Suppose I have two sets of integers $A$ and $B$ and I have a sketch data structure described by a function $\mathsf{sketch}_n : \mathcal{P}(\mathbb{Z}) \to 2^n$ that returns a bitstring of size $n$. These sketches can be checked for disjointness with a function $\mathsf{disjoint}_n : 2^n \times 2^n \to 2$, where if $\mathsf{disjoint}_n(\mathsf{sketch}_n(A), \mathsf{sketch}_n(B)) = 1$, then $A \cap B = \emptyset$ (but $A \cap B = \emptyset$ may also be true when $\mathsf{disjoint}$ returns 0, i.e.: no false positives).
This can easily be accomplished with a bloom filter, where $\mathsf{sketch}_n$ is the result of inserting each element of the set into a bloom filter and $\mathsf{disjoint}_n(x, y) = (x \,\&\, y) \equiv \mathbf{0}$ ($\&$ being bitwise AND).
Now, suppose that I want there to exist a function $\mathsf{shift}_n : 2^n \times \mathbb{Z} \to 2^n$ such that $\mathsf{shift}_n(\mathsf{sketch}_n(X), \delta) = \mathsf{sketch}_n(\{x + \delta \mid x \in X\})$. Is there a data structure that supports this operation, perhaps using some variety of homomorphic hashing?
Alternatively, $\mathsf{shift}_n$ need not exactly return the same sketch as $\{x + \delta \mid x \in X\}$, but rather a different sketch that still has the no-false-positives property when checked for disjointness with another set.
I'm also willing to consider answers where we can't ensure no false positives, but there is a good bound on the probability of a false positive.
Also, wherever I've said $\mathbb{Z}$ here, feel free to interpret that as some finite set of integers (e.g.: 32-bit integers).
Bonus points if there is a way to compute the union of two sketches, and if there is a way to extend this to sets of tuples in $\mathbb{Z} \times 2^p$ where $\mathsf{shift}$ only affects the first element of each tuple.