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mJournal of Number Theory � 1889

journal of number theory 56, 1�3 (1996)

A Note on B2k Sequences

Sheng Chen

Department of Mathematics, Southwest Texas State University, San Marcos, Texas 78666

Communicated by R. L. Graham

Received July 14, 1992

Let h�2 be an integer. A set A of positive integers is called aBh-sequence if all sums a1+a2+ } } } +ah , where ai # A (i=1, 2, ..., h), aredistinct up to rearrangements of the summands. A Bh-sequence is alsocalled a Sidon sequence of order h [5].

Let A be a B2k-sequence. Denote by A(n) the cardinality of A & [0, n].When k=1, Erdo� s [1] showed that

lim infn � �

A(n) �log nn

<�.

Recently, Jia [2] showed that, if A(n2)�A(n2), then

lim infn � �

A(n) 2k�log nn

<�.

As mentioned in [2], the result holds when k=2 without the extra con-dition A(n2)�A(n)2 and this condition does not always hold for aB2k -sequence.

Here we show that

Theorem. Let A be a B2k-sequence (k�2). Then

lim infn � �

A(n)n1�2k (log n)1�(4k&4)<�.

Corollary. Let A=[a1<a2<a3< } } } <an< } } } ] be an infiniteB2k -sequences. Then

lim supn � �

an

n2k(log n)1�2=�.

article no. 0001

10022-314X�96 �12.00

Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

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The Corollary follows from the Theorem when k�2. When k=1, it isa weaker statement of the result of Erdo� s.

Proof of Theorem. Let B=kA. For any positive integer n, define

;(n)= minn�m�kn2

A(m)m1�2k ,

and Ds(kn)=B(skn)&B((s&1) kn), 1�s�n. Then

\ :n

s=1

Ds(kn)

- s +2

�2 log n :n

s=1

Ds(kn)2.

On the other hand,

:n

s=1

Ds(kn)

- s= :

n

s=1

B(skn)&B((s&1) kn)

- s

= :n

s=1

B(skn) \ 1

- s&

1

- s+1++B(kn2)

- n+1

� :n

s=1

B(skn)s3�2

� :n

s=1

A(sn)k

s3�2

�;(n)k } - n } log n.

So,

n;(n)2k log n�2 :n

s=1

Ds(kn)2. (1)

Now for any integer r (1�r�k) define V(r, n) to be the set of 2r-tuples(a1 , a2 , ..., ar , b1 , b2 , ..., br) such that

(1) &n<�ri=1 ai&�r

i=1 bi<n,

(2) ai , bi # A & [0, n2], and

(3) [a1 , a2 , ..., ar] & [b1 , b2 , ..., br]{<.

And, for convenience, set V(0, n)=[1].Then,

:n

s=1

Ds(kn)2� :k

r=0\k

r+2

\A(kn2)k&r + } |V(r, kn)|

= :k

r=0

A(kn2)k&r } |V(r, kn)| } O(1).

2 SHENG CHEN

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Lemma. (Jia, [2]).

A(n)2k&2r } |V(r, n)|=O(n), and |V(1, n)|=O(n1�2k).

Using the lemma and the fact that A(m)=O(m1�2k) and ;(n)�A(n)�n1�2k=O(1),

;(n)2k&4 :n

s=1

Ds(kn)2= :k

r=0

;(n)2k&4 A(kn2)k&r } |V(r, kn)| } O(1)

= :k

r=0

;(n)2k&4 |V(r, kn)| O(n(2k&2r)�2k)

=O(n)+ :k

r=2\A(n)

n1�2k+2k&2r

|V(r, kn)| O(n (2k&2r)�2k)

=O(n).

Combining this with inequality (1),

;(n)4k&4 } n log n�O(n).

Therefore,

minm�n

A(m)m1�2k (log m)1�(4k&4))� min

n�m�kn2

A(m)m1�2k (log m)1�(4k&4)

�;(n)(log(kn2))1�(4k&4)�O(1).

This completes the proof of Theorem. K

Acknowledgment

The author thanks Xing-De Jia for his helpful discussion.

References

1. H. Haberstam and K. F. Roth, ``Sequences,'' Springer-Verlag, New York, 1983.2. X.-D. Jia, On B2k -sequences, J. Number Theory 48 (1994), 183�196.3. F. Kruckeberg, B2 -Folgen und verwandte Aanhlenfolgen, J. Reine Angew. Math. 206

(1961), 53�60.4. J. C. M. Nash, On B4 -sequences, Canad. Math. Bull. 32 (1989), 446�449.5. S. Sidon, Ein Satz u� ber trigonometrische Polynome und seine Anwendungen in der

Theorie der Fourier-Reihen, Math. Ann. 106 (1932), 536�539.6. A. Sto� hr, Gelo� ste und ungelo� ste Fragen u� ber Basen der natu� rlichen Zahlenreihe, II,

J. Reine Angew. Math. 194 (1995), 111�140.

3B2k SEQUENCES