Continued Fractions and Heavy Sequences
David Ralson, Rice

A sequence of points in the unit interval is said to be heavy with respect to a set A if the average number of points in the sequence in A dominates the measure of A for all finite averages. We will prove that for x irrational, the sequence {nx mod(1)}, for n=1,2,3,..., is heavy for the interval [0,1/k) if and only if all odd-indexed terms in the continued fraction expansion of x are zero modulo k. We will prove this result using simple ergodic theoretic techniques, and it is a rederivation of a result from joint work with M. Boshernitzan.