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


Author: Bernt Lindström
Journal: Proc. Amer. Math. Soc. 128 (2000), 657-659
MSC (2000): Primary 11B75, 11P99
DOI: https://doi.org/10.1090/S0002-9939-99-05122-9
Published electronically: September 9, 1999
MathSciNet review: 1636907
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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 $n\to \infty$.


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Additional Information

Bernt Lindström
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Email: bernt@math.kth.se

DOI: https://doi.org/10.1090/S0002-9939-99-05122-9
Keywords: $B_h$-sequence, Sidon sequence
Received by editor(s): April 17, 1998
Published electronically: September 9, 1999
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society

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