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Small doubling in prime-order groups: from 2.4 to 2.6. (English) Zbl 1470.11259

Let \(p\) be a prime. According to the celebrated 2.4-Theorem by G. A. Freïman [Sov. Math., Dokl. 2, 1520–1522 (1961; Zbl 0109.27203); translation from Dokl. Akad. Nauk SSSR 141, 571–573 (1961)], if \(A\subset\mathbb{Z}/p\mathbb{Z}\) satisfies \(|A+A|< 2.4|A|-3\) and \(|A|< p/35\), then \(A\) is contained in an arithmetic progression with at most \(|A+A|- |A|+1\) terms. As to the constants 2.4 and 1/35, B. Green and I. Z. Ruzsa [Bull. Lond. Math. Soc. 38, No. 1, 43–52 (2006; Zbl 1155.11307)] obtained 3 (the best one) and \(1/96^{108}\). Ø. J. Rødseth [Skr., K. Nor. Vidensk. Selsk. 2006, No. 4, 11–18 (2006; Zbl 1162.11010)] proved 2.4 and 1/10.7. P. Candela et al. [J. Théor. Nombres Bordx. 32, No. 1, 275–289 (2020; Zbl 1459.11202)] obtained 2.48 (but with \(-7\) instead of \(-3\)) and \(1/10^{10}\).
In the paper under review, the authors prove 2.59 and 0.0045 for \(|A|>100\). The improvement comes from using the properties of higher energies. They also obtain an analogous result for \(|A-A|\) with 2.6 and 0.0045.

MSC:

11P70 Inverse problems of additive number theory, including sumsets
11B13 Additive bases, including sumsets
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References:

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