Combinatorica

, Volume 28, Issue 4, pp 401–414 | Cite as

Perfect difference sets constructed from Sidon sets

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Abstract

A set \( \mathcal{A} \) of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of \( \mathcal{A} \). We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set \( \mathcal{A} \) such that
$$ A(x) \gg x^{\sqrt 2 - 1 - o(1)} $$
.

Also we prove that there exists a perfect difference set \( \mathcal{A} \) such that \( \mathop {\lim \sup }\limits_{x \to \infty } \) A(x)/\( \sqrt x \)≥ 1/\( \sqrt 2 \).

Mathematics Subject Classification (2000)

11B13 11B34 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma de Madrid Ciudad Universitaria de CantoblancoMadridSpain
  2. 2.Department of MathematicsLehman College (CUNY)New YorkUSA

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