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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/\( \sqrt 2 \).

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain

    Javier Cilleruelo

  2. Department of Mathematics, Lehman College (CUNY), 250 Bedford Park Boulevard West Bronx, New York, 10468, USA

    Melvyn B. Nathanson

Authors
  1. Javier Cilleruelo
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  2. Melvyn B. Nathanson
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Corresponding author

Correspondence to Javier Cilleruelo.

Additional information

The work of J. C. was supported by Grant MTM 2005-04730 of MYCIT (Spain).

The work of M. B. N. was supported in part by grants from the NSA Mathematical Sciences Program and the PSC-CUNY Research Award Program.

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Cilleruelo, J., Nathanson, M.B. Perfect difference sets constructed from Sidon sets. Combinatorica 28, 401–414 (2008). https://doi.org/10.1007/s00493-008-2339-4

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  • DOI: https://doi.org/10.1007/s00493-008-2339-4

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