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/\( \sqrt 2 \).