On two new kinds of restricted sumsets

Abstract

Let $A_1,\dots ,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots =|A_n|\ge 2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\dots ,A_n)=\{a_1+\cdots +a_n:a_1\in A_1,\dots ,a_n\in A_n,\text{and}{a}_i\ne a_{i+1}\text{for}1\le i<n\}$$ and $$C(A_1,\dots ,A_n)=\{a_1+\cdots +a_n:a_i\in A_i(1\le i\le n),\text{and}{a}_i\ne a_{i+1}\text{for}1\le i<n,\text{and}{a}_n\ne a_1\}$$ recently introduced by the second author, when $G$ is the additive group of a field we obtain lower bounds for $|L(A_1,\dots ,A_n)|$ and $|C(A_1,\dots ,A_n)|$ via the polynomial method. Moreover, when $G$ is torsion-free and $A_1=\cdots =A_n$, we determine completely when $|L(A_1,\dots ,A_n)|$ or $|C(A_1,\dots ,A_n)|$ attains its lower bound.

19 pages