Angular Measure of Near-Maximum Sets and Normalized Deficit Convergence for Finite Exponential Sums

Abstract

For a finite exponential sum with nonzero coefficients and pairwise distinct nonzero exponents, we study the gap between the maximum of log |f| on a circle and its angular values, normalized by the radius. We prove that this normalized deficit converges in normalized angular measure to the deterministic envelope deficit. This yields convergence against bounded continuous test functions, convergence of finite interval measures, and an explicit limiting distribution described by finitely many cosine arcs. For each fixed positive threshold, the normalized angular measure of the near-maximum set converges to the corresponding envelope sublevel measure. At fixed regular levels avoiding tie directions and critical points, we obtain an explicit first-order expansion F(C) + G(C)/r + o(r^{-1}). At nondegenerate quadratic critical values and at cusp values, we obtain the global estimate F(C) + O(r^{-1/2}), without a signed leading coefficient. No pointwise, almost-everywhere, Hausdorff, or uniform convergence is asserted, and no uniformity in the threshold is claimed.