Oscillatory Integrals and Fractal Dimension

TL;DR

This paper investigates the geometric properties of curves generated by oscillatory integrals with analytic phases, providing explicit formulas for their fractal dimensions using advanced mathematical techniques.

Contribution

It introduces explicit formulas for the box dimension and Minkowski content of curves defined by oscillatory integrals, based on critical point analysis and singularity resolution.

Findings

Formulas for fractal dimensions of oscillatory integral curves

Dependence of geometry on critical points of the phase

Application of Newton diagrams and singularity resolution

Abstract

We study geometrical representation of oscillatory integrals with an analytic phase function and a smooth amplitude with compact support. Geometrical properties of the curves defined by the oscillatory integral depend on the type of a critical point of the phase. We give explicit formulas for the box dimension and the Minkowski content of these curves. Methods include Newton diagrams and the resolution of singularities.