Let B be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial S (α;N) =₁ ₁, ₁ ₍ e (bα) satisfies \ S (α; N) N/ N<<₀, ₀' 1ϕ (q) + q{N} (N) ^7 +1 (N) A \ for any A, A' 0 and when (a, q) =1 and |qα-a| (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.
Ramaré et al. (Sat,) studied this question.