@article{20367,
  abstract     = {We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.},
  author       = {Browning, Timothy D and Sawin, Will and Wang, Victor},
  issn         = {1432-1807},
  journal      = {Mathematische Annalen},
  pages        = {1863–1880},
  publisher    = {Springer Nature},
  title        = {{Pairs of commuting integer matrices}},
  doi          = {10.1007/s00208-025-03285-5},
  volume       = {393},
  year         = {2025},
}