## 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

.

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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## 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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### Cite this article

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