On the representation of an integer in Ostrowski and recurrence numeration systems

TL;DR

This paper establishes bounds on the representations of integers with limited Hamming weight across Ostrowski and recurrence numeration systems, including cases involving two different Ostrowski systems, advancing understanding of their combinatorial properties.

Contribution

It provides effective upper bounds for integers with bounded Hamming weights in both recurrence and Ostrowski numeration systems, including cross-system representations.

Findings

Bounded Hamming weight representations in recurrence systems

Bounded Hamming weight in Ostrowski-$eta$ systems

Representation bounds for two different Ostrowski systems

Abstract

We provide an effective upper bound for positive integers with bounded Hamming weights with respect to both a linear recurrence numeration system and an Ostrowski-$\alpha$ numeration system, where $\alpha$ is a quadratic irrational. We prove a similar result for the representation of an integer in two \textit{different} Ostrowski numeration systems.