Abstract
We prove upper and lower estimates on the Hausdorff dimension of sets of infinite complex continued fractions with finitely many prescribed Gaussian integers. Particulary we will conclude that the dimension of theses sets is not zero or two and there are such sets with dimension greater than one and smaller than one.
Highlights
Continued fractions were studied in a number of theories since the work of Wallis in the 17th century; see [1]. e rst dimensional theoretical perspective on in nite real continued fractions can be found in the work of Jarnik [2], who introduced upper and lower estimates on the Hausdorff dimension of sets of continued fractions with bounded digits. e problem of calculating the dimension of these sets has been addressed by several authors [3,4,5,6,7]
We are able to state our main result on dimHH C(AAA
Corollary 2. or all nite sets AA: C (AA) A AAAAA one has dimHH C(AAA A 2; on the other hand dimHH C(AAA A A if AA has more than one element
Summary
Continued fractions were studied in a number of theories since the work of Wallis in the 17th century; see [1]. e rst dimensional theoretical perspective on in nite real continued fractions can be found in the work of Jarnik [2], who introduced upper and lower estimates on the Hausdorff dimension of sets of continued fractions with bounded digits. e problem of calculating the dimension of these sets has been addressed by several authors [3,4,5,6,7]. Given a sequence zznn ∈ C for nn nn0 of Gaussian integers we de ne the in nite complex continued fraction by We are able to state our main result on dimHH C(AAA. A nite set AA A AAAAA let DDD DD D D+ be the uni ue real numbers ful llin
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