On Finite Sidon Sequences

✍ Scribed by X.D. Jia

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 211 KB
• Volume: 44
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1993.1037

✦ Synopsis

A set (A) of integers is called a (B_{h})-sequence if all sums (a_{1}+\cdots+a_{h}), where (a_{i} \in A), are distinct up to rearrangement of the summands. Let (F_{h}(n)) (resp. (\left.f_{h}(n)\right)) denote the size of a largest (B_{h})-sequence (resp. (B_{h})-sequence for (\mathbf{Z} /(n)) ). It is proved that, for every (r \geqslant 1) as (n \rightarrow \infty, F_{2 r}(n) \leqslant r^{1 /(2 r)}(r!)^{1 / r} n^{1 / 2 r)}+O\left(n^{1 / 4 r}\right), f_{2 r}(n) \leqslant(r!)^{1 / r} n^{1 / 2 r)}+) (O\left(n^{1 /(4 r)}\right)). Some open problems concerning (B_{h})-sequences are also discussed in this paper. 1993 Academic Press, Inc.