Phase Measure Near Maximum Modulus Sets of Exponential Sums: Power Laws and Critical Corrections

For a finite exponential sum f(z) = Σ a_k e^{λ_k z} with nonzero coefficients and distinct exponents, let ν_r(w) be the normalized angular measure of the set where log |f(re^{iθ})| lies within wr of its circle maximum. Put Λ = max_k |λ_k| > 0 and let F be the distribution function of the envelope gap d(θ) = Λ − max_k Re(λ_k e^{iθ}). For every fixed C in (0, Λ) we prove ν_r(C) − F(C) = κ(C)r^{-1/2} + O((1 + log r)/r), including levels through ties of arbitrary multiplicity. The explicit coefficient κ(C) ≥ 0 assigns full weight to interior quadratic minima and half weight to quadratic tie sides, whose common curvature is Λ − C. At regular levels the correction is G(C)/r + o(1/r) with an explicit boundary-displacement coefficient. We also obtain a transition law at thresholds C + u/r, uniformly for u in each fixed compact interval, including the resonances where a quadratic contribution appears or disappears. Its coefficient may have either sign. A two-term example has a correction asymptotic to (log r)/(2π√2 r) at a moving resonance along specified radii, proving that the uniform remainder order is sharp. For shrinking thresholds we establish ν_r(Cr^{-s}) ≍ r^{-s/2} for every s > 0, with explicit rescaled limits and a treatment of degenerate resonances at s = 1. The proofs use a measure-theoretic comparison that remains valid on circles through zeros. Error constants and radius thresholds are effectively computable, though the bounds need not be numerically sharp.