A burden for the verifier is that it needs to store all the primes and all P_k constants, to verify the proof.
For a light verifier, we can save on storage space, when the prover supplies both P_k and p_k. The verifier can check P_k^{p_k} \equiv P_{\emptyset} \mod N, where P_{\emptyset} = g^{\prod\limits_{k=0}^n p_{k}}. This way the light verifier only has to store P_{\emptyset}.
For a bulk proof the prover can provide P_B and all primes corresponding the the set B. The verifier can check the correctness of P_B, by checking P_B^{\prod\limits_{k\in B} p_{k}} \equiv P_{\emptyset} \mod N and calculate all needed P_k.
The disadvantage of this trick is that the verifier can only know that a value x is one of the n values in the accumulator, but not the specific spot.
For applications that do not need the specific spot, this light verifier trick can be used.
I was wondering whether zk-starks proofs (for which we can shrink the proof a lot by having a smaller merkle tree replacement) actually need the spot, or that putting the spot in the hashed data is sufficient. From what I understand of zk-stark proofs, it would not weaken the proof as long as the number of values in the accumulator is limited to n. A forger could try to put multiple values in the same spot to increase the chance to be able to give the correct answer for that spot. But now the forger increases the chance of not being able to answer when an empty spot is chosen.