953–970, 2001."},"year":"2001","publisher":"Springer Nature","_id":"8524","publication_status":"published","issue":"5","status":"public","day":"01"}