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Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...

Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$. For suitable functions $g \geqslant 0$, define $$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...