Let $F$ be a non-singular quadratic form in four variables and let $w: \mathbb{R}^4 \to \mathbb{R}$ be a non-negative compactly supported function satisfying certain suitable conditions. Set $$N(F,w,m) = \sum_{x \in \mathbb{Z}^4, F(x) = m} w\left( \frac{x}{\sqrt{m}} \right).$$ Then Heath-Brown proved in the 90's as $m \to \infty$, $$N(F,w,m) = C_{F,w}\sigma(F,m)m + O_{F,w,\varepsilon}(m^{\frac{3}{4} + \varepsilon}),$$ where $C_{F,w}$ is a positive constant depending only on $F$ and $w$ and $\sigma(F,m)$ is the singular series.
I am interested to know, whether up till today there are improvements on the error rate of this result. More precisely, is there any hope to improve the error rate to $O_{F,w,\varepsilon}(m^{\frac{1}{2} + \varepsilon})$. If this is unknown, what is the conjectured error term?
For the application I have in mind, it would be already interesting to understand the case $$F(x) = x_1^2 + x_2^2 + x_3^2 + x_4^2.$$ Are there any results for this case or any other specific quadratic form?
