A Romanoff-type theorem for a multiset of products of powers

Let b₁,,bd≥2 be fixed integers, and let k be their multiplicative rank. We study representations n=a+b₁u₁⋯ bdud, where a belongs to a set A of positive integers and u₁,,ud are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on A, we obtain a lower bound for the number of integers with many such representations, in terms of d and k. In particular, when A is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers n≤ x have at least c₁(log x)ᵈ⁻¹ or c₁(log x)d-1/2 representations, respectively, for some c₁>0 and all sufficiently large x. No multiplicative independence or coprimality assumptions on the bases are required.