1 Introduction

For a function $f \in L^1(\mathbb {R}^d)$ , its Fourier transform is defined by

$$\begin{align*}\widehat{f}(\xi) = \int_{\mathbb{R}^d} f(x)\, e^{-2\pi i \langle x, \xi \rangle} \, dx, \quad \xi \in \mathbb{R}^d. \end{align*}$$

If $f \in L^2(\mathbb {R}^d)$ , then, since $L^1(\mathbb {R}^d)$ is dense in $L^2(\mathbb {R}^d)$ , the Fourier transform extends naturally to $L^2(\mathbb {R}^d)$ .

The Fourier transform is a fundamental mathematical tool that allows us to analyze functions and signals in terms of their frequency components. It decomposes a function into oscillatory modes, revealing its spectral structure and providing a powerful framework for studying differential equations and signal processing. For further details, see [Reference Bracewell2, Reference Fourier3, Reference Stein and Shakarchi11].

Let $A \subseteq \mathbb {R}^d$ . The characteristic function (or indicator function) of A is defined by

$$\begin{align*}1_A(x) = \begin{cases} 1, & \text{if } x \in A, \\ 0, & \text{if } x \notin A. \end{cases} \end{align*}$$

Thus, $\widehat {1}_A$ denotes the Fourier transform of the characteristic function of the set A.

The measurable set $A \subset \mathbb {R}^d$ is said to k-tile $\mathbb {R}^d$ with the translation set $\Lambda $ if almost every point in $\mathbb {R}^d$ is covered exactly k times by the translates $A+\lambda $ for $\lambda \in \Lambda $ , that is,

(1.1) $$ \begin{align} \sum_{\lambda \in \Lambda} 1_{A}(x-\lambda) = k \quad \text{a.e. in } \mathbb{R}^d. \end{align} $$

A set with this property is called a multi-tiler. In the special case $k=1$ , we simply say that A tiles $\mathbb {R}^d$ . Tilings and multi-tilings naturally arise in diverse areas, such as quasicrystals, Fourier analysis, and the theory of lattices. For further details, see [Reference Agora, Antezana and Cabrelli1, Reference Fuglede4, Reference Grepstad and Lev6, Reference Iosevich and Kolountzakis7, Reference Kolountzakis and Lev9] and the references therein.

In this article, we construct a counterexample to the following conjecture.

Next, we briefly describe the motivation behind the conditions in the conjecture. The Fourier transform at integer points is equivalent to the equality of the corresponding Fourier series. In particular, for two measures supported on $[0,1]^d$ , equality at ${\xi \in \mathbb {Z}^d}$ implies that the measures themselves coincide. More generally, for two measures $\mu _1$ and $\mu _2$ with unbounded support, equality of their Fourier series implies that they agree on the factor space $\mathbb {R}^d / \mathbb {Z}^d$ . In the case of measures uniformly distributed on convex bodies $B_1$ and $B_2$ , this condition can be interpreted as a geometric or combinatorial property.

Furthermore, the class of convex and centrally symmetric bodies is a natural class of geometric objects to study, appearing in classical results, such as Minkowski’s theorem and the second Minkowski theorem.

Finally, without the condition on multiple tilers, any two convex polytopes that tile $\mathbb {R}^d$ with the same lattice $\Lambda \subset \mathbb {Z}^d$ , as described in Lemma 2.1, provide a natural source of counterexamples to the conjecture.

In this article, we only address one direction of the equivalence:

$$ \begin{align*}\widehat{1}_{\mathcal{P}} (\xi)=\widehat{1}_{\mathcal{Q}} (\xi) \text{ for all } \xi\in\mathbb{Z}^d \Rightarrow \mathcal{P}=\mathcal{Q} \text{ up to rigid motions}. \end{align*} $$

The main result of the article can be formulated as the following theorem.

The article is organized as follows. Section 2 presents the proof of Theorem 1.2 together with related remarks, while Section 3 states and proves a result illustrating the importance of convexity in Conjecture 1.1.