M-Cantorvals of Ferens type

Abstract

A special family of multigeometric series is considered from the point of view of behaviour of their sets of subsums. A sufficient condition for their sets of subsums to be M-Cantorvals is proven. The Lebesgue measure of those special M-Cantorvals is computed and it is shown to be equal to the sum of lengths of all component intervals of the M-Cantorvals. A new sufficient condition for the set of subsums of a series to be a Cantor set is formulated and it is used to demonstrate that the discussed multigeometric series always have Cantor sets as their sets of subsums for sufficiently small ratios of the series.

Topological and measure properties of some self-similar sets

“…Obviously, for n greater then some n 0 2 n−1 s > n and hence 2 n−1 s n > n s n−1 which proves (2). , then there is a decreasing sequence (q n ) ∞ n=1 tending to 1 |Σ| such that, for any n ∈ N, the self-similar set K(Σ, q n ) has Lebesgue measure zero.…”

“…with m ≥ k we will call, after [2], Ferens-like sequences. The achievement set E (x) for a Ferens-like sequence coincides with the self-similar set K(Σ; q) for the set…”

the center of distances for some multigeometric series

“…Such Cantorvals do exist. For example, it suffices to take the Ferens type series F (2, 5; 1 16 ) (see [2,Lemma 7]). Since the set n C(F n ) is countable for any series a n , the equality C(E) = n C(F n ) does not hold for any series satisfying the assumptions of the Fact 2.6.…”

“…A series a n is said to be fast convergent if a n > r n for all n. Fast convergence of a n is a sufficient condition for E to be a Cantor set ( [13], [14], [20]). Some other sufficient conditions for E to be a Cantor set can be found in [1], [2] and [7].…”

Conditions for the difference set of a central Cantor set to be a Cantorval

“…Recently, there have appeared a lot of interesting papers that explore properties of achievement sets, i.a. [5], [7], [27], [9], and [6].…”