Abstract
AbstractLet $$C(\lambda )\subset [0,1]$$
C
(
λ
)
⊂
[
0
,
1
]
denote the central Cantor set generated by a sequence $$\lambda =(\lambda _{n})\in \left( 0,\frac{1}{2} \right) ^{\mathbb {N}}$$
λ
=
(
λ
n
)
∈
0
,
1
2
N
. By the known trichotomy, the difference set $$ C(\lambda )-C(\lambda )$$
C
(
λ
)
-
C
(
λ
)
of $$C(\lambda )$$
C
(
λ
)
is one of three possible sets: a finite union of closed intervals, a Cantor set, or a Cantorval. Our main result describes effective conditions for $$(\lambda _{n})$$
(
λ
n
)
which guarantee that $$C(\lambda )-C(\lambda )$$
C
(
λ
)
-
C
(
λ
)
is a Cantorval. We show that these conditions can be expressed in several equivalent forms. Under additional assumptions, the measure of the Cantorval $$C(\lambda )-C(\lambda )$$
C
(
λ
)
-
C
(
λ
)
is established. We give an application of the proved theorems for the achievement sets of some fast convergent series.
Funder
Austrian Science Foundation (FWF) - Czech Science Foudation
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)
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