On $B_{2k}$-sequences

• Autorzy: Martin Helm
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1993
• Tom: 63
• Numer: 4
• Strony: 367-371

Daty

wydano

1993

otrzymano

1992-07-10

poprawiono

1992-09-24

Twórcy

autor

Martin Helm

• Fachbereich Mathematik, Johannes Gutenberg-Universität Mainz, Saarstr. 21, D-6500 Mainz, Germany

Bibliografia

• [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