On strictly sparse subsets of a free group

This paper is motivated by needs of practical computations in finitely generated groups. In the most of the computations in finitely generated groups $G$ the elements are represented as freely reduced words in the free group $F$. In [1] a family of probability measures was used for estimating the complexity of algorithms on groups and subsets of $F$ were classified according to these measures. We find out which regular sets are sparse, i.e. small with respect to the probability measures.