A Note onB2kSequences

✍ Scribed by Sheng Chen

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 182 KB
• Volume: 56
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0001

✦ Synopsis

Let h 2 be an integer. A set A of positive integers is called a B h -sequence if all sums a 1 +a 2 + } } } +a h , where a i # A (i=1, 2, ..., h), are distinct up to rearrangements of the summands. A B h -sequence is also called a Sidon sequence of order h [5].

Let A be a B 2k -sequence. Denote by

Recently, Jia [2] showed that, if A(n 2 ) A(n 2 ), then

As mentioned in [2], the result holds when k=2 without the extra condition A(n 2 ) A(n) 2 and this condition does not always hold for a B 2k -sequence.

Here we show that Theorem. Let A be a B 2k -sequence (k 2). Then

Corollary. Let A=[a 1 <a 2 <a 3 < } } } <a n < } } } ] be an infinite B 2k -sequences. Then lim sup n Ä a n n 2k (log n) 1Â2 = .