---
_id: '8524'
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_processing_charge: No
article_type: original
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: I.
  full_name: Rodnianski, I.
  last_name: Rodnianski
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>
  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>
  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.
  ista: Kaloshin V, Rodnianski I. 2001. Diophantine properties of elements of SO(3).
    Geometric And Functional Analysis. 11(5), 953–970.
  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>.
  short: V. Kaloshin, I. Rodnianski, Geometric And Functional Analysis 11 (2001) 953–970.
date_created: 2020-09-18T10:50:11Z
date_published: 2001-12-01T00:00:00Z
date_updated: 2021-01-12T08:19:52Z
day: '01'
doi: 10.1007/s00039-001-8222-8
extern: '1'
intvolume: '        11'
issue: '5'
language:
- iso: eng
month: '12'
oa_version: None
page: 953-970
publication: Geometric And Functional Analysis
publication_identifier:
  issn:
  - 1016-443X
  - 1420-8970
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Diophantine properties of elements of SO(3)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2001'
...