The trigonometric polynomial on sums of two squares, an additive problem and generalisation

Let B be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial S(α;N)=∑b∈ B,b≤ N e(bα) satisfies \[ {S(α; N)}{N/√log N}<<A,A' 1/ϕ(q) + √q/N(log N)⁷ +1/(log N)^A \] for any A,A'≥ 0 and when $(a,q)=1$ and |qα-a|≤ (log N)A'/N. We use this estimate together with a variant of the circle method influenced by Green and Tao's Transference Principle to obtain the number of representations of a large enough odd integer N as a sum b+b₁+b₂, where b∈ B while b₁ (resp. b₂) belongs to a general subset B₁ (resp. B₂) of B of relative positive density. We further show that the above bound is effective when 0≤ A<1/2.