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Large sets with small doubling modulo \(p\) are well covered by an arithmetic progression. (English) Zbl 1247.11130

Summary: We prove that there is a small but fixed positive integer \(\varepsilon\) such that for every prime \(p\) larger than a fixed integer, every subset \(S\) of the integers modulo \(p\) which satisfies \(|2S|\leq (2+\varepsilon)|S|\) and \(2(|2S|)-2|S|+3\leq p\) is contained in an arithmetic progression of length \(|2S|-|S|+1\). This is the first result of this nature which places no unnecessary restrictions on the size of \(S\).

MSC:

11P70 Inverse problems of additive number theory, including sumsets
11B30 Arithmetic combinatorics; higher degree uniformity
11B25 Arithmetic progressions
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