Zihuan Xu

Critical Lorentz Bounds for Hankel Multipliers with Slow Phases

For compactly supported radial symbols with a logarithmic slow phase near a sphere, we characterize the bounds L^{p,q}_rad → L^{p,r}_rad at p = 2d/(d + 1) and p = 2d/(d − 1) for every 0 < q,r ≤ ∞. We assume Φ is C², a = Φ′ > 0, a(v) → 0, and |a′(v)| ≲ a(v)/v; neither monotonicity...

Maximum Modulus Points of Entire Functions: Angular Oscillation and Circle Reflection

Let f be a non-monomial entire function and let v_f(r) denote the number of points on |z| = r where |f| attains its maximum. We study the finite-lower-order form of the Erdős maximum-modulus problem. For 0 < η < 1, an elementary variation argument gives an M(r,f)-relative bound f...

Exact Angular Critical-Point Counts for e^z ± 1

For f_a(z) = e^z + a, a = ±1, we count the distinct angular critical points of log |f_a| on every zero-free circle of radius r ≥ 12. Writing N_p(r) for the number of zeros in the disk, we prove that the critical-point count N_osc(r) satisfies 0 ≤ N_osc(r) − 2N_p(r) ≤ 4, with both...

Unbounded Least Repelling Periods in Degree Four

We construct degree-four rational maps of positive equilibrium Lyapunov exponent whose least classical repelling period is arbitrarily large. For every N ≥ 2, the map F_N(z) = z² − t^{2^{N−1}}z + t^{4·2^{N−1}+4}/z² has repelling cycles of exactly the periods n ≥ N. All its shorte...

Phase-Resolved Asymptotics for Proximity Functions of Finite Exponential Sums

For a finite exponential sum with constant coefficients, we determine the first-order asymptotic of the proximity function at every fixed value different from its constant term. The leading term is the angular measure of the strict decay region multiplied by the negative logarith...

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

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 meas...

A Planar Counterexample to the Sphere-Wedge Conjecture for Matching Complexes

The matching complex of a finite simple graph consists of its sets of pairwise disjoint edges. We prove that the matching complex of the planar pentagonal bipyramid simplicially collapses to a triangulated torus. This disproves the conjecture that matching complexes of planar gra...