Suppose $f : \mathbb{T}^d \to \mathbb{R}$ has Fourier coefficients $\hat{f}(j) = (2\pi)^{-d}\int_{\mathbb{T}^d}f(\theta)e^{-ij^T\theta}$. If $\hat{f}(j) = \hat{f}(j')$ whenever $\|j\|^2 = \|j'\|^2$, what can we say about $f$?. We can conclude that if $R$ is any orthogonal matrix mapping $\mathbb{Z}^d$ into $\mathbb{Z}^d$, then $f(\theta) = f(R\theta)$. But this isn't sufficient as this only implies $\hat{f}$ is invariant under negation of coordinates and permutation of coordinates, so we can't deduce $\hat{f}(1, 1, 1, 1) = \hat{f}(4, 0, 0, 0)$.
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