One-dimensional crystals and quadratic residues (Q1360713)

The distribution \(\rho= \sum^N_{j=1} b_j \sum_{x\in\mathbb{Z}} \delta_{x+x_j}\), where \(\delta_y\) denotes Dirac's measure at \(y\), has the Fourier transform \(\widehat \rho= \sum_{y \in \mathbb{Z}} a(y) \cdot \delta_y\) with coefficients \(a(y) =\sum^N_{j=1} b_j \cdot e^{2 \pi ix_j \cdot y}\). For \(b_j \in \mathbb{N}\) and \(0\leq x_j <1\), the article describes sufficient conditions to recover the numbers \(a(y)\) from the knowledge of their absolute values, which is a special case of the phase reconstruction problem in crystallography. The connection is through Gaussian sums which are unique among a class of exponential sums by the behaviour of their absolute value.