Nice partition of $\mathbb{R}$ into uncountably many uncountable sets

Question

A recent issue of American Math. Monthly has a paper that partitions $\mathbb{R}$ into an arbitrary finite number of uncountable sets such that every real number is a condensation point of all the sets in the partition. In fact, we can find uncountably many uncountable subsets $S_\alpha$ of $\mathbb{R}$ that are (1) pairwise disjoint, (2) have union $\mathbb{R}$, and (3) every real number is a condensation point of all the sets. One way to do this is the following. For any infinite binary sequence $\alpha=(a_1, a_2, \dots)$, let $S_\alpha$ be the set of all real numbers whose fractional part has the binary expansion $0.b_1b_2\cdots$ such that the sequence $(b_2,b_4,b_6,\dots)$ differs from $\alpha$ in only finitely many terms. When $\alpha=(1,1,1,\dots)$ we must remove from $S_\alpha$ all numbers whose binary expansion ends in infinitely many 1's, due to the nonuniqueness of the binary expansion. The set of distinct $S_\alpha$'s then has the desired properties. This partition of $\mathbb{R}$ also has the property that any two of the sets $S_\alpha$ and $S_\beta$ are translates of each other, except for $S_{1,1,1,\dots}$. This suggests the question: does there exist a partition of $\mathbb{R}$ satisfying (1), (2), and (3) such that every pair of sets in the partition are translates?

Answer 1

I think the following works:

Fix an uncountable subgroup $G$ of $(\mathbb{R}, +)$ - note that any such is dense in $\mathbb{R}$, and in fact intersects each nonempty open interval uncountably often - of uncountable index, and consider the set of cosets of $G$.

(Such a group can be produced by transfinite recursion using a well-ordering of $\mathbb{R}$: alternate between throwing reals into the group, and barring reals from entering the group.)

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Note that the argument above relies on the axiom of choice. However, this isn't necessary: provably in ZF, there is a $\mathbb{Q}$-linearly independent set $S$ of reals of size continuum. We can split this into two disjoint sets $S_1, S_2$ of size continuum (pick some appropriate real $a$ and consider the points to the left of $a$ versus the right of $a$). Now consider the $\mathbb{Q}$-vector space generated by $S_1$.