Hermite neighborhood entry bounds

Determine sharp bounds, as functions of n and δ>0, for the number of normalized finite free self-convolutions required to bring every increasingly ordered simple degree-n root configuration of mean zero and variance n−1 within Euclidean distance δ of the increasing root vector of He_n, particularly when the target neighborhood is required for the nonlinear local contraction theorem to apply.

Background

The appendix proves local Euclidean contraction near the Hermite root configuration for the normalized self-convolution map T_n, with an asymptotic one-step factor 2^{-1/2} and exact dyadic-rate estimates inside a sufficiently small neighborhood.

What is not established is a global entry-time estimate: the paper does not determine how many iterations are needed for an arbitrary normalized simple configuration to reach that local neighborhood. The requested bounds should clarify their dependence on n and on the radius δ, and should account for the neighborhood size required by the nonlinear contraction theorem.