格点计数函数的傅里叶分析唯一性定理
A Fourier-analytic Uniqueness Theorem for Lattice-point Enumerators
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中文总结 AI 辅助
该研究针对有界可测集，通过傅里叶分析方法证明其格点计数函数的唯一性，推广了Royer的相关结果，为凸体的唯一确定提供了统一的理论框架。
中文摘要 AI 辅助
我们考虑有界集 $P \subset \mathbb{R}^d$ 及实 $t>0$ 的格点计数函数 $L_P(t) = |tP \cap \mathbb{Z}^d|$。证明：若两个边界测度为零的有界可测集对所有整数平移具有相同的实参数格点计数函数，则它们的指示函数几乎处处一致；由此推论，任意凸体可由该数据唯一确定。证明简短且基于傅里叶分析，关键工具是周期点计数函数，其傅里叶系数在稠密集上恢复指示函数的傅里叶变换。该证明通过统一论证，恢复并推广了Royer（arXiv:1712.01973、arXiv:1712.03937）建立的有理多面体与对称凸体的唯一性结果，Royer的证明依赖复杂的特定几何构造。
英文摘要
We consider a bounded set $P \subset \mathbb{R}^d$ and the lattice-point enumerator $L_P(t) = |tP \cap \mathbb{Z}^d|$ for real $t > 0$. We show that if two bounded measurable sets with boundary of measure zero have the same real-parameter lattice-point enumerators for all integer translates, then their indicator functions agree almost everywhere. As a corollary, any convex body is uniquely determined by this data. Our proof is short and Fourier-analytic, where the key device is a periodic point-counting function whose Fourier coefficients recover the Fourier transform of the indicator function on a dense set. This recovers and extends, with a unified argument, the uniqueness results for rational polytopes and symmetric convex bodies established by Royer [arXiv:1712.01973, arXiv:1712.03937], whose proofs relied on intricate case-specific geometric constructions.
