I am trying to read this letter by Sarnak, where, among other things, he discusses the problem of intrinsic diophantine approximation on $S^3 = \left\{\mathbf{x} \in \mathbb{R}^4 : x_1^2+x_2^2+x_3^2+x_4^2 = 1\right\}$. In particular, on page 22, he studies diophantine approximation of $S^3$ by points on $S^3(\mathbb{Z}[\tfrac{1}{5}])$. I am aware that the circle method can be used to obtain these approximations.

My question is as follows: Is there a more direct way to prove that $S^3(\mathbf{Z}[\tfrac{1}{5}])$ is dense in $S^3$? Also, is the statement true if we replace $5$ with any other prime?

I am trying to read this letter by Sarnak, where, among other things, he discusses the problem of intrinsic diophantine approximation on $S^3 = \left\{\mathbf{x} \in \mathbb{R}^4 : x_1^2+x_2^2+x_3^2+x_4^2 = 1\right\}$. In particular, on page 22, he studies diophantine approximation of $S^3$ by points on $S^3(\mathbb{Z}[\tfrac{1}{5}])$. I am aware that the circle method can be used to obtain these approximations.

My question is as follows: Is there a more direct way to prove that $S^3(\mathbf{Z}[\tfrac{1}{5}])$ is dense in $S^3$? Also, is the statement true if we replace $5$ with any other prime $p \neq 2$?

Edit: Following the comments of Dodd and Wojowu, the condition $p \neq 2$ has been added, since this is clearly necessary.