Abstract In this paper it is shown that there is an absolute constant κ > 0 with the following property. For any prime p and nonempty subsets A , B of Z / p Z such that 1 | A | p 2 , B ∩ − B = ∅ and | B | κ | A | ln ⁡ ( | A | ) , we have that max b ∈ B ⁡ | ( A + b ) ∖ A | ≥ | B | . In 2011, V. Lev proved the former statement assuming also that | B | ≤ p 8 ; in the same paper, Lev suggested that this technical condition could be eliminated or weakened. In this paper his conjecture is confirmed.

中文翻译：

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关于 Z/pZ 中流行差异的数量

摘要 本文证明存在一个绝对常数κ > 0，其性质如下。对于任何质数 p 和非空子集 A , B 的 Z / p Z 使得 1 | 一个 | p 2 , B ∩ − B = ∅ 和 | 乙 | κ | 一个 | ln ⁡ ( | A | ) ，我们有 max b ∈ B ⁡ | ( A + b ) ∖ A | ≥ | 乙 | . 2011 年，V. Lev 证明了前面的陈述，还假设 | 乙 | ≤ p 8 ；在同一篇论文中，Lev 建议可以消除或削弱这种技术条件。在这篇论文中，他的猜想得到了证实。