Dual Borel conjecture in Laver's model

Question

A set $X\subseteq 2^\omega$ of reals is of strong measure zero (smz) if $X+M\not=2^\omega$ for every meager set $M$. (This is a theorem of Galvin-Mycielski-Solovay, but for the question I am going to ask we may as well take it as a definition.)

A set $Y$ is strongly meager (sm) if $Y+N\not=2^\omega$ for every Lebesgue null set $N$.

The Borel conjecture (BC) says that every smz set is countable; the dual Borel conjecture (dBC) says that every sm set is countable.

In Laver's model (obtained by a countable support iteration of Laver reals of length $\aleph_2$) the BC holds. Same for the Mathias model.

In a paper that I (with Kellner+Shelah+Wohofsky) just sent to arxiv.org, we claim that it is not clear if Laver's model satisfies the dBC.

QUESTION: Is that correct? Or is it perhaps known that Laver's model has uncountable sm sets?

Additional remark 1: Bartoszynski and Shelah (MR 2020043) proved in 2003 that in Laver's model there are no sm sets of size continuum ($\aleph_2$). (The MR review states that the paper proves that the sm sets are exactly $[\mathbb R] ^{\le \aleph_0}$. This is obviously a typo in the review.)

Additional remark 2: If many random reals are added to Laver's model (either during the iteration, or afterwards), then BC still holds, but there will be sm sets of size continuum, so dBC fails in a strong sense.

Answer 1

In analogy with the fact that Lusin set are strongly null, Pawlikowski has shown that Sierpinksi sets are strongly meagre, so one might ask if there are are Sierpinski sets in the iterated Laver Model. The answer is no; indeed any set of positive outer measure contains an uncountable null set after adding a Laver real

Let $\psi:\omega\to\omega$ be a function growing so quickly that $$\lim_{n\to\infty} \prod_{j=n}^\infty \left(1- 2^{-\psi(j)} \right) =1 $$ Then let $L:\omega\to \omega$ be the generic real added by $\mathbb L$ and define $A$ to be the set of those $f\in 2^\omega $ such that $f$ is constantly $0$ on the interval $[L(n), L(n) + \psi(n)] $ for all but finitely many $n$. $A$ is Lebesgue null and it will be shown that $|A\cap X|=\aleph_1$.

Suppose not and let $T\in \mathbb L$ and $C\in [X]^{\aleph_0}$ be such that $T$ forces that $X\cap A\subseteq C$. For $s\in T$ extending the root of $T$ define $S_s $ to be the set of all $f\in 2^\omega$ for which there exists infinitely many $ n\in \omega$ such that $ s^\frown n\in T$ and $ f(n+j) = 0 $ for all $j\leq \psi(|s|) $. Then each $S_s$ is of measure one. Since $X$ has positive outer measure there is some $x\in \bigcap_{s\supseteq \text{root}(T)}S_s\cap(X\setminus C)$. It follows that there is then $\tilde{T}\subseteq T$ such that $\tilde{T}\in \mathbb L$ and $x$ is constantly $0$ on the interval $[m,m+\psi(|s|)]$ for all $s^\frown m\supseteq \text{root}(\tilde{T})$. In other words, $\tilde{T}$ forces that $x$ is in $A$ contradicting that $x\in X\setminus C$.

Answer 2

This is not an answer, but is too long for a comment, and may be useful still.

Unfortunately, the notion of strongly meager is not well-understood. Do we have any other example of strongly meager sets beyond Sierpinski sets (and some mild generalizations of Sierpinski sets)?

As long as we cannot construct interesting examples, the notion is ill-motivated to me. The only two motivations seem to be the analogy you cited with GMS Theorem, and the technical difficulty of handling this notion (which thus makes it challenging for fonds of technique development).

There are other natural "duals" of strong measure zero. From the point of view of selection principles, an important dual is Borel-Hurewicz sets: sets having all Borel images in the Baire space bounded. Here too, the main example is a Siepinski set, but also some kinds of $\gamma$-sets (constructed by Todor\v{c}evic using CH) form examples.

Interestingly, in Laver's model (where there are no uncountable SMZ set), there are also no nontrivial Borel-Hurewicz sets, that is, ones of cardinality at least the critical cardinality of this property (namely, $\mathfrak{b}$): Point-cofinite covers in Laver's model (with A. Miller), Proceedings of the American Mathematical Society 138 (2010), 3313-3321. Notice that, since $\mathfrak{b}=\aleph_2$ in Laver's model, there are uncountable Borel-Hurewicz sets there, but they are all trivial in the above sense. The modern perspective and accumulated experience suggest that the right question for a notion of smallness for real sets is whether there are nontrivial examples (as defined above, that is, ones of cardinality at least the critical cardinality of this property), not whether there are (possibly trivial) uncountable examples.

Thus, in this sense, BC+dBC does hold in Laver's model. :)