Lagrange-like spectrum of perfect additive complements

TL;DR

This paper introduces a Lagrange-like spectrum for perfect additive complements of non-negative integers, characterizes its smallest accumulation point, and proves that the spectrum set is closed, advancing understanding of unique additive representations.

Contribution

It defines a new spectrum for perfect additive complements, identifies its smallest accumulation point, and proves the spectrum set is closed, providing new insights into additive number theory.

Findings

Identified the smallest accumulation point of the spectrum

Proved the spectrum set is closed

Established properties of the Lagrange-like spectrum

Abstract

Two infinite sets $A$ and $B$ of non-negative integers are called \emph{perfect additive complements of non-negative integers}, if every non-negative integer can be uniquely expressed as the sum of elements from $A$ and $B$. In this paper, we define a Lagrange-like spectrum of the perfect additive complements ($\mathfrak{L} $ for short). As a main result, we obtain the smallest accumulation point of the set $\mathfrak{L} $ and prove that the set $\mathfrak{L} $ is closed. Other related results and problems are also contained.