关于稀疏采样猜想的一则注记
Fourier sampling on a lattice
浏览论文内容
中文总结 AI 辅助
本文针对傅里叶变换在格点上一致的凸体相关猜想，证明其在2、3维仍不成立，并给出了更具一般性的正面结论：满足格点同余条件的凸体连通并集若傅里叶变换在对偶格一致则仅相差格点平移。
中文摘要 AI 辅助
第二作者曾提出一个猜想：两个具有正测度的凸中心对称体，若均非多重平铺体，则只要它们的指示函数的傅里叶变换在\\(\Z^d\\)上一致，二者就仅相差一个刚体运动。第一作者在维度\\(d\geq4\\)时构造出反例，表明格点\\(\Z^d\\)可能过于稀疏。本文证明该初始猜想在剩余的2维和3维情形下同样不成立。\n 不过，我们证明了一个相关的正面结果，其结论更强，且既不要求中心对称性，也不要求非多重平铺假设。设\\(\mathcal L\subset\R^d\\)为满秩格，\\(\mathcal P,Q\subset\R^d\\)为凸体的连通有限并集，且每个集合中任意两个不同点模\\(\mathcal L\\)均不同余。若它们的指示函数的傅里叶变换在对偶格\\(\mathcal L^*\\)上一致，则存在\\(\ell\in \mathcal L\\)使得\\(Q=\mathcal P+\ell\\)。特别地，若两个集合均关于原点对称，则\\(\mathcal P=Q\\)。
英文摘要
We characterize the ambiguity in reconstructing bounded measurable sets from their Fourier transforms sampled on a lattice. For a full-rank lattice \(\mathcal L\subset\R^d\), we prove that two bounded measurable sets have identical indicator Fourier transforms on \(\mathcal L^*\) if and only if they are equidecomposable, modulo null sets, into finitely many measurable pieces by translations in \(\mathcal L\). If the sets are connected finite unions of convex bodies and no two distinct points of either set are congruent modulo \(\mathcal L\), we show that this equidecomposition reduces to a single translation by a vector of \(\mathcal L\). In particular, two such sets that are symmetric about the origin must coincide. We also construct counterexamples in dimensions two and three to a conjecture of the second author, which had a similar hypothesis, but lacked the anti-aliasing hypothesis. Together with the first author's counterexamples in dimensions at least four, these show that the conjecture fails in every dimension $d\geq 2$. The positive results here are an outgrowth of refinements of this conjecture.
