论文标题
在两种新型的限制性总和
On two new kinds of restricted sumsets
论文作者
论文摘要
令$ a_1，\ ldots，a_n $为$ | a_1 | = \ cdots = | a_n | \ ge2 $的添加剂abelian $ g $的有限子集。关于两种新型限制性总和$$ l（a_1，\ ldots，a_n）= \ {a_1+\ cdots+a_n：\ a_1 \ in A_1，\ ldots，\ ldots，a_n \ in a_n in a_n in a_n，a_n in a_n，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a_i \ a_i \ a_i} 1 \ le i <n \} $$和$$ c（a_1，\ ldots，a_n）= \ {a_1+\ cdots+a_n：\ a_i \ in a_i \ in a_i \（1 \ le i \ le i \ le i \ le n），\ \ \ \ \ \ \\ text {和} 1 \ le i <n，\ \ text {and} \ a_n \ not = a_1 \} $$最近是由第二作者引入的，当$ g $是$ | l（a_1，\ ldots，a_n）| $和$ c（a_1 | c（a_1 c（a_1 c（a_1 c（a_1），a_1，a_1，\ ldots a_ y _ poloyn）的$ g $我们获得的下限我们获得下限。此外，当$ g $不含扭转和$ a_1 = \ cdots = a_n $时，我们完全确定$ | l（a_1，\ ldots，a_n）| $或$ | c（a_1，\ ldots，a_n）|获得其下限。
Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n\not=a_1\}$$ recently introduced by the second author, when $G$ is the additive group of a field we obtain lower bounds for $|L(A_1,\ldots,A_n)|$ and $|C(A_1,\ldots,A_n)|$ via the polynomial method. Moreover, when $G$ is torsion-free and $A_1=\cdots=A_n$, we determine completely when $|L(A_1,\ldots,A_n)|$ or $|C(A_1,\ldots,A_n)|$ attains its lower bound.
