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 $S(\alpha ;N)={\sum }_{b\in B,b\le N}e(b\alpha )$ satisfies $$\frac{S(\alpha ;N)}{N\mathrm{/}\sqrt{\log N}}<{<}_{A,A^{\mathrm{\prime }}}\frac{1}{\phi (q)}+\sqrt{\frac{q}{N}}(\log N{)}^7+\frac{1}{(\log N{)}^A}$$ for any $A,A^{\mathrm{\prime }}\ge 0$ and when $(a,q)=1$ and $\mathrm{\mid }q\alpha -a\mathrm{\mid }\le (\log N{)}^{A^{\mathrm{\prime }}}\mathrm{/}N$. 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 $b+b_1+b_2$, where $b\in B$ while $b_1$ (resp. $b_2$) belongs to a general subset $B_1$ (resp. $B_2$) of $B$ of relative positive density. We further show that the above bound is effective when $0\le A<1\mathrm{/}2$.