Strong Measure Zero Sets Research Articles

Strong measure zero and meager-additive sets through the prism of fractal measures

We develop a theory of sharp measure zero sets that parallels Borel's strong measure zero, and prove a theorem analogous to Galvin--Mycielski--Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^{\omega}$ is meager-additive if and only if it is $\mathcal E$-additive; if $f\colon 2^{\omega}\to 2^{\omega}$ is continuous and $X$ is meager-additive, then so is $f(X)$.

There are no very meager sets in the model in which both the Borel Conjecture and the dual Borel Conjecture are true

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.

Borel conjecture and dual Borel conjecture

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.

Dual Borel Conjecture and Cohen reals

AbstractWe construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size ℵ1 that does not have measure zero.

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \boldsymbol \aleph _1$ and find some topological characterizations of perfectly meager and universally meager sets.

Function Spaces and Strong Measure Zero Sets

In this paper we study how the classical concept of being a set of reals of strong measure zero manifests itself in the context of function spaces.

Strong Measure Zero and Strongly Meager Sets

Strong Measure Zero and Strongly Meager Sets

Finite support iteration and strong measure zero sets

AbstractAny finite support iteration of posets with precalibre ℵ1 which has the length of cofinahty greater than ω1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ1 strong measure zero sets.

MA(σ-centered): Cohen reals, strong measure zero sets and strongly meager sets

We prove thatMA(σ-centered) + the Dual Borel Conjecture is consistent; and thatMA(σ-centered) + the non-additivity of the ideal of the strong measure zero sets also is consistent.