Semi-fast convergent sequences and k -sums of central Cantor sets
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Semi-fast convergent sequences and k-sums of central Cantor sets Artur Bartoszewicz1 · Małgorzata Filipczak2
· Franciszek Prus-Wi´sniowski3
Received: 23 June 2019 / Revised: 31 May 2020 / Accepted: 5 October 2020 / Published online: 17 November 2020 © The Author(s) 2020
Abstract We study k-sums of central Cantor sets and present a full characterization of the case of positive measure for such k-sums. This generalizes the celebrated Sannami’s counterexample for the Palis hypothesis. Moreover, we find a characterization of all semi-fast convergent sequences generating central Cantor sets. Keywords Central Cantor set · Achievement set · Fast convergent sequence · Semi-fast convergent series · Lebesgue measure Mathematics Subject Classification 40A05 · 11B05 · 28A75 Suppose that (xn ) = (x1 , x2 , . . .) is an absolutely summable sequence. The set E(xn ) ..=
∞
εn xn : (εn ) ∈ {0, 1}N
n=1
of all subsums of the series ∞ n=1 x n is called the achievement set of (x n ), see [10]. Since E(xn ) is a translate of E(|xn |), it suffices to study only summable sequences
B
Małgorzata Filipczak [email protected] Artur Bartoszewicz [email protected] Franciszek Prus-Wi´sniowski [email protected]
1
Institute of Mathematics, Łód´z University of Technology, ul. Wólcza´nska 215, 93-005 Łód´z, Poland
2
Faculty of Mathematics and Computer Science, Łód´z University, ul. Stefana Banacha 22, 90-238 Łód´z, Poland
3
Institute of Mathematics, University of Szczecin, ul. Wielkopolska 15, 70-453 Szczecin, Poland
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of nonincreasing terms. It is easy to see that for (xn ) = 23 , 322 , . . . the set E(xn ) is equal to the classic Cantor ternary set C. By a Cantor set we mean any subset of reals homeomorphic to C. Denote by rn the number ∞ k=n+1 x n and recall that sequences satisfying x n > rn for any n ∈ N are called fast convergent, and sequences with xn rn are called slowly convergent. Probably Kakeya was the first who observed that for any fast convergent sequence (xn ) the set E(xn ) is a Cantor set. He also showed that in this case the Lebesgue measure μ of E(xn ) is equal to limn 2n rn [11]. Moreover, Kakeya proved that the achievement set E(xn ) of a monotonic sequence is an interval if and only if (xn ) is slowly convergent. It turned out that achievement sets may take a form different from Cantor sets or finite unions of closed intervals. Recall that a set is called a Cantorval if it is homeomorphic to the set T = C ∪ S2n−1 , where Sn denotes the union of 2n−1 open middle thirds which are removed from [0, 1] at the n-th step in the standard construction of the Cantor ternary set C. It can be shown based on [3, Theorem 21.17] that a Cantorval (more precisely, an M-Cantorval, see [13]) is a nonempty compact subset of the real line that is the closure of its interior, and both endpoints of any of its components with nonempty interior are accumulation points of one-point components. For other characterizations of Cantorvals see
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