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This is a Quantitative Study study.

October 13, 2025Open Access

Irregularities of distribution and Fourier transforms of multi-dimensional convex bodies

Key Points

This extension shows new properties of irregularities of distribution in general convex bodies.
Key findings include a geometric inequality for the Fourier transform relating to convex shapes.
The research incorporates convex bodies with a smooth boundary and finite order of contact conditions.
Results may influence further studies on distribution properties in higher-dimensional geometry.

Abstract

W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to d-dimensional balls. In this paper, we extend their result to any d-dimensional convex body with a smooth boundary and finite order of contact. As an intermediate step, we prove a geometric inequality for the Fourier transform of the characteristic function of a convex body.

Irregularities of distribution and Fourier transforms of multi-dimensional convex bodies

Key Points

Abstract

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