On the number of ordered pairs of $l$-balanced sets of length $n$

We introduce the notion of ordered pairs of $l$-balanced binary vectors of length $n$. This notion characterizes the closeness of the pair of vectors in a natural way. We give exact and asymptotic formulae for the cardinality of the set of all ordered pairs of $l$-balanced vectors of length $n$.
This work is supported by the Russian Foundation for Fundamental Investigations, grant 93–01–1527.