Some existence results on Cantor sets

On intrinsic and extrinsic rational approximation to Cantor sets

“…we have that ord b mod s is the lowest common multiple of the orders ord b mod q α i i over 1 ≤ i ≤ v and q i are distinct primes not dividing b, we see that ord b mod s is divisible by s/Q O (1) and thus ord b mod s s. On the other hand, according to Proposition 4.10 the period length of r/s ∈ C b,W equals ord b mod s, which is C s D = s by Theorem 4.3. For large s we get a contradiction as < 1.…”

“…Assume ξ 1 , ξ 2 are two ξ as in (29) arising from sequences (a 1 n ) n≥1 and (a 2 n ) n≥1 , respectively. We claim that if they are ordered a 1 1 < a 2 1 < a 1 2 < a 2 2 < a 1 3 < • • • with very large gaps between two consecutive elements, then ξ 1 = ξ 2 . If true, this clearly implies that we get an uncountable family of suitable ξ .…”

“…Intuitively, (similarity) Cantor sets containing no Liouville vector should exist when we drop the rationality condition. For d = 1 and the class of topological Cantor sets, the existence was shown in [1]. See also [4] for fat Cantor sets without rational elements.…”

“…If true, this clearly implies that we get an uncountable family of suitable ξ . The claim is obvious if there is some coordinate i for which ξ 1 i is rational and ξ 2 i is irrational, or vice versa. Thus, as we have shown above that both vectors ξ 1 , ξ 2 are irrational, we may assume that there is an index i with both ξ 1 i , ξ 2 i irrational.…”

“…To show the claim, observe that the respective rational approximations θ j k,i = p j k,i /q j k,i as in (30) are again very good approximating convergents to ξ j i . If we choose the gaps between two consecutive elements of a 1 1 , a 2 1 , a 1 2 , a 2 2 , . .…”