Translator Disclaimer
2018 Magic Sets
Lorenz Halbeisen, Marc Lischka, Salome Schumacher
Real Anal. Exchange 43(1): 187-204 (2018). DOI: 10.14321/realanalexch.43.1.0187

## Abstract

In this paper we study magic sets for certain families $$\mathcal{H}\subseteq {^\mathbb{R}\mathbb{R}}$$ which are subsets $$M\subseteq\mathbb{R}$$ such that for all functions $$f,g\in\mathcal{H}$$ we have that $$g[M]\subseteq f[M]\Rightarrow f=g$$. Specifically we are interested in magic sets for the family $$\mathcal{G}$$ of all continuous functions that are not constant on any open subset of $$\mathbb{R}$$. We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than $$\mathfrak{c}$$ meager sets is still meager, we can also add and remove sets of cardinality less than $$\mathfrak{c}$$ without destroying the magic set.

Then we will enlarge the family $$\mathcal{G}$$ to a family $$\mathcal{F}$$ by replacing the continuity with symmetry and assuming that the functions are locally bounded. A function $$f:\mathbb{R}\to\mathbb{R}$$ is symmetric iff for every $$x\in\mathbb{R}$$ we have that $$\lim_{h\downarrow 0}\frac{1}{2}\left(f(x+h)+f(x-h)\right )=f(x)$$. For this family of functions we will construct $$2^\mathfrak{c}$$ pairwise different magic sets which cannot be destroyed by adding and removing a set of cardinality less than $$\mathfrak{c}$$. We will see that under the continuum hypothesis magic sets and these more stable magic sets for the family $$\mathcal{F}$$ are the same. We shall also see that the assumption of local boundedness cannot be omitted. Finally, we will prove that for the existence of a magic set for the family $$\mathcal{F}$$ it is sufficient to assume that the union of less than $$\mathfrak{c}$$ meager sets is still meager. So for example Martin’s axiom for $$\sigma$$-centered partial orders implies the existence of a magic set.

## Citation

Lorenz Halbeisen. Marc Lischka. Salome Schumacher. "Magic Sets." Real Anal. Exchange 43 (1) 187 - 204, 2018. https://doi.org/10.14321/realanalexch.43.1.0187

## Information

Published: 2018
First available in Project Euclid: 2 May 2018

zbMATH: 06924880
MathSciNet: MR3816438
Digital Object Identifier: 10.14321/realanalexch.43.1.0187

Subjects:
Primary: 03E20
Secondary: 03E35 , 03E50 , 26A99

Keywords: Continuum hypothesis , magic sets , Martin's Axiom , sets of range uniqueness , symmetric functions  