article published yes Ali Mohammadi author Thang Pham author Yiting Wang author 1917d194-076e-11ed-97cd-837255f887850000-0002-2856-767X GradSch department 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). Cambridge University Press2023 eng 0008-4395 1496-4287 2106.07328 00101196300000110.4153/S000843952300036X 6641280-1295 Mohammadi, Ali, et al. “An Energy Decomposition Theorem for Matrices and Related Questions.” <i>Canadian Mathematical Bulletin</i>, vol. 66, no. 4, Cambridge University Press, 2023, pp. 1280–95, doi:<a href="https://doi.org/10.4153/S000843952300036X">10.4153/S000843952300036X</a>. A. Mohammadi, T. Pham, Y. Wang, Canadian Mathematical Bulletin 66 (2023) 1280–1295. Mohammadi A, Pham T, Wang Y. 2023. An energy decomposition theorem for matrices and related questions. Canadian Mathematical Bulletin. 66(4), 1280–1295. Mohammadi, A., Pham, T., &#38; Wang, Y. (2023). An energy decomposition theorem for matrices and related questions. <i>Canadian Mathematical Bulletin</i>. Cambridge University Press. <a href="https://doi.org/10.4153/S000843952300036X">https://doi.org/10.4153/S000843952300036X</a> Mohammadi, Ali, Thang Pham, and Yiting Wang. “An Energy Decomposition Theorem for Matrices and Related Questions.” <i>Canadian Mathematical Bulletin</i>. Cambridge University Press, 2023. <a href="https://doi.org/10.4153/S000843952300036X">https://doi.org/10.4153/S000843952300036X</a>. Mohammadi A, Pham T, Wang Y. An energy decomposition theorem for matrices and related questions. <i>Canadian Mathematical Bulletin</i>. 2023;66(4):1280-1295. doi:<a href="https://doi.org/10.4153/S000843952300036X">10.4153/S000843952300036X</a> A. Mohammadi, T. Pham, and Y. Wang, “An energy decomposition theorem for matrices and related questions,” <i>Canadian Mathematical Bulletin</i>, vol. 66, no. 4. Cambridge University Press, pp. 1280–1295, 2023. 131282023-06-11T22:00:40Z2026-04-08T13:04:49Z