  A number alpha in R is diophantine if it is not well approximable by
rationals, i.e. for some C, nu>0 and any relatively prime p, q in Z we have
|alpha q -p|>C q^{-1-\vu}. It is well-known and easy to prove that almost every
alpha in R is diophantine. In this paper we address a noncommutative version of
the question about diophantine properties. Consider a pair A,B in SO(3) and for
each n in Z_+ take all possible words in A, A^{-1}, B, and B^{-1} of length n,
i.e. for a multiindex I=(i_1,j_1,..., i_s,j_s) put |I|=\sum_{k=1}^s
(|i_k|+|j_k|) and W_{A,B}(n)={W_{I}(A,B)= A^{i_1}B^{j_1}...
A^{i_s}B^{j_s}}_{|I|=n}.
  Gamburd-Jakobson-Sarnak raised the problem: prove that for Haar almost every
pair A, B in SO(3) the closest distance of the word of length n to the
identity, i.e. s_{A,B}(n)=min_{|I|=n} |W_{I}(A,B)-Id|, is bounded from below by
an exponential function in n. This is an analog of diophantine property for
elements of SO(3). In this paper we make first step toward understanding
diophantine properties of SO(3). We prove that s_{A,B}(n) is bounded from below
by an exponential function in n^2. We also exhibit obstructions to prove a
simple exponential in n estimate.
