On Kakeya Conditions for Achievement Sets
AbstractWe prove that for each infinite subset C of $${\mathbb {N}}$$ N there exists a sequence $$(x_n)$$ ( x n ) such that $$\{n: x_n>r_n\}=C$$ { n : x n > r n } = C and the achievement set $$A(x_n)$$ A ( x n ) is a Cantor set. Moreover, we show that it is possible to construct a sequence $$(x_n)$$ ( x n ) such that the set $$\{n: x_n>r_n\}$$ { n : x n > r n } has asymptotic density $$\alpha $$ α for each $$\alpha \in [0,1)$$ α ∈ [ 0 , 1 ) and $$A(x_n)$$ A ( x n ) is a Cantorval.
2017 ◽
Vol 28
(10)
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pp. 1750073
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2001 ◽
Vol 21
(06)
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ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE
2004 ◽
Vol 04
(01)
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pp. 63-76
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2019 ◽
Vol 2019
(746)
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pp. 149-170
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2008 ◽
Vol 28
(5)
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pp. 1509-1531
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2001 ◽
Vol 28
(6)
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pp. 367-373
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Keyword(s):
Keyword(s):