Sharp Extremal Bounds for Angular Occupancy

This paper is a sequel to my earlier preprint On $\varepsilon$-angle classes determined by planar point sets. In that work, I introduced the angular occupancy function $\Feps(A)$, which measures how many $\varepsilon$-sized bins in $[0,\pi]$ contain an angle determined by a finite planar point set $A$. The first paper established universal upper bounds

\Feps(A)=O ⁣(min⁡{1/ε, n2}),\Feps(A) = O\!\big(\min\{1/\varepsilon,\,n^2\}\big),\Feps(A)=O(min{1/ε,n2}),

and analyzed several extremal examples.

In this follow-up, I develop new methods to study the sharpness of these bounds:

• I extend the basic energy method by introducing higher-moment inequalities, which give stronger lower bounds when the angle distribution is uneven.

• I analyze the multiplicity structure of angles, proving connections between maximum multiplicity and occupancy, and working out examples for regular polygons, convex configurations, and circle constructions.

• I introduce a probabilistic perturbation model, showing that in typical (non-degenerate) point sets, angle collisions are rare, leading to occupancy growth of order $\Theta(n^2)$ even in sparse regimes.

The paper concludes with several open problems on multiplicity bounds, extremal occupancy, and computational aspects.

Overall, this work refines the theoretical framework for angular occupancy, connects it to classical extremal geometry, and develops new tools (energy methods, multiplicity, random perturbations) that open directions for further research.