The Furstenberg-Sárközy theorem for polynomials in one or more prime variables

We establish upper bounds on the size of the largest subset of {1,2,…,N} lacking nonzero differences of the form h(p_1,…,p_ℓ), where h∈ℤ[x_1,…,x_ℓ] is a fixed polynomial satisfying appropriate conditions and p_1,…,p_ℓ are prime. The bounds are of the same type as the best-known analogs for unrestricted integer inputs, due to Bloom-Maynard and Arala for ℓ=1, and to the authors for ℓ≥ 2.