Upper and lower bounds on $B_k^{+}$-sets

Abstract

Let $G$ be an abelian group. A set $A\subset G$ is a \emph{$B_k^{+}$-set} if whenever $a_1+\cdots +a_k=b_1+\cdots +b_k$ with $a_i,b_j\in A$ there is an $i$ and a $j$ such that $a_i=b_j$. If $A$ is a $B_k$-set then it is also a $B_k^{+}$-set but the converse is not true in general. Determining the largest size of a $B_k$-set in the interval $\{1, 2, \dots, N \} \subset \integers$ or in the cyclic group $\integers_N$ is a well studied problem. In this paper we investigate the corresponding problem for $B_k^{+}$-sets. We prove non-trivial upper bounds on the maximum size of a $B_k^{+}$-set contained in the interval $\{1,2,\dots ,N\}$. For odd $k\ge 3$, we construct $B_k^{+}$-sets that have more elements than the $B_k$-sets constructed by Bose and Chowla. We prove a $B_3^{+}$-set $A \subset \integers_N$ has at most $(1+o(1))(8N{)}^{1/3}$ elements. Finally we obtain new upper bounds on the maximum size of a $B_k^{\ast }$-set $A\subset \{1,2,\dots ,N\}$, a problem first investigated by Ruzsa.

arXiv comment26 pages