There is no clustering of the solutions of $a^2+b^2+c^2=n$, even for individual $n$'s, assuming the number of solutions is large (e.g. when $n\equiv 1\pmod{4}$ and $n$ is large). This was proved by William Duke (Invent. Math. 92 (1988), 73-90), his paper is available (for free) here.

For more recent results, e.g. what happens beyond equidistribution, see the work of Bourgain-Sarnak-Rudnick here and here.

There is no clustering of the solutions of $a^2+b^2+c^2=n$, even for individual $n$'s, assuming the number of solutions is large (e.g. when $n\equiv 1,2,3,5,6\pmod{8}$ and $n$ is large). This was proved by William Duke (Invent. Math. 92 (1988), 73-90), his paper is available (for free) here.

For more recent results, e.g. what happens beyond equidistribution, see the work of Bourgain-Sarnak-Rudnick here and here.

There is no clustering, even for fixed $n$, assuming the number of solutions is large (e.g. when $n\equiv 1\pmod{4}$ and $n$ is large). This was proved by William Duke (Invent. Math. 92 (1988), 73-90).

There is no clustering of the solutions of $a^2+b^2+c^2=n$, even for individual $n$'s, assuming the number of solutions is large (e.g. when $n\equiv 1\pmod{4}$ and $n$ is large). This was proved by William Duke (Invent. Math. 92 (1988), 73-90), his paper is available (for free) here.

For more recent results, e.g. what happens beyond equidistribution, see the work of Bourgain-Sarnak-Rudnick here and here.