{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T04:06:00Z","timestamp":1750910760811,"version":"3.41.0"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T00:00:00Z","timestamp":1747008000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T00:00:00Z","timestamp":1747008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,6]]},"abstract":"Abstract<\/jats:title>\n The Liebeck\u2013Nikolov\u2013Shalev conjecture (Bull Lond Math Soc 44(3):469\u2013472, 2012) asserts that, for any finite simple non-abelian group G<\/jats:italic> and any set \n \n $$A\\subseteq G$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \u2286<\/mml:mo>\n G<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with \n \n $$|A|\\ge 2$$<\/jats:tex-math>\n \n \n |<\/mml:mo>\n A<\/mml:mi>\n |<\/mml:mo>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, G<\/jats:italic> is the product of at most \n \n $$N\\frac{\\log |G|}{\\log |A|}$$<\/jats:tex-math>\n \n \n N<\/mml:mi>\n \n \n log<\/mml:mo>\n |<\/mml:mo>\n G<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n log<\/mml:mo>\n |<\/mml:mo>\n A<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> conjugates of A<\/jats:italic>, for some absolute constant N<\/jats:italic>. For G<\/jats:italic> of Lie type, we prove that for any \n \n $$\\varepsilon >0$$<\/jats:tex-math>\n \n \n \u03b5<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> there is some \n \n $$N_{\\varepsilon }$$<\/jats:tex-math>\n \n \n N<\/mml:mi>\n \u03b5<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for which G<\/jats:italic> is the product of at most \n \n $$N_{\\varepsilon }\\left( \\frac{\\log |G|}{\\log |A|}\\right) ^{1+\\varepsilon }$$<\/jats:tex-math>\n \n \n \n N<\/mml:mi>\n \u03b5<\/mml:mi>\n <\/mml:msub>\n \n \n \n \n log<\/mml:mo>\n |<\/mml:mo>\n G<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n log<\/mml:mo>\n |<\/mml:mo>\n A<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mfenced>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n \u03b5<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> conjugates of either A<\/jats:italic> or \n \n $$A^{-1}$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. For symmetric sets, this improves on results of Liebeck et al. (2012) and Gill et al. (Groups Geom Dyn 7(4):867\u2013882, 2013). During the preparation of this paper, the proof of the Liebeck\u2013Nikolov\u2013Shalev conjecture was completed by Lifshitz (Completing the proof of the Liebeck\u2013Nikolov\u2013Shalev conjecture, 2024, https:\/\/arxiv.org\/abs\/2408.10127<\/jats:ext-link>). Both papers use Gill et al. (Initiating the proof of the Liebeck\u2013Nikolov\u2013Shalev conjecture, 2024, https:\/\/arxiv.org\/abs\/2408.07800<\/jats:ext-link>) as a starting point. Lifshitz\u2019s argument uses heavy machinery from representation theory to complete the conjecture, whereas this paper achieves a more modest result by rather elementary combinatorial arguments.<\/jats:p>","DOI":"10.1007\/s00493-025-00155-1","type":"journal-article","created":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T15:51:25Z","timestamp":1747065085000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Writing Finite Simple Groups of Lie Type as Products of Subset Conjugates"],"prefix":"10.1007","volume":"45","author":[{"given":"Daniele","family":"Dona","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,12]]},"reference":[{"issue":"1","key":"155_CR1","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1137\/0209018","volume":"9","author":"L Babai","year":"1980","unstructured":"Babai, L.: On the complexity of canonical labeling of strongly regular graphs. 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