@article{13128,
  abstract     = {Given 𝐴 ⊆𝐺⁡𝐿2⁡(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 ⊆𝐺⁡𝐿2⁡(𝔽𝑞), we have
|𝐴⁢𝐵+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, 𝐹⁡𝑜⁢𝑟⁢𝑢⁢𝑚⁢𝑀⁢𝑎⁢𝑡⁢ℎ., 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},
}