Perfect difference sets constructed from Sidon sets

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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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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Mathematics Subject Classification (2000)

  • 11B13
  • 11B34