Upper bound on the number of lattice points on the intersection of a hyperplane and a sphere

Question

Let $R>0$, $\overrightarrow{\alpha} \in \mathbb{R}^{d}$. Consider the intersection $T$of $RS^{d-1}$ and the hyperplane $\overrightarrow{\alpha} \cdot \overrightarrow{x} = n$. What is the best known (asymptotic) upper bound for the number of lattice points (integer points) in this region? That is, an upper bound depending only on $R$ that is valid for all hyperplanes. $T$ is a $(d-1)$-dimentional sphere in the hyperplane, so I would expect the bound to be of the order of the number of lattice points in $RS^{d-2}$, which for big enough $d$ is $R^{d-3+\epsilon}$.