Fourier-analytic expansions that incorporate set geometry in high dimensions

Develop Fourier-analytic techniques for Edgeworth or related asymptotic expansions that explicitly incorporate the geometry of rectangles and similar sets to achieve improved dimension-dependent error bounds for P(S_n ∈ A) when both n and d grow.

Background

The authors explain that in high-dimensional problems, the geometry of the index set (e.g., rectangles) critically affects achievable error rates. While Stein's method can leverage such geometry, Fourier-analytic approaches have difficulty doing so, limiting their effectiveness for high-dimensional accuracy improvements.

A Fourier-analytic framework that captures geometric structure would complement existing techniques and potentially yield refined bounds comparable to those now available via Stein's method.