## Real Analysis Exchange

### Magic Sets

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 43, Number 1 (2018), 187-204.

Dates
First available in Project Euclid: 2 May 2018

https://projecteuclid.org/euclid.rae/1525226429

Digital Object Identifier
doi:10.14321/realanalexch.43.1.0187

Mathematical Reviews number (MathSciNet)
MR3816438

Zentralblatt MATH identifier
06924880

#### Citation

Halbeisen, Lorenz; Lischka, Marc; Schumacher, Salome. Magic Sets. Real Anal. Exchange 43 (2018), no. 1, 187--204. doi:10.14321/realanalexch.43.1.0187. https://projecteuclid.org/euclid.rae/1525226429