{"_id":"18930","OA_place":"publisher","publication":"The Quarterly Journal of Mathematics","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"department":[{"_id":"TiBr"}],"publication_status":"published","doi":"10.1093/qmath/haae044","publisher":"Oxford University Press","abstract":[{"text":"We study sumsets 𝒜 + ℬ in the set of squares 𝒮 (and, more generally, in the set of kth powers 𝒮k, where k ≥2 is an integer). It is known by a result of Gyarmati that 𝒜 + ℬ ⊂ 𝒮k ∩[1,N] implies that min(|𝒜|,|ℬ|) =Ok(logN). Here, we study how the upper bound on |ℬ| decreases, when the size of |𝒜| increases (or vice versa). In particular, if |𝒜| ≥ Ck1m m(logN)1m , then |ℬ| = Ok(m2logN), for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with m ∼ loglogN this gives: If |𝒜| ≥ CkloglogN,then |ℬ| = Ok(logN(loglogN)2).","lang":"eng"}],"file":[{"date_created":"2025-01-28T07:03:51Z","file_size":424645,"content_type":"application/pdf","file_name":"2024_QuarterlyJourMath_Elsholtz.pdf","date_updated":"2025-01-28T07:03:51Z","creator":"dernst","success":1,"access_level":"open_access","relation":"main_file","checksum":"1a06e052761d3f1e873463d6f529dd82","file_id":"18931"}],"citation":{"mla":"Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.” <i>The Quarterly Journal of Mathematics</i>, vol. 75, no. 4, Oxford University Press, 2024, pp. 1243–54, doi:<a href=\"https://doi.org/10.1093/qmath/haae044\">10.1093/qmath/haae044</a>.","chicago":"Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.” <i>The Quarterly Journal of Mathematics</i>. Oxford University Press, 2024. <a href=\"https://doi.org/10.1093/qmath/haae044\">https://doi.org/10.1093/qmath/haae044</a>.","short":"C. Elsholtz, L. Wurzinger, The Quarterly Journal of Mathematics 75 (2024) 1243–1254.","ista":"Elsholtz C, Wurzinger L. 2024. Sumsets in the set of squares. The Quarterly Journal of Mathematics. 75(4), 1243–1254.","ama":"Elsholtz C, Wurzinger L. Sumsets in the set of squares. <i>The Quarterly Journal of Mathematics</i>. 2024;75(4):1243-1254. doi:<a href=\"https://doi.org/10.1093/qmath/haae044\">10.1093/qmath/haae044</a>","ieee":"C. Elsholtz and L. Wurzinger, “Sumsets in the set of squares,” <i>The Quarterly Journal of Mathematics</i>, vol. 75, no. 4. Oxford University Press, pp. 1243–1254, 2024.","apa":"Elsholtz, C., &#38; Wurzinger, L. (2024). Sumsets in the set of squares. <i>The Quarterly Journal of Mathematics</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/qmath/haae044\">https://doi.org/10.1093/qmath/haae044</a>"},"date_published":"2024-12-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","date_updated":"2025-01-28T07:04:45Z","corr_author":"1","month":"12","type":"journal_article","author":[{"first_name":"Christian","full_name":"Elsholtz, Christian","last_name":"Elsholtz"},{"first_name":"Lena","full_name":"Wurzinger, Lena","id":"50c57d72-32a8-11ee-aeea-d652094d2ccd","last_name":"Wurzinger"}],"article_type":"original","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"4","license":"https://creativecommons.org/licenses/by/4.0/","volume":75,"acknowledgement":"This manuscript grew out of the second author’s MSc Thesis at Graz University of Technology [34]. C. Elsholtz is supported by a joint FWF-ANR project ArithRand, grant numbers FWF I 4945-N and ANR-20-CE91-0006. Both authors would like to thank Igor Shparlinski for drawing our attention to related character sum estimates. Furthermore, we would like to thank the referee for a careful reading of the paper.","scopus_import":"1","date_created":"2025-01-28T06:55:31Z","quality_controlled":"1","has_accepted_license":"1","day":"01","language":[{"iso":"eng"}],"publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"page":"1243-1254","intvolume":"        75","status":"public","oa":1,"oa_version":"Published Version","OA_type":"hybrid","file_date_updated":"2025-01-28T07:03:51Z","title":"Sumsets in the set of squares","year":"2024","ddc":["510"]}