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On Sidon sequences of even orders. (English) Zbl 0781.11010

Let \(h\geq 2\) be an integer. A set of positive integers \(A\) is called a \(B_ h\)-sequence, or a Sidon sequence of order \(h\), if all sums \(a_ 1+a_ 2+ \cdots+a_ h\), where \(a_ i\in B\) (\(i=1,2,\dots,h\)), are distinct up to rearrangements of the summands. Let \(A(n)\) be the cardinality of the set \(A\cap [0,n]\). It is proved that, for any \(B_{2k}\)-sequence \(A\), \[ \liminf_{n\to\infty} A(n) \root 2k \of {{{\log n} \over n}}<\infty, \] which extends the results of P. Erdős (k=1) and J. C. M. Nash (k=2) [Can. Math. Bull. 32, 446-449 (1989; Zbl 0692.10046)] and improves the results of others.

MSC:

11B83 Special sequences and polynomials
11B50 Sequences (mod \(m\))
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)

Citations:

Zbl 0692.10046
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