Diophantine properties of elements of SO(3)

Abstract

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.