Representations of Real Numbers by Alternating Perron Series and Their Geometry

TL;DR

This paper explores the properties of alternating Perron series representations of real numbers, generalizing several classical series representations, and investigates their topological, metric, and measure-theoretic relationships.

Contribution

It establishes fundamental topological and metric properties of $P^-$-representation and connects it with $P$-representation in measure theory contexts.

Findings

Proved basic topological properties of $P^-$-representation.

Analyzed metric properties of alternating Perron series.

Established relationships between $P$- and $P^-$-representations in measure theory.

Abstract

We consider the representation of real numbers by alternating Perron series ($P^-$-representation), which is a generalization of representations of real numbers by Ostrogradsky-Sierpi\'nski-Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating L\"{u}roth series, etc. Namely, we prove the basic topological and metric properties of $P^-$-representation and find the relationship between $P$-representation and $P^-$-representation in some measure theory problems.