**Marcin Kysiak, Enrico Zoli**

## Set-Theoretic Properties of
Schmidt's Ideal

**Abstract:**

We study some set-theoretic properties of Schmidt's $\sigma$-ideal on $\mathbb{R}$,
emphasizing its analogies and dissimilarities with both the classical
$\sigma$-ideals on $\mathbb{R}$ of Lebesgue measure zero sets and of Baire first
category sets. We highlight the strict analogy between Schmidt's ideal on $\mathbb{R}$
and Mycielski's ideal on $2^{\mathbb{N}}$.

**Keywords:**

$\sigma$-ideals, $(\alpha,\beta)$-games, Mycielski $\sigma$-ideal, Borel sets,
countable chain condition, Steinhaus property, Ruziewicz property, Sierpi\'nski-Erd\H
os maps, badly approximable numbers.

**MSC 2000:** 03E50, 11K60, 28A05, 91A44