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We show that the Cantorvals connected with the geometric Cantor sets are not achievement sets of any series. However many of them are attractors of IFS consisting of affine functions.

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The Shao-Sablin index of a Λ-sequence Λ = (λ i ) is defined by S Λ := lim sup n→∞The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if S Λ < 2. Also, ΛBV = ΛBV c if and only if S Λ < 2, where ΛBV c denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia-Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.Functions of bounded Λ-variation are a natural generalization of functions of bounded variation and were introduced by D. Waterman in order to extend a number of results related to Fourier series of a function, in particular, to essentially generalize the Dirichlet-Jordan Test (see [17,18,20]).