An energy decomposition theorem for matrices and related questions

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).