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On Freiman’s 2.4-Theorem. (English) Zbl 1162.11010

Summary: Gregory Freiman’s celebrated “2.4-Theorem” says that if \(A\) is a set of residue classes modulo a prime \(p\) satisfying \(|2A|\leq 2.4\;|A|-3\) and \(|A| < p/35\), then \(A\) is contained in an arithmetic progression of length \(|2A|-|A|+1\). Without much extra effort, the bound \(|A| < p/35\) can be relaxed to \(|A|\leq p/11.3\). A result of Freiman on the distribution of points on the circle plays an important rôle in the proof. This result was recently refined by Lev. By using Lev’s result instead of Freiman’s, we increase the bound on the size of \(A\) to \(|A| \leq p/10.7\).

MSC:

11B25 Arithmetic progressions
11B75 Other combinatorial number theory
11P70 Inverse problems of additive number theory, including sumsets
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