# Diophantine properties of elements of SO(3)

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A real number x is called diophantine if its distance to rationals p/q is large relative to q -- more precisely, if for every d > 0 there is a positive C such that for every reduced rational p/q, we have |x - p/q| > Cq^{-2-d}, or equivalently |qx-p| > Cq^{-1-d}. Almost all reals have this property. Furthermore, almost every pair (x_1, x_2) has the property that for every d > 0 there is a C such that |q_1x_1+q_2x_2 -p| > C||q||^{-2(1+d)} for all p, q_1, q_2. In this talk, we discuss a noncommutative analog of this property for elements of SO(3). Namely, a pair (A,B) is called diophantine if there exists a constant D such that for every positive integer n and every reduced word W of length n in A, B, A^{-1}, B^{-1}, we have ||W - E|| > D^{-n}, where E is the identity matrix. It is conjectured that almost every such pair (in the sense of Haar measure) is diophantine. We will present a paper of Kaloshin and Rodnianski, in which the weaker bound D^{-n^2} is obtained.