Rachel Greenfeld

activity

20162026

most citedFuglede's spectral set conjecture for convex polytopes

50 citations · 96 across the 16 of their papers we have counts for

collaborators

19 papers

math.DS2026

Translational tilings by axis-parallel polygonal sets

Jaume de Dios Pont, Jan Grebík, Rachel Greenfeld +1

The periodic tiling conjecture asserts that if a bounded set $Ω \subset R^{d}$ of finite measure tiles $R^{d}$ by translations, then it admits at least one fully periodi…

math.CA2025

Translational tilings: structured or wild?

Rachel Greenfeld

The study of the structure of translational tilings has captivated mathematicians, scientists, and the general public for centuries and continues to thrive at the crossroads of ana…

math.CA2025

Some variants of the periodic tiling conjecture

Rachel Greenfeld, Terence Tao

The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$ , that if there is a set $A$ that solves the tiling equation…

math.CO2024

Periodicity and decidability of translational tilings by rational polygonal sets

Jaume de Dios Pont, Jan Grebík, Rachel Greenfeld +1

The periodic tiling conjecture asserts that if a region $Σ \subset R^{d}$ tiles $R^{d}$ by translations then it admits at least one fully periodic tiling. This conjectur…

math.NT2024★ 1 cited

On integer distance sets

Rachel Greenfeld, Marina Iliopoulou, Sarah Peluse

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Eu…

math.CO2023★ 3 cited

Undecidability of translational monotilings

Rachel Greenfeld, Terence Tao

In the 60's, Berger famously showed that translational tilings of $Z^{2}$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability…

math.CA2023

Tiling, spectrality and aperiodicity of connected sets

Rachel Greenfeld, Mihail N. Kolountzakis

Let $Ω \subset R^{d}$ be a set of finite measure. The periodic tiling conjecture suggests that if $Ω$ tiles $R^{d}$ by translations then it admits at least one periodi…

math.CO2022★ 18 cited

A counterexample to the periodic tiling conjecture

Rachel Greenfeld, Terence Tao

The periodic tiling conjecture asserts that any finite subset of a lattice $Z^{d}$ which tiles that lattice by translations, in fact tiles periodically. In this work we disp…

math.CO2022

A counterexample to the periodic tiling conjecture (announcement)

Rachel Greenfeld, Terence Tao

The periodic tiling conjecture asserts that any finite subset of a lattice $Z^{d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a d…

math.DS2022

Measurable tilings by abelian group actions

Jan Grebík, Rachel Greenfeld, Václav Rozhoň +1

Let $X$ be a measure space with a measure-preserving action $(g, x) \mapsto g \cdot x$ of an abelian group $G$ . We consider the problem of understanding the structure of measurable…

math.CO2021★ 1 cited

Additive energies on discrete cubes

Jaume de Dios Pont, Rachel Greenfeld, Paata Ivanisvili +1

We prove that for $d \geq 0$ and $k \geq 2$ , for any subset $A$ of a discrete cube ${0, 1}^{d}$ , the $k -$ higher energy of $A$ (the number of $2 k -$ tuples $(a_{1}, a_{2}, \dots, a_{2 k})$ in $A…

math.DS2021★ 1 cited

Generalized Collatz Maps with Almost Bounded Orbits

Felipe Gonçalves, Rachel Greenfeld, Jose Madrid

If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable condit…

math.CO2021

Undecidable translational tilings with only two tiles, or one nonabelian tile

Rachel Greenfeld, Terence Tao

We construct an example of a group $G = Z^{2} \times G_{0}$ for a finite abelian group $G_{0}$ , a subset $E$ of $G_{0}$ , and two finite subsets $F_{1}, F_{2}$ of $G$ , such that it is u…

math.DS2021

An uncountable ergodic Roth theorem and applications

Polona Durcik, Rachel Greenfeld, Annina Iseli +2

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous…

math.CA2020★ 1 cited

Decoupling for fractal subsets of the parabola

Alan Chang, Jaume de Dios Pont, Rachel Greenfeld +3

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2} L^{p}$ decoupling for a fractal subset on the parabola ${(t, t^{2}) : 0 \leq t \leq 1}$ to stu…

math.CA2020★ 6 cited

The structure of translational tilings in $Z^{d}$

Rachel Greenfeld, Terence Tao

We obtain structural results on translational tilings of periodic functions in $Z^{d}$ by finite tiles. In particular, we show that any level one tiling of a periodic set in…

math.CA2018

Spectrality of product domains and Fuglede's conjecture for convex polytopes

Rachel Greenfeld, Nir Lev

A set $Ω \subset R^{d}$ is said to be spectral if the space $L^{2} (Ω)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "…

math.CA2016★ 50 cited

Fuglede's spectral set conjecture for convex polytopes

Rachel Greenfeld, Nir Lev

Let $Ω$ be a convex polytope in $R^{d}$ . We say that $Ω$ is spectral if the space $L^{2} (Ω)$ admits an orthogonal basis consisting of exponential functions. There is a conject…

math.CA2016★ 15 cited

Spectrality and tiling by cylindric domains

Rachel Greenfeld, Nir Lev

A bounded set $Ω \subset R^{d}$ is called a spectral set if the space $L^{2} (Ω)$ admits a complete orthogonal system of exponential functions. We prove that a cylindric set $Ω$ …

Translational tilings by axis-parallel polygonal sets

Translational tilings: structured or wild?

Some variants of the periodic tiling conjecture

Periodicity and decidability of translational tilings by rational polygonal sets

On integer distance sets

Undecidability of translational monotilings

Tiling, spectrality and aperiodicity of connected sets

A counterexample to the periodic tiling conjecture

A counterexample to the periodic tiling conjecture (announcement)

Measurable tilings by abelian group actions

Additive energies on discrete cubes

Generalized Collatz Maps with Almost Bounded Orbits

Undecidable translational tilings with only two tiles, or one nonabelian tile

An uncountable ergodic Roth theorem and applications

Decoupling for fractal subsets of the parabola

The structure of translational tilings in Zd

Spectrality of product domains and Fuglede's conjecture for convex polytopes

Fuglede's spectral set conjecture for convex polytopes

Spectrality and tiling by cylindric domains

The structure of translational tilings in $Z^{d}$