Representations of rational numbers and Minkowski dimension
Abstract
In this paper, we investigate the representations of rational numbers via continued fraction, Egyptian fraction, and Engel fraction expansions. Given m ∈ N, denote by Cm, Em, Em* the sets of rational numbers whose continued fraction, Egyptian fraction, and Engel fraction expansions have length m, respectively. We first establish the Minkowski dimensions of these sets, which implies that their global scaling properties are different. We also apply the results to sumsets of decreasing sequences.
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