Multiplication on self-similar sets with overlaps

Let K be the attractor of the following IFSThe main results of this paper are as follows:By virtue of Lemma 2.3, it follows thatLemma 2.12. If c ≥ (1 − λ) 2 , then K + K = [0, 2]. Proof. By Lemmas 2.1 and 2.11. Take I = J = [c − λ, c] ∪ [1 − λ, 1]. Therefore, for f (x, y) = x + y, we have f (I, J) = [2(c − λ), 2c] ∪ [c + 1 − 2λ, 1 + c] ∪ [2(1 − λ), 2]. By Lemma 2.3, we conculde that 2c − (c + 1 − 2λ) = c + 2λ − 1 ≥ 0, 1 + c − (2(1 − λ)) = c + 2λ − 1 ≥ 0.

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“…Very useful is the following lemma. It comes from [3] and [42]. For the convenience of readers, we give the detailed proof.…”

Section: Proofs Of Theorem 11 and Corollary 13mentioning

confidence: 99%

“…In [28], Jiang, Wang and Xi gave a necessary condition such that when c = λ − λ 2 the self-similar set K is bi-Lipschitz equivalent to another self-similar set with the strong separation condtion. In [42]…”

Section: Introductionmentioning

confidence: 99%

Multiple Representations of Real Numbers on Self-Similar Sets With Overlaps

et al. 2019

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“…Representation of real numbers is a topic of great interest in number theory. There are many approaches which can represent real numbers, for instance, the β-expansions [17,1,26,3,5,23], the continued fractions [10,9], multiplication (division, quadratic sum) on fractal sets [28,29], the Lüroth expansions [4], and so forth. These representations are related to many different mathematical aspects, for instance, the matrix theory, ergodic theory, fractal geometry, Diophantine approximation, combinatorics, and so on.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, they also proved that the division on C, denoted by C C , is exactly the union of some closed sets. In [28], Tian et al defined a class of overlapping self-similar sets as follows: let K be the attractor of the IFS…”

Section: Introductionmentioning

confidence: 99%

Arithmetic representations of real numbers in terms of self-similar sets

Jiang¹,

Lifeng²

2019

Ann. Acad. Sci. Fenn. Math.

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Suppose n ≥ 2 and A i ⊂ {0, 1, · · · , (n − 1)} for i = 1, · · · , l, let K i = a∈A i n −1 (K i + a) be self-similar sets contained in [0, 1]. Given m 1 , · · · , m l ∈ Z with i m i = 0, we let Sx = {(y 1 , · · · , y l ) : m 1 y 1 + · · · + m l y l = x with y i ∈ K i ∀i} .In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set Ur = {x : Card(Sx) = r}, where Card(Sx) denotes the cardinality of Sx, and r ∈ N + . We prove under the so-called covering condition that the Hausdorff dimension of U 1 can be calculated in terms of some matrix. Moreover, if r ≥ 2, we also give some sufficient conditions such that the Hausdorff dimension of Ur takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of Ur is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.2000 Mathematics Subject Classification. Primary 28A80.

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“…For more results, see [14]. In [25], Tian et al defined a class of overlapping self-similar sets as follows: let K be the attractor of the IFS…”

Section: Introductionmentioning

confidence: 99%