[{"_id":"13128","year":"2023","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2106.07328","open_access":"1"}],"article_type":"original","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.4153/S000843952300036X","date_created":"2023-06-11T22:00:40Z","publisher":"Cambridge University Press","language":[{"iso":"eng"}],"publication":"Canadian Mathematical Bulletin","date_updated":"2026-04-08T13:04:49Z","isi":1,"abstract":[{"lang":"eng","text":"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\r\nmax⁡{𝐸+⁡(𝐵),𝐸×⁡(𝐶)}≪|𝐴|3/𝑀⁡(|𝐴|), where\r\n𝑀⁡(|𝐴|)=min⁡{𝑞4/3/|𝐴|1/3⁢(log⁡|𝐴|)2/3, |𝐴|4/5/𝑞13/5⁢(log⁡|𝐴|)27/10}.\r\n \r\nWe also study some related questions on moderate expanders over matrix rings, namely, for 𝐴,𝐵,𝐶 ⊆𝐺⁡𝐿2⁡(𝔽𝑞), we have\r\n|𝐴⁢𝐵+𝐶|, |(𝐴+𝐵)⁢𝐶|≫𝑞4,\r\n 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)."}],"publication_status":"published","arxiv":1,"quality_controlled":"1","issue":"4","scopus_import":"1","page":"1280-1295","external_id":{"isi":["001011963000001"],"arxiv":["2106.07328"]},"title":"An energy decomposition theorem for matrices and related questions","publication_identifier":{"issn":["0008-4395"],"eissn":["1496-4287"]},"intvolume":"        66","volume":66,"department":[{"_id":"GradSch"}],"citation":{"ista":"Mohammadi A, Pham T, Wang Y. 2023. An energy decomposition theorem for matrices and related questions. Canadian Mathematical Bulletin. 66(4), 1280–1295.","apa":"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>","short":"A. Mohammadi, T. Pham, Y. Wang, Canadian Mathematical Bulletin 66 (2023) 1280–1295.","chicago":"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>.","mla":"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>.","ieee":"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.","ama":"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>"},"type":"journal_article","author":[{"first_name":"Ali","last_name":"Mohammadi","full_name":"Mohammadi, Ali"},{"first_name":"Thang","last_name":"Pham","full_name":"Pham, Thang"},{"full_name":"Wang, Yiting","orcid":"0000-0002-2856-767X","last_name":"Wang","first_name":"Yiting","id":"1917d194-076e-11ed-97cd-837255f88785"}],"month":"12","day":"01","oa_version":"Preprint","article_processing_charge":"No","date_published":"2023-12-01T00:00:00Z","status":"public"}]