# Diophantine properties of elements of SO(3)

Kaloshin V, Rodnianski I. 2001. Diophantine properties of elements of SO(3). Geometric And Functional Analysis. 11(5), 953–970.

Download

**No fulltext has been uploaded. References only!**

*Journal Article*|

*Published*|

*English*

Author

Kaloshin, Vadim

^{IST Austria}^{}; Rodnianski, I.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

### Cite this

Kaloshin V, Rodnianski I. Diophantine properties of elements of SO(3).

*Geometric And Functional Analysis*. 2001;11(5):953-970. doi:10.1007/s00039-001-8222-8Kaloshin, V., & Rodnianski, I. (2001). Diophantine properties of elements of SO(3).

*Geometric And Functional Analysis*. Springer Nature. https://doi.org/10.1007/s00039-001-8222-8Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of SO(3).”

*Geometric And Functional Analysis*. Springer Nature, 2001. 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. Springer Nature, pp. 953–970, 2001.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.