### 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-∈}. 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 script I sign = (i_{1}, j_{1}, . . . , i_{m}, j_{m}) define |script I sign| = ∑^{m}_{k=1}(|i_{k}|+|j_{k}|) = n and W_{n}(A, B) = {W_{script I sign}(A, B) = A^{i1}B^{j1} . . . A^{im}B^{jm}|_{|script I sign|=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. s_{A, B}(n) = min_{|script I sign|=n} ||W_{script I sign}(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.

Original language | English (US) |
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Pages (from-to) | 953-970 |

Number of pages | 18 |

Journal | Geometric and Functional Analysis |

Volume | 11 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2001 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

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## Cite this

*Geometric and Functional Analysis*,

*11*(5), 953-970. https://doi.org/10.1007/s00039-001-8222-8