# On ${B}_{2k}$-sequences

Acta Arithmetica (1993)

- Volume: 63, Issue: 4, page 367-371
- ISSN: 0065-1036

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topMartin Helm. "On $B_{2k}$-sequences." Acta Arithmetica 63.4 (1993): 367-371. <http://eudml.org/doc/206528>.

@article{MartinHelm1993,

abstract = {Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a $B_r$-sequence A satisfies
$lim inf_\{n→ ∞\} (A(n)/(n^\{1/r\}))=0$.
The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3].
Here we present a different, very short proof of Erdős’ hypothesis for all even r=2k which we developped independently of Jia’s version.},

author = {Martin Helm},

journal = {Acta Arithmetica},

keywords = {-sequence; Sidon sets},

language = {eng},

number = {4},

pages = {367-371},

title = {On $B_\{2k\}$-sequences},

url = {http://eudml.org/doc/206528},

volume = {63},

year = {1993},

}

TY - JOUR

AU - Martin Helm

TI - On $B_{2k}$-sequences

JO - Acta Arithmetica

PY - 1993

VL - 63

IS - 4

SP - 367

EP - 371

AB - Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a $B_r$-sequence A satisfies
$lim inf_{n→ ∞} (A(n)/(n^{1/r}))=0$.
The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3].
Here we present a different, very short proof of Erdős’ hypothesis for all even r=2k which we developped independently of Jia’s version.

LA - eng

KW - -sequence; Sidon sets

UR - http://eudml.org/doc/206528

ER -

## References

top- [1] H. Halberstam and K. F. Roth, Sequences, Springer, New York 1983.
- [2] X.-D. Jia, On B₆-sequences, J. Qufu Norm. Univ. Nat. Sci. 15 (3) (1989), 7-11.
- [3] X.-D. Jia, On ${B}_{2k}$-sequences, J. Number Theory, to appear.
- [4] J. C. M. Nash, On B₄-sequences , Canad. Math. Bull. 32 (1989), 446-449.
- [5] A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. II, J. Reine Angew. Math. 194 (1955), 111-140

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