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Error recognition in the Cantor cube

Paweł Pasteczka

TL;DR

It is proved that all protocols detecting errors are Baire spaces and generic ones are not neither Borel nor meager and it is shown that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit.

Abstract

Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~Głąb, E.~Jabłońska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube $\mathbb{Z}_2^ω$). Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager. We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D.~Niwiński and E.~Kopczyński in 2014.

Error recognition in the Cantor cube

TL;DR

It is proved that all protocols detecting errors are Baire spaces and generic ones are not neither Borel nor meager and it is shown that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit.

Abstract

Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~Głąb, E.~Jabłońska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube ). Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager. We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D.~Niwiński and E.~Kopczyński in 2014.

Paper Structure

This paper contains 7 sections, 16 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Few properties of thin sets
  4. Banach-Mazur game
  5. Main result
  6. Applications to Xor-sets
  7. Generalization to $k$-thin sets

Key Result

Lemma 1

Let $T \subset \mathbb{Z}_2^n$ ($n \in{\bar{\mathbb{N}}}$). Then, for all $k \in \mathbb{N}$,

Figures (2)

  • Figure 1: Capturing Ego's strategy
  • Figure 2: Capturing Alter's strategy

Theorems & Definitions (27)

  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Theorem 1: Banach-Mazur
  • Corollary 1
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 17 more