Generalizations of certain representations of real numbers

TL;DR

This paper introduces and investigates generalized representations of real numbers, extending classical positive and alternating representations, and establishes key metric properties and a representation theorem for certain intervals.

Contribution

It presents new generalized real number representations, analyzes their metric properties, and proves a theorem on representing numbers within specific intervals.

Findings

Established metric relations for the new representations

Proved properties of cylinder sets associated with these representations

Formulated a theorem on representing real numbers in certain intervals

Abstract

In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are proved. The theorem on the representation of real numbers from a certain interval is formulated.