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