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On additive complement with special structures

Mohan, Bhuwanesh Rao Patil, Ram Krishna Pandey

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. $α\leq 1$ 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 $α$.

On additive complement with special structures

Abstract

Let be a set of natural numbers. A set , a set of natural numbers, is said to be an additive complement of the set if all sufficiently large natural numbers can be represented in the form , where and . This article describes various types of additive complements of the set such as those additive complement of that does not intersects , 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. and finite set , we investigate a set such that it can be written as union of disjoint infinite arithmetic progression and density of is .
Paper Structure (13 sections, 19 theorems, 51 equations)

This paper contains 13 sections, 19 theorems, 51 equations.

Table of Contents

  1. Notations
  2. Introduction
  3. Preliminary
  4. Primitive root
  5. Base representation of real number
  6. Euler's totient function
  7. Union of disjoint arithmetic progressions
  8. Primes in arithmetic progresssions
  9. Proof of Theorem \ref{['Thm-1']}
  10. Proof of Theorem \ref{['Thm-2']}
  11. Proof of Theorem \ref{['Thm-3']}
  12. Proof of Theorem \ref{['Thm-4']} and Corollary \ref{['THM-4Cor1']}
  13. Conclusion

Key Result

Proposition 2.1

Let $A$ be an infinite set of natural numbers. Then there exists a subset $B$ of $\mathbb{N}$ such that $B$ is an additive complement of $A$ and where $C$ is an absolute constant and the terms of the sum with $|A \cap [1,k]| = 0$ are to be replaced by one.

Theorems & Definitions (38)

  • Definition 2.1: Additive complement
  • Definition 2.2
  • Proposition 2.1: Lorentz, Lorentz
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Corollary 2.1
  • Definition 3.1
  • Conjecture 3.2
  • ...and 28 more