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Homometry and direct-sum decompositions of lattice-convex sets

Two sets in $\mathbb{R}^d$ are called homometric if they have the same covariogram, where the covariogram of a finite subset $K$ of $\mathbb{R}^d$ is the function associating to each $u \in \mathbb{R}^d$ the cardinality of $K \cap (K+u)$. Understanding the structure of homometric sets is important for a number of areas of mathematics and applications. If two sets are homometric but do not coincide up to translations and point reflections, we call them nontrivially homometric. We study nontrivially homometric pairs of lattice-convex sets, where a set $K$ is called lattice-convex with respect to a lattice $\mathbb{M} \subseteq \mathbb{R}^d$ if $K$ is the intersection of $\mathbb{M}$ and a convex subset of $\mathbb{R}^d$. This line of research was initiated in 2005 by Daurat, Gérard and Nivat and, independently, by Gardner, Gronchi and Zong. All pairs of nontrivially homometric lattice-convex sets that have been known so far can essentially be written as direct sums $S \oplus T$ and $S \oplus (-T)$, where $T$ is lattice-convex, the underlying lattice~$\mathbb{M}$ is the direct sum of $T$ and some sublattice $\mathbb{L}$, and $S$ is a subset of $\mathbb{L}$. We study pairs of nontrivially homometric lattice-convex sets assuming this particular form and establish a necessary and a sufficient condition for the lattice-convexity of $S \oplus T$. This allows us to explicitly describe all nontrivially homometric pairs in dimension two, under the above assumption, and to construct examples of nontrivially homometric pairs of lattice-convex sets for each $d \ge 3$.