Multiplicative irreducibility of shifted multiplicative subgroups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912964206919680 |
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| author | Kim, Seoyoung Yip, Chi Hoi Yoo, Semin |
| author_facet | Kim, Seoyoung Yip, Chi Hoi Yoo, Semin |
| contents | In a recent breakthrough, Kalmynin resolved conjectures of Lev--Sonn and Sárközy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of Sárközy, we prove multiplicative analogues of Kalmynin's results. We show that for every proper multiplicative subgroup $G$, the shifted set $(G-1)\setminus\{0\}$ cannot be written as a product set nontrivially, addressing a conjecture of Sárközy. In addition, we prove that no nonzero shift of any coset of a proper multiplicative subgroup is a ratio set of the form $A/A$. Our results substantially sharpen previous theorems of Shkredov and the authors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_20919 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative irreducibility of shifted multiplicative subgroups Kim, Seoyoung Yip, Chi Hoi Yoo, Semin Combinatorics Number Theory 11P70, 11B30 In a recent breakthrough, Kalmynin resolved conjectures of Lev--Sonn and Sárközy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of Sárközy, we prove multiplicative analogues of Kalmynin's results. We show that for every proper multiplicative subgroup $G$, the shifted set $(G-1)\setminus\{0\}$ cannot be written as a product set nontrivially, addressing a conjecture of Sárközy. In addition, we prove that no nonzero shift of any coset of a proper multiplicative subgroup is a ratio set of the form $A/A$. Our results substantially sharpen previous theorems of Shkredov and the authors. |
| title | Multiplicative irreducibility of shifted multiplicative subgroups |
| topic | Combinatorics Number Theory 11P70, 11B30 |
| url | https://arxiv.org/abs/2602.20919 |