arXiv subjects
Publications and source records attributed to Emily R. Korfanty.
Is there a translation-bounded measure whose diffraction is spherically symmetric and concentrated on a single sphere? This note constructively answers this question of Strungaru in the affirmative.
We review the diffraction theory for plane waves and establish its connection to the diffraction of Besicovitch almost periodic functions, extending the theory to an unbounded setting and providing explicit formulas. Then, we give an alternative proof that the diffraction of a spherical wave in $\mathbb{R}^d$ is a single sphere, which was recently shown in \cite{BKM25}. After developing a suitable framework for working with spherically symmetric measures, including a radial analogue of the usual Lebesgue decomposition, we introduce the notion of radial almost periodicity. In particular, we define a space of Besicovitch radially almost periodic functions and show that this space contains precisely the functions whose radial part is Besicovitch almost periodic. The paper concludes with a diffraction analysis of these functions.
We construct new Delone sets associated with badly approximable numbers which are expected to have rotationally invariant diffraction. We optimize the discrepancy of corresponding tile orientations by investigating the linear equation $x+y+z=1$ where $πx$, $πy$, $πz$ are three angles of a triangle used in the construction and $x$, $y$, $z$ are badly approximable. In particular, we show that there are exactly two solutions that have the smallest partial quotients by lexicographical ordering.
Given a Fourier transformable measure in two dimensions, we find a formula for the intensity of its Fourier transform along circles. In particular, we obtain a formula for the diffraction measure along a circle in terms of the autocorrelation measure. We look at some applications of this formula.