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arXiv · 2402.16636

Convexity, Fourier transforms, and lattice point discrepancy

Michael Greenblatt
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Abstract

In a well-known paper by Bruna, Nagel and Wainger [BNW], Fourier transform decay estimates were proved for smooth hypersurfaces of finite line type bounding a convex domain. In this paper, we generalize their results in the following ways. First, for a surface that is locally the graph of a convex real analytic function, we show that a natural analogue holds even when the surface in question is not of finite line type. Secondly, we show a result for a general surface that is locally the graph of a convex $C^2$ function, or a piece of such a surface defined through real analytic equations, that implies an analogous Fourier transform decay theorem in situations where the oscillatory index is less than $1$. In such situations, for a compact surface the exponent provided is sharp. This result has implications for lattice point discrepancy problems, which we describe.

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Michael Greenblatt. 2024-02-26. Convexity, Fourier transforms, and lattice point discrepancy. https://arxiv.org/abs/2402.16636

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