Borel sets without perfectly many overlapping translations, III

Borel sets without perfectly many overlapping translations IV

We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no perfect set P such that |(eta+B) cap (nu+B)|>5 for any distinct eta,nu from P. This answers two questions from our previous works.

Borel sets without perfectly many overlapping translations

For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134

Borel sets without perfectly many overlapping translations, II

For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .

$F_σ$ ideals of perfectly bounded sets

Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\bf x})$ iff there is a non pathological lower semicontinuous submeasure $φ$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(φ)=\{A\subseteq \mathbb{N}: \;φ(A)<\infty\}$. We address the questions of when $FIN(φ)$ is a tall ideal and has a Borel selector. W…

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \aleph_1$ and find some topological characterizations of perfectly meager and universally meager sets.

可数完全贫乏集与可数完全零集

Countably perfectly meager and countably perfectly null sets

本文通过引入可数完全贫乏集和可数完全零集以强化普遍贫乏集与普遍零集概念，并在连续统假设弱化条件下证明了二者不等价。