Representations of rational numbers and Minkowski dimension

TL;DR

The paper analyzes the size of sets of rational numbers determined by fixed-length representations: continued fractions, Egyptian fractions, and Engel fractions, using Minkowski dimension as a coarse-scale measure. It derives exact dimension results for each family: dim_M(C_m)=1/2, dim_M(E_m)=dim_M(A_m)=1-1/2^m, and dim_M(E_m^*)=dim_M(A_m^*)=m/(m+1); it also connects Egyptian-fraction sums to iterated sumsets of decreasing sequences and discusses asymptotic behavior as m grows. The methods combine symbolic coding of representations with precise covering estimates, comparing the different expansion types and highlighting distinct global scaling properties. The results have implications for the arithmetic properties of discrete rational sets and their fractal-like dimensional behavior.

Abstract

In this paper, we investigate the representations of rational numbers via continued fraction, Egyptian fraction, and Engel fraction expansions. Given $m \in \mathbb{N}$, denote by $C_m, E_m, E_m^*$ 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.

Theorems & Definitions (37)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Lemma 2.2