On additive complement with special structures

Abstract: Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is said to be an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. This article describes various types of additive complements of the set $A$ such as those additive complement of $A$ that does not intersects $A$, additive complements of the form of the union of disjoint infinite arithmetic progressions, additive complement having various density etc. As an application of this study, we also focus on the structure of sumset of arithmetic progression and geometric progression. Apart from this, for given positive real no. $\alpha \leq 1$ and finite set $A$, we investigate a set $B$ such that it can be written as union of disjoint infinite arithmetic progression and density of $A+B$ is $\alpha$.