Vinogradov’s Theorem with Piatetski-Shapiro Primes

We prove that, for any c₁,c₂,c₃∈ (1,41/35), every sufficiently large odd number N can be represented as the sum of three primes, N = p₁ + p₂ +p₃, such that pᵢ = nᵢ^cᵢ for some nᵢ ∈ N for each 1 ≤ i ≤ 3. Our arguments are based on a variant of Green’s transference principle due to Matomäki, Maynard, and Shao. We prove a necessary restriction estimate using Bourgain’s strategy and employ Harman’s sieve method to optimize our upper bound for cᵢ.