Some Properties on the Error-Sum Function of Alternating Sylvester Series

Abstract

The error-sum function of alternating Sylvester series is introduced. Some elementary properties of this function are studied. Also, the hausdorff dimension of the graph of such function is determined.

On the error-sum function of Pierce expansions

“…We prove that E : I → R enjoys an intermediate value property in some sense, which is an analogue of Theorem 4.3 in [23]. Similar results can also be found in Theorem 5 of [14], Theorem 2.6 of [36], and Theorem 2.5 of [28]. In fact, every result aforementioned is a consequence of the following theorem.…”

“…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.…”