Diophantine properties of elements of SO(3)

V. Kaloshin, I. Rodnianski, Geometric And Functional Analysis 11 (2001) 953–970.

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Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2001-12-01
Journal Title
Geometric And Functional Analysis
Volume
11
Issue
5
Page
953-970
IST-REx-ID

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Kaloshin V, Rodnianski I. Diophantine properties of elements of SO(3). Geometric And Functional Analysis. 2001;11(5):953-970. doi:10.1007/s00039-001-8222-8
Kaloshin, V., & Rodnianski, I. (2001). Diophantine properties of elements of SO(3). Geometric And Functional Analysis, 11(5), 953–970. https://doi.org/10.1007/s00039-001-8222-8
Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of SO(3).” Geometric And Functional Analysis 11, no. 5 (2001): 953–70. https://doi.org/10.1007/s00039-001-8222-8.
V. Kaloshin and I. Rodnianski, “Diophantine properties of elements of SO(3),” Geometric And Functional Analysis, vol. 11, no. 5, pp. 953–970, 2001.
Kaloshin V, Rodnianski I. 2001. Diophantine properties of elements of SO(3). Geometric And Functional Analysis. 11(5), 953–970.
Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of SO(3).” Geometric And Functional Analysis, vol. 11, no. 5, Springer Nature, 2001, pp. 953–70, doi:10.1007/s00039-001-8222-8.

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