Binomial-coefficient multiples of irrationals
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Denote by $x$ a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by $d_n(x)$ the binomial coefficient at the $n$'th level along the path $x$. Then for a dense $G_δ$ set of $θ$ in the unit interval, $\{d_n(x)θ\}$ is almost surely dense but not uniformly distributed modulo 1.
10 pages, to appear in Monatshefte f. Math
10 pages, to appear in Monatshefte f. Math