{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T09:40:01Z","timestamp":1746524401867,"version":"3.40.4"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T00:00:00Z","timestamp":1744070400000},"content-version":"vor","delay-in-days":7,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100006919","name":"Massachusetts Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100006919","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,4]]},"abstract":"Abstract<\/jats:title>\n A subset S<\/jats:italic> of real numbers is called bi-Sidon<\/jats:italic> if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of S<\/jats:italic> are distinct. Imre Ruzsa asked the following question: What is the maximum number f<\/jats:italic>(N<\/jats:italic>) such that every set S<\/jats:italic> of N<\/jats:italic> real numbers contains a bi-Sidon subset of size at least f<\/jats:italic>(N<\/jats:italic>)? He proved that \n \n $$f(N)\\geqslant cN^{\\frac{1}{3}}$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n \n (<\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2a7e<\/mml:mo>\n c<\/mml:mi>\n \n N<\/mml:mi>\n \n 1<\/mml:mn>\n 3<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, for a constant \n \n $$c>0$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. In this note, we improve this bound to \n \n $$N^{\\frac{1}{3}+\\frac{7}{78}+o(1)}$$<\/jats:tex-math>\n \n \n N<\/mml:mi>\n \n \n 1<\/mml:mn>\n 3<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n \n 7<\/mml:mn>\n 78<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n o<\/mml:mi>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00493-025-00151-5","type":"journal-article","created":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T21:29:51Z","timestamp":1744147791000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ruzsa\u2019s Problem on Bi-Sidon Sets"],"prefix":"10.1007","volume":"45","author":[{"given":"J\u00e1nos","family":"Pach","sequence":"first","affiliation":[]},{"given":"Dmitrii","family":"Zakharov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,8]]},"reference":[{"issue":"1","key":"151_CR1","first-page":"207","volume":"68","author":"A Balog","year":"2017","unstructured":"Balog, A., Wooley, T.D.: A low-energy decomposition theorem. 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