$B_h[g]$-sequences from $B_h$-sequences
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- by Bernt Lindström PDF
- Proc. Amer. Math. Soc. 128 (2000), 657-659 Request permission
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 $n\to \infty$.References
- D. Hajela, Some remarks on $B_h[g]$ sequences, J. Number Theory 29 (1988), no. 3, 311–323. MR 955956, DOI 10.1016/0022-314X(88)90109-6
- D. Hajela, Some remarks on $B_h[g]$ sequences, J. Number Theory 29 (1988), no. 3, 311–323. MR 955956, DOI 10.1016/0022-314X(88)90109-6
- H. Halberstam and K. F. Roth, Sequences. Vol. I, Clarendon Press, Oxford, 1966. MR 0210679
- Xingde Jia, $B_h[g]$-sequences with large upper density, J. Number Theory 56 (1996), no. 2, 298–308. MR 1373553, DOI 10.1006/jnth.1996.0019
- Torleiv Kløve, Constructions of $B_h[g]$-sequences, Acta Arith. 58 (1991), no. 1, 65–78. MR 1111091, DOI 10.4064/aa-58-1-65-78
- Mihail N. Kolountzakis, The density of $B_h[g]$ sequences and the minimum of dense cosine sums, J. Number Theory 56 (1996), no. 1, 4–11. MR 1370193, DOI 10.1006/jnth.1996.0002
Additional Information
- Bernt Lindström
- Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
- Email: bernt@math.kth.se
- Received by editor(s): April 17, 1998
- Published electronically: September 9, 1999
- Communicated by: David E. Rohrlich
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1636907