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Abundance of progression in large set for non commutative semigroup

Sujan Pal

Abstract

The notion of abundance of certain type of configuration in certain large sets was first proved by Furstenberg and Glazner in 1998. After that many author investigate abundance of different types of configurations in different types of large sets. Hindman, Hosseini, Strauss and Tootkaboni recently introduced another notion of large sets called $CR$ sets. Then Debnath and De proved abundance of arithmetic progression in $CR$ sets for commutative semigroups. In the present article we investigate abundance of progressions in for non-commutative semigroups.

Abundance of progression in large set for non commutative semigroup

Abstract

The notion of abundance of certain type of configuration in certain large sets was first proved by Furstenberg and Glazner in 1998. After that many author investigate abundance of different types of configurations in different types of large sets. Hindman, Hosseini, Strauss and Tootkaboni recently introduced another notion of large sets called sets. Then Debnath and De proved abundance of arithmetic progression in sets for commutative semigroups. In the present article we investigate abundance of progressions in for non-commutative semigroups.
Paper Structure (2 sections, 8 theorems, 26 equations)

This paper contains 2 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. ABUNDANCE OF CR SET

Key Result

Theorem 1.1

Let $k\in\mathbb{N}$ and assume that $S\subset\mathbb{Z}$ is piecewise syndetic. Then $\left\{ \left(a,d\right):\left\{ a,a+d,...,a+kd\right\} \subset S\right\}$ is piecewise syndetic in $\mathbb{Z}^{2}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • proof
  • Definition 1.6
  • Theorem 1.7
  • proof
  • Theorem 2.1
  • ...and 12 more