Integer set disjointness query on sketches with something like homomorphic hashing
Here is one simple idea -- not sure if it's practical.
Define $\mathsf{sketch}_n(X)$ to be a length-$n$ bitstring in which bit $i$ is set iff there exists $x \in X$ such that $x = i \mod n$. In practice, this means that inserting an element $x$ into a set involves turning on bit $x \mod n$. $\mathsf{disjoint}_n(\cdot)$ is defined as before. Then $\mathsf{shift}_n(\cdot, \delta)$ can be found by rotating the sketch right by $\delta$ bits (which can be done implicitly by storing a &quot;rotation amount&quot; in the data structure).
To reduce the false positive rate for membership tests (but increase the false negative rate for disjointness tests), you can store multiple bitstrings using different moduli -- e.g., if the moduli 100, 101 and 105 are used, there would be 306 bits in total. Intuition suggests relatively prime moduli would work best, but I have not tried looking into this in detail.