Let $A=\{\mathbf{v} \in \mathbb{Z}^{n}: \|\mathbf{v}\|^2= m \}$ and a fixed $\mathbf{y}\in \mathbb{R}^n$, the norm here refers to the Euclidean norm.

Suppose $\mathbf{x}$ is a uniform distribution on the set $A$, then what's the distribution of $\ \langle\mathbf{x},\mathbf{y}\rangle$. I hope it can be expressed in terms of $m,n,\mathbf{y}$.

I know that for large $n$, even given the size of $A$ is hard, but I'm wondering if there's some way I can learn something about this distribution. Thanks for any help or reference.

I found related papers AES16, EH99

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    $\begingroup$ Maybe this is also of interest: Bourgain, Sarnak,Rudnick: Local statistics of lattice points on the sphere. Modern trends in constructive function theory, 269–282, Contemp. Math., 661, Amer. Math. Soc., Providence, RI, 2016. $\endgroup$ May 16 at 10:06
  • $\begingroup$ Thanks @ Kurisuto Asutora $\endgroup$ May 17 at 8:35


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