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We investigate some self-similar Cantor sets $C(l,r,p)$, which we callS-Cantor sets, generated by numbers $l,r,p \in \mathbb{N}$, $l+r<p$. We give afull characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take oneof the form: the interval $[-1,1]$, a Cantor set, an L-Cantorval, anR-Cantorval or an M-Cantorval. As corollaries we give examples of Cantor setsand Cantorvals, which can be easily described using some positional numeralsystems.

Characterization of the algebraic difference of special affine Cantor sets