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B2[g] Sets and a Conjecture of Schinzel and Schmidt

Published online by Cambridge University Press:  01 November 2008

JAVIER CILLERUELO  and
CARLOS VINUESA
JAVIER CILLERUELO
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain (e-mail: franciscojavier.cilleruelo@uam.es, c.vinuesa@uam.es)
CARLOS VINUESA
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain (e-mail: franciscojavier.cilleruelo@uam.es, c.vinuesa@uam.es)
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Abstract

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A set of integers is called a B2[g] set if every integer m has at most g representations of the form m = a + a′, with a ≤ a′ and a, a′ ∈ . We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1,. . .,n}. More precisely, we prove that infn→∞$\frac{F(g, n)}{\sqrt{gn}}\geq \frac 2{\sqrt \pi}-\e_g$ where ϵg → 0 when g → ∞. We show a connection between this problem and another one discussed by Schinzel and Schmidt, which can be considered its continuous version.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 6 , November 2008 , pp. 741 - 747
Copyright
Copyright © Cambridge University Press 2008

References

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