Convexity, Fourier transforms, and lattice point discrepancy
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916413125427200 |
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| author | Greenblatt, Michael |
| author_facet | Greenblatt, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convexity, Fourier transforms, and lattice point discrepancy Greenblatt, Michael Classical Analysis and ODEs Number Theory 42B20, 42B10, 26E05, 11P21, 11K38 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. |
| title | Convexity, Fourier transforms, and lattice point discrepancy |
| topic | Classical Analysis and ODEs Number Theory 42B20, 42B10, 26E05, 11P21, 11K38 |
| url | https://arxiv.org/abs/2402.16636 |