n-polynomial convexity of symmetric circular arcs at and below the (n−1)π/n threshold

Determine whether the set A_α = {z ∈ S^1 : −α ≤ arg z ≤ α} is n-polynomially convex when α ≤ (n−1)π/n.

Background

Section 4 analyzes subsets of the unit circle with respect to n-polynomial convexity. Proposition 16 proves that A_α is not n-polynomially convex when α > (n−1)π/n, giving a sharp obstruction on one side of a threshold.

The complementary direction—establishing n-polynomial convexity when α ≤ (n−1)π/n—remains unresolved, though the paper notes it is straightforward for the special case n = 2 when α < π/2.