Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples

[2309.09124] Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2309.09124 (math) [Submitted on 17 Sep 2023 ( v1 ), last revised 12 Feb 2025 (this version, v4)] Title: Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples Authors: Seoyoung Kim , Chi Hoi Yip , Semin Yoo View a PDF of the paper titled Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples, by Seoyoung Kim and 2 other authors View PDF HTML (experimental) Abstract: In this paper, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup $G$ contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is $n$ less than a $k$-th power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress towards a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes $p$, the set $\{x^2-1: x \in \mathbb{F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in $\mathbb{F}_p$ non-trivially. Comments: 33 pages, 1 figure. Title changed, revised based on referee comments Subjects: Number Theory (math.NT) MSC classes: Primary 11B30, 11D72, Secondary 11D45, 11N36, 11L40 Cite as: arXiv:2309.09124 [math.NT] (or arXiv:2309.09124v4 [math.NT] for this version) https://doi.org/10.48550/arXiv.2309.09124 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Canad. J. Math., 2026 Related DOI : https://doi.org/10.4153/S0008414X25000136 Focus to learn more DOI(s) linking to related resources Submission history From: Chi Hoi Yip [ view email ] [v1] Sun, 17 Sep 2023 01:21:39 UTC (86 KB) [v2] Sun, 8 Oct 2023 02:46:29 UTC (85 KB) [v3] Tue, 3 Sep 2024 00:53:49 UTC (83 KB) [v4] Wed, 12 Feb 2025 14:59:39 UTC (87 KB) Full-text links: Access Paper: View a PDF of the paper titled Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples, by Seoyoung Kim and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2023-09 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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