---
res:
  bibo_abstract:
  - "Given \U0001D434 ⊆\U0001D43A⁡\U0001D43F2⁡(\U0001D53D\U0001D45E), we prove that
    there exist disjoint subsets \U0001D435,\U0001D436 ⊆\U0001D434 such that \U0001D434
    =\U0001D435 ⊔\U0001D436 and their additive and multiplicative energies satisfying\r\nmax⁡{\U0001D438+⁡(\U0001D435),\U0001D438×⁡(\U0001D436)}≪|\U0001D434|3/\U0001D440⁡(|\U0001D434|),
    where\r\n\U0001D440⁡(|\U0001D434|)=min⁡{\U0001D45E4/3/|\U0001D434|1/3⁢(log⁡|\U0001D434|)2/3,
    |\U0001D434|4/5/\U0001D45E13/5⁢(log⁡|\U0001D434|)27/10}.\r\n \r\nWe also study
    some related questions on moderate expanders over matrix rings, namely, for \U0001D434,\U0001D435,\U0001D436
    ⊆\U0001D43A⁡\U0001D43F2⁡(\U0001D53D\U0001D45E), we have\r\n|\U0001D434⁢\U0001D435+\U0001D436|,
    |(\U0001D434+\U0001D435)⁢\U0001D436|≫\U0001D45E4,\r\n whenever |\U0001D434|⁢|\U0001D435|⁢|\U0001D436|
    ≫\U0001D45E10+1/2. These improve earlier results due to Karabulut, Koh, Pham,
    Shen, and Vinh ([2019], Expanding phenomena over matrix rings, \U0001D439⁡\U0001D45C⁢\U0001D45F⁢\U0001D462⁢\U0001D45A⁢\U0001D440⁢\U0001D44E⁢\U0001D461⁢ℎ.,
    31, 951–970).@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Ali
      foaf_name: Mohammadi, Ali
      foaf_surname: Mohammadi
  - foaf_Person:
      foaf_givenName: Thang
      foaf_name: Pham, Thang
      foaf_surname: Pham
  - foaf_Person:
      foaf_givenName: Yiting
      foaf_name: Wang, Yiting
      foaf_surname: Wang
      foaf_workInfoHomepage: http://www.librecat.org/personId=1917d194-076e-11ed-97cd-837255f88785
    orcid: 0000-0002-2856-767X
  bibo_doi: 10.4153/S000843952300036X
  bibo_issue: '4'
  bibo_volume: 66
  dct_date: 2023^xs_gYear
  dct_identifier:
  - UT:001011963000001
  dct_isPartOf:
  - http://id.crossref.org/issn/0008-4395
  - http://id.crossref.org/issn/1496-4287
  dct_language: eng
  dct_publisher: Cambridge University Press@
  dct_title: An energy decomposition theorem for matrices and related questions@
...