Special additive complements of a set of natural numbers

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 $x\in A$ and $y\in B$. We shall describe various types of additive complements of the set $A$ such as those additive complements of $A$ that do or do not intersect $A$, additive complements which are the union of disjoint infinite arithmetic progressions, and additive complements having various densities etc. We estabilish that if $A=\{a_i: i\in \mathbb{N}\}$ is a set of natural numbers such that $a_{i} < a_{i+1} $ for $i \in \mathbb{N}$ and $\liminf_{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb{N}$ such that $B\cap A = \varnothing$ and $B$ is a sparse additive complement of the set $A$. Besides this, for a given positive real number $\alpha \leq 1$ 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 $\alpha$.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)