We determine all triples $(α, β, λ)$ such that $C_α- λ C_β $ forms a closed interval, where $C_α$ and $C_β$ are middle Cantor sets. This follows from a new recurrence type result for certain renormalization operators. We also consider the affine Cantor sets $K$ and $K'$ defined by two increasing maps which the product of their thicknesses is bigger than one. Then we construct a recurrent set for their renormalization operators. This leads us to characterize all $λ$ that $K- λ K' $ is a closed interval.
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Figure 1. Blue region and curves determine all $\alpha, \beta$ that $C_\alpha+C_\beta= [0, ~2] $. For instant, the curves $r_{3, 1}, r_{2, 1}, r_{3, 2}, r_{4, 3}, r_{5, 4}$ are selected, that we characterized them by the functions $\beta=\alpha ^\frac{3}{1}, \alpha ^\frac{2}{1}, \alpha ^\frac{3}{2}, \alpha ^\frac{4}{3}, \alpha ^\frac{5}{4}$, respectively. The yellow Curves are the graph of these functions from downside to upside which have been drawn by Maple program
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Blue region and curves determine all
Gray region illustrates the recurrent set
For middle Cantor sets, the sets
The graph of map
The functions