114 research outputs found
Diophantine property in the group of affine transformations of the line
We investigate the Diophantine property of a pair of elements in the group of
affine transformations of the line. We say that a pair of elements g_1,g_2 in
this group is Diophantine if there is a number A such that a product of length
l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element
or of distance at least A^{-l} from the unit element. We prove that the set of
non-Diophantine pairs in a certain one parameter family is of Hausdorff
dimension 0.Comment: 12 pages, no figures, reference to [ABRS] update
Random walks in compact groups
Let X_1,X_2,... be independent identically distributed random elements of a
compact group G. We discuss the speed of convergence of the law of the product
X_l*...*X_1 to the Haar measure. We give poly-log estimates for certain finite
groups and for compact semi-simple Lie groups. We improve earlier results of
Solovay, Kitaev, Gamburd, Shahshahani and Dinai.Comment: 35 pages, no figures, revision based on referee's report, results and
proofs unchange
- …