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-sequences
Authors:Bernt Lindströ  m
Institution:Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Abstract:A sequence $A$ of positive integers is called a $B_hg]$-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 $B_h1]$-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 $B_hg]$-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 $B_hg]$-sequence of size $(gn)^{1/h}(1+o(1))$, when $n\to \infty$.

Keywords:$B_h$-sequence  Sidon sequence
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