On the uniform distribution of remainders in the expression of real numbers over a multiplicative system numbers

In this paper theorems on the expression of real numbers on multiplicative number system. It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations. Here is found that the sequence remainders in this expansion has the uniform distribution. The given statement generalises the known result of Hardy–Littlewood for a positional system of calculus. On the base of proof lie two statement: the Weyl's criteria of the uniform distribution of a sequence modulo unit and the theoretic-probability lemma of Borel–Cantelli.