Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

中文翻译：

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通过傅里叶分析的加法碱基

将 Alon、Linial 和 Meshulam 的结果推广到阿贝尔群，我们证明如果G是指数的有限阿贝尔群米和小号是一系列元素G使得任何子序列小号至少包括$$|S| - m\ln |G|$$元素生成G， 然后小号是一个加法基础G. 我们还证明了任何的加性跨度l发电机组G至少包含一个大小为的子群的陪集$$|G{|^{1 - c{ \in ^l}}}$$对于某些C=C(米） 和$$ \in = \in (m) < 1$$; 我们使用概率方法给出更清晰的值C(米） 和$$ \in (m)$$在这种情况下G是向量空间；我们给出了相关已知结果的新证明。