Almost convergent $0$-$1$-sequences and primes

This paper is devoted to $0$-$1$-sequences. We establish the connection between values that Sucheston functional can take on $0$-$1$-sequence and multiplicative structure of the support of the sequence. If the set of all the divisors of support elements is finite, then the sequence is almost convergent to zero. Then we consider characteristic sequences of sets of multiples and establish necessary and sufficient conditions for the upper Sucheston functional to be $1$ on such sequence. We prove that there are infinitely many sets of pairwise relative prime numbers such that the lower Sucheston functional evaluates to $1$ on the corresponding set of multiples (and lower Sucheston functional never evaluates to $0$ on a set of multiples).