An additive problem for powers of primes
Let \(Q_ k\) be the set of k-th powers of nonnegative integers and \(P_ k\) the set of k-th powers of prime numbers. The question of this paper is how the density of a set increases by adding \(Q_ k\) or \(P_ k\) to it. The essentially optimal result states that \[ \underline d(A+Q_ k)\quad \geq \quad \underline d(A+P_ k)\quad \geq \quad K^{-c/\log \log K}, \] where ḏ(A) is the lower asymptotic density, \(K=1/\underline d(A)\) and c is a positive constant depending on k. The proof is based on the Hardy- Littlewood method.
scientific article; zbMATH DE number 4122078