Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T07:41:21.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Bh [g] sequences

Part of: Sequences and sets

Published online by Cambridge University Press:  26 February 2010

Javier Cilleruelo  and
Jorge Jiménez-Urroz
Javier Cilleruelo
Affiliation:
Departamento de Matematicas, Universidad Autónoma de Madrid, Madrid-28049, Spain. E-mail: franciscojavier.cilleruelo@uam.es
Jorge Jiménez-Urroz
Affiliation:
Departamento de Matematicas, Universidad Autónoma de Madrid, Madrid-28049, Spain. E-mail: jorge.jimenez@uam.es
Get access

Abstract

New upper and lower bounds are given for Fh(g, N), the maximum size of a Bh[g] sequence contained in [1, N]. It is proved that and that

and that, for any ε > 0 and g > g(ε, h),

MSC classification

Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 109 - 115
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[B-Ch]Bose, R. C. and Chowla, S.. Theorems in the additive theory of numbers. Comment. Math. Helv., 37 (1962/1963), 141–147.CrossRefGoogle Scholar
[C-R-T]Cilleruelo, J., Ruzsa, I. and Trujillo, C.. Upper and lower bounds for Bh[g] sequences. J. Number Theory (to appear).Google Scholar
[Cl]Cilleruelo, J.. New upper bounds for finite Bh sequences. Advances Math. 159 (2001), 1–17.Google Scholar
[C2]Cilleruelo, J.. An upper bound for B2[2] sequences. J. Combinat. Theory A, 89 (2000), 141–144.Google Scholar
[E-T]Erdős, P. and Turan, P.. On a problem of Sidon in additive number theory and on some related problems. J. London Math. Soc., 16 (1941), 212–215; Addendum (by P. Erdős), J. London Math. Soc., 19 (1944), 208.CrossRefGoogle Scholar
[H-R]Halberstam, H. and Roth, K. F.. Sequences (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
[H]Helm, M.. Upper bounds for B2[g]-sets (preprint).Google Scholar
[K]Kolountzakis, M.. Problems in the Additive Number Theory of General Sets, I. Sets with distinct sums. (1996, Available at http://www.math.uiuc.edu/~kolount/surveys.html).Google Scholar
[L-N]Lesieur, L. and Nicolas, J. L.. On the Eulerian numbers Mn = max1≤k≤nA(n, k). Europ. J. Comb., 13 (1992), 379–399.Google Scholar
[L]Lindstrom, B.. B2[g]-sequences from Bh sequences. Proc. Amer. Math. Soc., 128 (2000), 657–659.Google Scholar
[N]Nicolas, J. L.. An integral representation for Eulerian numbers. Coll. Math. Soc. Jdnos Bolyai 60: Sets, Graphs and Numbers (Budapest, 1991),Google Scholar
[R]Ruzsa, I.. Solving a linear equation I. Acta Arithmetica LXV.3 (1993).Google Scholar
[S-S]Sarkozy, A. and Sos, V. T.. The Mathematics of Paul Erdős, vol. I: Algorithms and Combinatorics, 13 (Springer, 1996).Google Scholar
[S]Singer, J.. A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc., 43 (1938), 377–385.CrossRefGoogle Scholar