This paper uses randomization to prove the existence of a subset of the circle which is small in the sense that its Lebesque measure is small, and which is large in the sense that a certain stochastic process almost surely visits the set infinitely often.
"Thin But Unavoidable Sets." Ann. Probab. 9 (1) 154 - 156, February, 1981. https://doi.org/10.1214/aop/1176994516