A finite set A = { a 1 < … < a n }⊂ ℝ is said to be convex if the sequence ( a i − a i −1 ) i =2 n is strictly increasing. Using an estimate of the additive energy of convex sets, one can estimate the size of the sumset as ∣ A + A ∣ ≳ ∣ A ∣ 102/65 , which slightly sharpens Shkredov’s latest result ∣ A + A ∣ ≳ ∣ A ∣ 58/37 .

中文翻译：

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锐化凸集和集大小的估计

如果序列 ( ai − ai −1 ) i =2 n 严格递增，则称有限集 A = { a 1 < … < an }⊂ ℝ 是凸集。使用对凸集的加性能量的估计，可以估计和集的大小为 ∣ A + A ∣ ≳ ∣ A ∣ 102/65 ，这稍微锐化了 Shkredov 的最新结果 ∣ A + A ∣ ≳ ∣ A ∣ 58/ 37 .