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.
Basic properties of the center of distances of a set are investigated. Computation of the center for achievement sets is particularly aimed at. A new sufficient condition for the center of distances of the set of subsums of a fast convergent series to consist of only 0 and the terms of the series is found. A complete description of the center of distances for some particular type of multigeometric series is provided. In particular, the center of distances C(E) is the union of all centers of distances of C(Fn) where Fn is the set of all ninitial subsums of the discussed multigeometric series satisfying some special requirements. Several open questions are raised.
We study a recently discovered metric invariant - the center of distances. The center of distances of a nonempty subset A of a metric space $$(X,\,d)$$ ( X , d ) is defined by $$S(A) :=\{ \alpha \in [0,\,+\infty ):\ \forall \ x\in A\ \ \exists \ y\in A d(x,\,y)=\alpha \} $$ S ( A ) : = { α ∈ [ 0 , + ∞ ) : ∀ x ∈ A ∃ y ∈ A d ( x , y ) = α } . Given a nonincreasing sequence $$(a_{n})$$ ( a n ) of positive numbers converging to 0, the set $$E(a_{n})\ :=\ \left\{ x\in {\mathbb {R}}:\ \exists A\subset {\mathbb {N}} \ \ x=\,\sum _{n\in A}a_{n}\right\} $$ E ( a n ) : = x ∈ R : ∃ A ⊂ N x = ∑ n ∈ A a n is called the achievement set of the sequence $$(a_{n})$$ ( a n ) . This new invariant is particularly useful in investigating achievability of sets on the real line. We concentrate on computing the centers of distances of central Cantor sets. Any central Cantor set C is an achievement set of exactly one fast convergent series $$ \sum a_{n}$$ ∑ a n , and consequently $$S(C)\supset \left\{ 0\right\} \cup \left\{ a_{n}:n\in {\mathbb {N}}\right\} $$ S ( C ) ⊃ 0 ∪ a n : n ∈ N . We try to check which central Cantor sets have the minimal possible center of distances and which have not.
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