Convergence exponent of Pierce expansion digit sequences

TL;DR

This paper studies the convergence properties of Pierce expansion digit sequences, analyzing the convergence exponent as a function on the unit interval and examining divergence sets for series of reciprocals raised to positive powers.

Contribution

It introduces new analysis of the convergence exponent function and characterizes divergence sets for series involving Pierce expansion digits.

Findings

Characterized the properties of the convergence exponent as a real-valued function.

Described the structure of level sets of the convergence exponent.

Identified subsets where the series of reciprocals diverge for positive powers.

Abstract

In this paper, we investigate the convergence exponent of Pierce expansion digit sequences. We explore some basic properties of the convergence exponent as a real-valued function defined on the closed unit interval, as well as those of the level sets of the function. Additionally, we further study subsets of the closed unit interval on which the series of positive $s$th powers of the reciprocals of the Pierce expansion digits diverges.