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On dynamical $C^{\star}$-set and its combinatorial consequences

Pintu Debnath, Sayan Goswami

Abstract

Using the methods from topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. Later D. De, Neil Hindman, and D. Strauss [Fund. Math.199 (2008), 155-175.] established a stronger version of the Central Sets Theorem and then introduced the notion of $C$-sets satisfying the Central Sets Theorem and studied the properties of these sets. For any weak mixing system $\left(X, \mathcal{B},μ, T\right),$ and $A_{0},A_{1}\in\mathcal{B}$, with $μ\left(A_{0}\right)μ\left(A_{1}\right)>0$, R. Kung and X.Ye [Disc. Cont. Dyn. sys., 18 (2007) 817-827.] proved that the set $N\left(A,B\right)= \left\{n:μ\left(A_{0}\cap T^{-n}A_{1}\right)>0\right\}$ intersects all sets of positive upper Banach density. However, later N. Hindman and D. Strauss [New York J. Math. 26 (2020) 230-260.] proved that there exist $C$-sets having zero upper Banach density. Inspired by this result, in this article, we prove that $N\left(A, B \right)$ intersects with all $C$-sets. Then we introduce the notion of a dynamical $C^{\star}$-set and then we study their combinatorial properties.

On dynamical $C^{\star}$-set and its combinatorial consequences

Abstract

Using the methods from topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. Later D. De, Neil Hindman, and D. Strauss [Fund. Math.199 (2008), 155-175.] established a stronger version of the Central Sets Theorem and then introduced the notion of -sets satisfying the Central Sets Theorem and studied the properties of these sets. For any weak mixing system and , with , R. Kung and X.Ye [Disc. Cont. Dyn. sys., 18 (2007) 817-827.] proved that the set intersects all sets of positive upper Banach density. However, later N. Hindman and D. Strauss [New York J. Math. 26 (2020) 230-260.] proved that there exist -sets having zero upper Banach density. Inspired by this result, in this article, we prove that intersects with all -sets. Then we introduce the notion of a dynamical -set and then we study their combinatorial properties.

Paper Structure

This paper contains 3 sections, 22 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Dynamical $C^{\star}$-set
  3. Zigzag structure in dynamical $C^{\star}$-set

Key Result

Theorem 1.1

[Central Sets Theorem] Let $l\in\mathbb{N}$, and $A\subseteq\mathbb{N}$ be a central set. For each $i\in\{1,2,\ldots,l\}$ let $\langle x_{i,m}\rangle_{m=1}^{\infty}$ be a sequence in $\mathbb{\mathbb{N}}$. Then there exists a sequence $\langle b_{m}\rangle_{m=1}^{\infty}$ in $\mathbb{N}$ and $\langl

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8: Dynamical $C^{\star}$- sets
  • Definition 1.9
  • Theorem 1.10
  • ...and 33 more