The ternary Goldbach problem in arithmetic progressions

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and ${\displaystyle (N,r)=\{b\in \mathbb{N}³:1\le b_j\le r,(b_j,r)=1}$ and b₁+b₂+b₃ ≡ N (mod r)}. It is known that ${\displaystyle \#(N,r)=r²{\prod }_{p\mid r}\_\left\{p\mid N\right\}((p-1)\frac{p-2}{p}²){\prod }_{p\mid r}\_\{p\nmid N\}(\frac{p²-3p+3}{p}²)}$. Let ε > 0 be arbitrary and ${\displaystyle R=N^{\frac{1}{8}-\epsilon }}$. We prove that for all positive integers r ≤ R, with at most ${\displaystyle O\left(R{\log }^{-A}N\right)}$ exceptions, the Diophantine equation N = p₁+p₂+p₃, ⎨ ${\displaystyle p_j\equiv b_j(modr),}$ j = 1,2,3,!$! ⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.