Repeated Angles in Three and Four Dimensions

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We show that the maximum number of occurrences of a given angle in a set of n points in $\mathbb{R}^3$ is $O(n^{7/3})$ and that a right angle can actually occur $\Omega(n^{7/3})$ times. We then show that the maximum number of occurrences of any angle different from $\pi/2$ in a set of n points in $\mathbb{R}^4$ is $O(n^{5/2}\beta(n))$, where $\beta(n) = 2^{O(\alpha(n)^2)}$ and $\alpha(n)$ is the inverse Ackermann function.