Card shuffling and Diophantine approximation

The .overlapping-cycles shuffle. mixes a deck of n cards by moving either the nth card or the (n.k)th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of k and n, has surprising behavior. For example, suppose k is the closest integer to .n for a fixed real ..(0, 1). Then for rational . the spectral gap is .(n.2), while for poorly approximable irrational numbers ., such as the reciprocal of the golden ratio, the spectral gap is .(n.3/2).