ON A PROBLEM OF NATHANSON ON NONMINIMAL ADDITIVE COMPLEMENTS

Let $C$ and $W$ be two subsets of $\mathbf{Z}$. The set $C$ is called an additive complement to $W$ in $\mathbf{Z}$ if $C+W=\mathbf{Z}$. An additive complement $C$ to $W$ is said to be minimal if no proper subset of $C$ is an additive complement to $W$ in $\mathbf{Z}$. In this paper, we deal with a problem of Nathanson on minimal additive complements and show that if $\inf W=-\infty $ and $\sup W=+\infty$, then there exists a minimal additive complement to $W$ in $\mathbf{Z}$. The conclusion is not true if $\inf W>-\infty $ or $\sup W<+\infty$.

A note on minimal additive complements of integers

On minimal additive complements of integers

Let C and W be two integer sets. If $C+W=\mathbb {Z}$ , then we say that C is an additive complement to W . If no proper subset of C is an additive complement to W , then we say that C is a minimal additive complement to W . We study the existence of a minimal additive complement to $W=\{w_i\}_{i=1}^{\infty}$ when W is not eventually periodic and $w_{i+1}-w_{i}\in \{2,3\}$ for all i .

Let A and B be two subsets of the nonnegative integers. We call A and B additive complements if all sufficiently large integers n can be written as a+b, where a∈A and b∈B. Let S={1 2 ,2 2 ,3 2 ,···} be the set of all square numbers. Ben Green was interested in the additive complement of S. He asked whether there is an additive complement B={b n } n=1 ∞ ⊆ℕ which satisfies b n =π 2 16n 2 +o(n 2 ). Recently, Chen and Fang proved that if B is such an additive complement, thenlim sup n→∞ π 2 16n 2 -b n n 1/2 logn≥2 π1 log4.They further conjectured thatlim sup n→∞ π 2 16n 2 -b n n 1/2 logn=+∞.In this paper, we confirm this conjecture by giving a much more stronger result, i.e.,lim sup n→∞ π 2 16n 2 -b n n≥π 4.

Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The notion was introduced by Nathanson and previous work by him, Chen--Yang, Kiss--S\`andor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this article, we study subsets which are not eventually periodic. We introduce the notion of "spiked subsets" and give necessary and sufficient conditions for the existence of minimal complements for them. This provides a partial answer to a problem of Nathanson.

On additive co-minimal pairs

Additive complements of the squares

On additive complement of a finite set

Sets arising as minimal additive complements in the integers

We say a subset $C$ of an abelian group $G$ arises as a minimal additive complement if there is some other subset $W$ of $G$ such that $C+W=\{c+w:c\in C,\ w\in W\}=G$ and such that there is no proper subset $C'\subset C$ such that $C'+W=G$. In their recent paper, Burcroff and Luntzlara studied, among many other things, the conditions under which eventually periodic sets, which are finite unions of infinite (in the positive direction) arithmetic progressions and singletons, arise as minimal additive complements in $\mathbb Z$. In the present paper we study this further and give, in the form of bounds on the period $m$, some sufficient conditions for an eventually periodic set to arise as a minimal additive complement; in particular we show that "all eventually periodic sets are eventually minimal additive complements''. Moreover, we generalize this to a framework in which "patterns'' of points (subsets of $\mathbb Z^2$) are projected down to $\mathbb Z$, and we show that all sets which arise this way are eventually minimal additive complements. We also introduce a formalism of formal power series, which serves purely as a bookkeeper in writing down proofs, and we prove some basic properties of these series (e.g. sufficient conditions for inverses to be unique). Through our work we are able to answer a question of Burcroff and Luntzlara (when does $C_1\cup(-C_2)$ arise as a minimal additive complement, where $C_1,C_2$ are eventually periodic sets?) in a large class of cases.

Inverse problems for minimal complements and maximal supplements

In this paper we consider the control problem of ensemble of trajectories for one In recent years, the importance of discrete mathematics has increased dramatically. This is not the name of a branch of mathematics such as number theory, algebra, calculus. This is a separate branch of knowledge, which is called “discrete” (individual). However, there are also processes of continuous mathematics, that is, it becomes a continuous whole. Discrete data can only take integer values, while continuous data can take any value. Discrete mathematics was created several decades ago and is the mathematical language of computer science. Scientists have found that the mathematical disciplines taught in universities are insufficient for learning a programming language. Therefore, they combine additional maths topics into one course, now called discrete mathematics. Discrete mathematics is the foundation of computer science and theoretical computer science. Without learning discrete mathematics, we miss the essence of the logic of computer science. Due to the multiplicity of objects within discrete mathematics, rational clarity in solving programming issues is possible. Every field in computer science is associated with discrete objects, be it databases, network components, computer organization, compilers, network programming, and so on. Essentially, discrete mathematics studies the concepts of counting, mathematical structures, logic, recursion, computational modeling, graph theory and algorithms, and many other disciplines. Evidence, a basic concept in discrete mathematics that includes deductive thinking, simplification, pattern identification, mathematical notation, working within constraints and conditions. This does not mean that a formal course in discrete mathematics is necessary or should be required for programmers. However, many of the concepts discussed in discrete mathematics are applied using this science. In this article, we will look at the basic postulates of discrete mathematics that are necessary to learn the basics of programming.

Let $W,W'\subseteq G$ be nonempty subsets in an arbitrary group $G$. The set $W'$ is said to be a complement to $W$ if $WW'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. We show that, if $W$ is finite then every complement of $W$ has a minimal complement, answering a problem of Nathanson. This also shows the existence of minimal $r$-nets for every $r\geqslant 0$ in finitely generated groups. Further, we give necessary and sufficient conditions for the existence of minimal complements of a certain class of infinite subsets in finitely generated abelian groups, partially answering another problem of Nathanson. Finally, we provide infinitely many examples of infinite subsets of abelian groups of arbitrary finite rank admitting minimal complements.

A novel Univariate Marginal Distribution Algorithm based discretization algorithm