子集和的渐近紧界
Asymptotically tight bounds on subset sums
摘要：
对于有限阿贝尔群G的子集A，定义Sigma（A）={ sum_{a\in B} a：B \subset A}.在Sigma（A）具有平凡稳定子的情况下，可以推导出Sigma（A）的大小在|A|中至少是二次的;|西格玛（A）|>= |A| ^{2}/64最近由De Vos、Goddyn、Mohar和Samal获得。我们将这个界改进到渐近最好的可能结果|西格玛（A）|>=（1/4-o（1））|A| ^{2}。我们还研究了一个相关的问题，其中A是任何子集的Z_{n}与所有元素互质的A的n;它最近已被证明，由Vu，如果这样的一个集合A的性质西格玛（A）是不是Z_{n}然后|A| =O（sqrt{n}）。De Vos，Goddyn，Mohar和Samal将此界改进为|A| <= 8 sqrt {n}，我们进一步改进此界，得到渐近最佳可能结果|A| <=（2+o（1））sqrt{n}.
英文摘要：
For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}.
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