Publications and preprints

• Periodicity and decidability of translational tilings by rational polygonal sets (with J. de Dios, J. Grebik, J. Madrid), preprint.
  
  
  
• On integer distance sets (with M. Iliopoulou, S. Peluse), preprint.
  
  
  Quanta Magazine: Merging Fields, Mathematicians Go the Distance on Old Problem
  



• Undecidability of translational monotilings (with T. Tao), to appear in the Journal of the European Mathematical Society.
  
  
  
• Tiling, spectrality and aperiodicity of connected sets (with M. Kolountzakis), to appear in Israel Journal of Mathematics.
  
  
  
• A counterexample to the periodic tiling conjecture (with T. Tao), Annals of Mathematics, 200 (2024), 1-63. [arXiv]
  
  
  Quanta Magazine: 'Nasty' Geometry Breaks Decades-Old Tiling Conjecture
  
  
  
• Measurable tilings by abelian group actions (with J. Grebik, V. Rozhon, T. Tao), International Mathematics Research Notices (2023). [arXiv]
  
  
  
• Additive energies on discrete cubes (with J. de Dios, P. Ivanisvili, J. Madrid), Discrete Analysis 2023:13, 16 pp.
  
  
  
• Generalized Collatz maps with almost bounded orbits (with F. Gonçalves, J. Madrid), to appear in Indiana University Mathematics Journal.
  
  
  
• Undecidable translational tilings with only two tiles, or one nonabelian tile (with T. Tao), Discrete & Computational Geometry 70 (4), 1652-1706 (2023). [arXiv]
  
  
  
• An uncountable ergodic Roth theorem and applications (with P. Durcik, A. Iseli, A. Jamneshan, J. Madrid), Discrete and Continuous Dynamical Systems, 2022, 42 (11) 5509-5540. [arXiv]
  
  
  
• Decoupling for fractal subsets of the parabola (with A. Chang, J. de Dios, A. Jamneshan, Z. Li, J. Madrid), Mathematische Zeitschrift, 301, 1851-1879 (2022). [arXiv]
  
  
  
• The structure of translational tilings in ℤᵈ (with T. Tao) Discrete Analysis 2021:16, 28 pp.
  
  
  
• Spectrality of product domains and Fuglede's conjecture for convex polytopes (with N. Lev) Journal d'Analyse Mathematique, 140 (2020) 409-441. [arXiv]
  
  
  
• Fuglede's spectral set conjecture for convex polytopes (with N. Lev) Analysis & PDE, 10 (2017), 1497-1538. [arXiv]
  



• Spectrality and tiling by cylindric domains (with N. Lev) Journal of Functional Analysis, 271 (2016), 2808-2821. [arXiv]