Hausdorff dimension of the graph of the error-sum function of $\alpha$-Lüroth series

Abstract

Let α be a countable partition of the unit interval [0, 1] . In this paper, we will introduce the error-sum function of α -Lüroth series and determine the Hausdorff dimension of its graph when the partition α is eventually decreasing. Some other properties of the error-sum function are also investigated.

On the error-sum function of Pierce expansions

“…2 are equal and so they have the same Pierce expansion, namely [2] P . It follows by the definition of fundamental intervals that I σ contains ϕ(σ), whereas I σ ′ fails to contain ϕ(σ ′ ).…”

“…We refer the reader to Chapters 2-4 of [11] for details on the Hausdorff measure, the Hausdorff dimension, and the box-counting dimension, and Chapters 1-2 of [3] for the covering dimension which is called the topological dimension in the book. It should be mentioned that the proof idea of the following theorem is borrowed from earlier studies, e.g., [2,4,14,23,28,30]. Proof.…”

“…Since there is, as is well known, a wide range of representations of real numbers (see [13] and [24] for details), it was natural for intrigued researchers to define the error-sum function for other types of representations and investigate its basic properties. To name but a few, the errorsum functions were defined and studied in the context of the decimal expansion [33], the integer p base expansion [36], the non-integer β base expansion [17], the classical Lüroth series [29,30], the alternating Lüroth series [28,31], the α-Lüroth series [2], the Engel continued fraction [32], and the alternating Sylvester series [14]. In the previous studies, the list of examined basic properties includes, but is not limited to, boundedness, continuity, integrality, and intermediate value property (or Darboux property) of the error-sum function, and the Hausdorff dimension of the graph of the function.…”