$B_h[g]$-sequences from $B_h$-sequences

Abstract:

A sequence $A$ of positive integers is called a $B_h[g]$-sequence if every integer $n$ has at most $g$ representations $n=a_1+a_2+\cdots +a_{h’}$ with all $a_i$ in $A$ and $a_1\le a_2\le \cdots \le a_h$. A $B_h[1]$-sequence is also called a $B_h$-sequence or Sidon sequence. The main result is the following Theorem. Let $A$ be a $B_h$-sequence and $g=m^{h-1}$ for an integer $m\ge 2$. Then there is a $B_h[g]$-sequence $B$ of size $|B|=m|A|$, where $B= \bigcup ^{m-1}_{i=0} \{ma+i| a\in A\}$. Corollary. Let $g=m^{h-1}$. The interval $[1,n]$ then contains a $B_h[g]$-sequence of size $(gn)^{1/h}(1+o(1))$, when $n\to \infty$.