On Additive Complements with Special Structures

Article Type

Research Article

Publication Title

Mediterranean Journal of Mathematics

Abstract

Let A be a set of natural numbers. A set B of natural numbers, is said to be an additive complement of the set A if all sufficiently large natural numbers can be represented as x+y for some x∈A and y∈B. This article describes various types of additive complements of the set A such as those additive complements of A that do not intersect A, additive complements which are the union of disjoint infinite arithmetic progressions, and additive complements having various densities etc. As an application, we also focus on the structure of the sumset of an arithmetic progression and a geometric progression. Besides this, for a given positive real number α≤1 and a finite set A, we investigate a set B such that B can be written as a union of disjoint infinite arithmetic progressions with the natural density of A+B equal to α.

DOI

10.1007/s00009-025-02825-2

Publication Date

5-1-2025

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