{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T06:32:03Z","timestamp":1760596323745},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2023,8]]},"abstract":"Abstract<\/jats:title>The main results of this paper concern growth in sums of a k<\/jats:italic>-convex function f<\/jats:italic>. Firstly, we streamline the proof (from Hanson et al. (Combinatorica 42:71\u201385, 2020)) of a growth result for f<\/jats:italic>(A<\/jats:italic>) where A<\/jats:italic> has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for $$\\begin{aligned} |2^k f(A) - (2^k-1)f(A)|. \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n f<\/mml:mi>\n \n (<\/mml:mo>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n -<\/mml:mo>\n \n (<\/mml:mo>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n f<\/mml:mi>\n \n (<\/mml:mo>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n |<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>We also generalise a recent result from Hanson et al. (J Lond Math Soc, 2021), proving that for any finite $$A\\subset \\mathbb {R}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n \u2282<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>$$\\begin{aligned} | 2^k f(sA-sA) - (2^k-1) f(sA-sA)| \\gg _s |A|^{2s} \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n f<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n A<\/mml:mi>\n -<\/mml:mo>\n s<\/mml:mi>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n -<\/mml:mo>\n \n (<\/mml:mo>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n f<\/mml:mi>\n \n (<\/mml:mo>\n s<\/mml:mi>\n A<\/mml:mi>\n -<\/mml:mo>\n s<\/mml:mi>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u226b<\/mml:mo>\n s<\/mml:mi>\n <\/mml:msub>\n \n \n |<\/mml:mo>\n A<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>where $$s = \\frac{k+1}{2}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n =<\/mml:mo>\n \n \n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This allows us to prove that, given any natural number $$s \\in \\mathbb {N}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there exists $$m = m(s)$$<\/jats:tex-math>\n \n m<\/mml:mi>\n =<\/mml:mo>\n m<\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that if $$A \\subset \\mathbb {R}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n \u2282<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then $$\\begin{aligned} |(sA-sA)^{(m)}| \\gg _s |A|^{s}. \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n \n |<\/mml:mo>\n \n (<\/mml:mo>\n s<\/mml:mi>\n A<\/mml:mi>\n -<\/mml:mo>\n s<\/mml:mi>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u226b<\/mml:mo>\n s<\/mml:mi>\n <\/mml:msub>\n \n \n |<\/mml:mo>\n A<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>This is progress towards a conjecture (Balog et al. in Electron J Comb 24(3):Paper No. 3.14, 17, 2017) which states that (1) can be replaced with $$\\begin{aligned} |(A-A)^{(m)}| \\gg _s |A|^{s}. \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n \n |<\/mml:mo>\n \n (<\/mml:mo>\n A<\/mml:mi>\n -<\/mml:mo>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u226b<\/mml:mo>\n s<\/mml:mi>\n <\/mml:msub>\n \n \n |<\/mml:mo>\n A<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>Developing methods of Solymosi, and Bloom and Jones, and using an idea from Bradshaw et al. (Electron J Comb 29, 2021), we present some new sum-product type results in the complex numbers $$\\mathbb {C}$$<\/jats:tex-math>\n C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and in the function field $$\\mathbb {F}_q((t^{-1}))$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n (<\/mml:mo>\n \n t<\/mml:mi>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00493-023-00035-6","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T14:03:00Z","timestamp":1685973780000},"page":"769-789","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Growth in Sumsets of Higher Convex Functions"],"prefix":"10.1007","volume":"43","author":[{"given":"Peter J.","family":"Bradshaw","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"35_CR1","doi-asserted-by":"crossref","unstructured":"Balog, A., Roche-Newton, O., Zhelezov, D.: Expanders with superquadratic growth. 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