Quantitative strong approximation for ternary quadratic forms I

We derive asymptotic formulas with a secondary term for the (smoothly weighted) count of number of integer solutions of height $⩽B$ with local conditions to the equation $F(x_1,x_2,x_3)=m$, where $F$ is a non-degenerate indefinite ternary integral quadratic form, and $m$ is a non-zero integer satisfying $-m{\mathrm{\Delta }}_F=\mathrm{\square }$ which can grow like $O(B^{2-\theta })$ for some fixed $\theta >0$. Our approach is based on the $\delta$-variant of the Hardy–Littlewood circle method developed by Heath-Brown.