Hi! I’m writing just to say that I’ve uploaded a post in the Portuguese version of this blog about the problem of Diophantine properties of elements of non-commutative groups (e.g., any subgroup of invertible matrices such as ). The question of how typical is the ”Diophantine behavior” (in an appropriate sense) among the elements of a certain non-commutative group (say or ) was proposed by Gamburd, Jakobson and Sarnak in their study of the Ruziewicz problem (about the characterization of the Lebesgue measure on the sphere as the unique finitely additive probability which is invariant by rotation). As far as I know, the question is still open although a important partial result was obtained by V. Kaloshin and I. Rodnianski (2001). Roughly speaking, Kaloshin and Rodnianski showed that typical elements of (with respect to the Haar measure) are weakly Diophantine. For the discussion of this topic, you can see this post here (if you can read Portuguese of course!).
It is quite interesting to compare this problem with the issue of the size of an unbalanced product of two (already discussed in this blog). Basically, these problems (as well as the partial results of Kaloshin-Rodnianski and Fayad-Krikorian) are very similar in spirit, as the reader can infer from the statements and the corresponding arguments.
That’s all folks! See you!