Storing a bit vector in uninitialized memory and minimal space
Yes, this can be done through bucketing. The representation in Briggs and Torczon can be used to store not just bits, but values associated with any set bits, by storing them next to the values in the dense array. For $k$-bit values, this leads to a total storage cost of $(2n+1)\lceil \lg n \rceil + kn$.

Now pick $k=\lceil \lg n \rceil$. Each $k$-bit value will represent a set of $\lceil \lg n \rceil$ contiguous bits in the bit array. We thus only need to track $\lceil n/\lceil \lg n \rceil \rceil$ of them with the dense and sparse arrays. This leads to a total cost of $\Theta(n)$ bits of storage, with $\Theta(\lg n)$ initialized bits.

Like the basic solution, this requires $O(1)$ operations on words of length $\Theta(\lg n)$ to set a single bit.