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We study the maximum size of Sidon sets in unions of integer intervals. If A ⊂ N is the union of two intervals and if | A | = n (where | A | denotes the cardinality of A ), we prove that A contains a Sidon set of size at least 0 , 876 n . On the other hand, by using the small differences technique, we establish a bound of the maximum size of Sidon sets in the union of k intervals. [ABSTRACT FROM AUTHOR]

Let h ≧ 2 be an integer. We say that a set $$ \mathcal{A} $$ of positive integers is an asymptotic basis of order h if every large enough positive integer can be represented as the sum of h terms from $$ \mathcal{A} $$. A set of positive integers $$ \mathcal{A} $$ is called a Sidon set if all the sums a + b with $$ a \in \mathcal{A},b \in \mathcal{A},a \leqq b $$, are distinct. In this paper we prove the existence of Sidon sets which are asymptotic bases of order 5 by using probabilistic methods. [ABSTRACT FROM AUTHOR]

We construct a Sidon set which is an asymptotic additive basis of order at most 7. [ABSTRACT FROM AUTHOR]