@article{13128,
  abstract     = {Given  A⊆GL2(Fq), we prove that there exist disjoint subsets  B,C⊆A such that  A=B⊔C and their additive and multiplicative energies satisfying max{E+(B),E×(C)}≪|A|3/M(|A|), where
M(|A|)=min{q4/3/|A|1/3(log|A|)2/3,|A|4/5/q13/5(log|A|)27/10}.
 We also study some related questions on moderate expanders over matrix rings, namely, for  A,B,C⊆GL2(Fq), we have |AB+C|, |(A+B)C|≫q4, whenever  |A||B||C|≫q10+1/2. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings,  ForumMath., 31, 951–970).
},
  author       = {Mohammadi, Ali and Pham, Thang and Wang, Yiting},
  issn         = {1496-4287},
  journal      = {Canadian Mathematical Bulletin},
  number       = {4},
  pages        = {1280--1295},
  publisher    = {Cambridge University Press},
  title        = {{An energy decomposition theorem for matrices and related questions}},
  doi          = {10.4153/S000843952300036X},
  volume       = {66},
  year         = {2023},
}