##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 841.11006

**Autor: ** Erdös, Paul; Sárközy, A.; Sós, V.T.

**Title: ** On sum sets of Sidon sets. II. (In English)

**Source: ** Isr. J. Math. 90, No.1-3, 221-233 (1995).

**Review: ** Let A\subseteq **N** = **{**1, 2,...**}** and S_{A} = **{**a+a' | a,a' in A**}**. If for every n in **N** the equation a+a' = n; a \leq a'; a,a' in A has at most one solution then A is called a Sidon set.

At first blocks of consecutive elements in S_{A} for Sidon sets A are studied. For n in **N** let H(n) = **max****{**h in **N** | **{**m+1, m+2, ..., m+h**}** \subseteq S_{A}, m \leq n**}** taken over all Sidon sets A\subseteq **{**1, 2, ..., n**}**. It is shown that n^{1/3} << H(n) << n^{ ½}. The lower bound is obtained by construction of a suitable infinite Sidon set while the upper bound is a consequence of the following much sharper result, choosing l = [200 n^{ ½}]: For all Sidon sets A\subseteq **{**1, 2,...,n**}** and all l in **N**, k in **Z** we have | S_{A}\cap [k+1, k+l]| < {style ^{1}/_{2}} l+7l^{ ½} n^{1/4}. Let n in **N**. For A\subseteq **N**, | A| = n the minimum of |S_{A}| is obtained by arithmetic progressions A and the maximum by Sidon sets A. Therefore one can expect that a well-covering of a Sidon set by arithmetic progressions is impossible, even by generalized arithmetic progressions P = **{**e+x_{1} f_{1}+···+x_{m} f_{m} | x_{i} in **{**1, ..., l_{i}**}** for i = 1,..., m**}**, where m,l_{1}, ..., l_{m} in **N**; e,f_{1}, ..., f_{m} in **Z**. Let **dim** P = m and Q(P) = l_{1} l_{2} ... l_{m} be the dimension and size of P. A measure of well-covering for A by generalized arithmetic progressions (g.a.p.) of dimension m is given by the minimum D_{m} (A) of the terms t**sum**^{t}_{j = 1} Q(P_{j}) taken over all coverings A\subseteq \bigcup^{t}_{j = 1} P_{j} where P_{j} are g.a.p. with **dim** P_{j} = m for j = 1, ..., t. If D_{m} (A) is close to |A| then A can be covered by ``few'' g.a.p.. If D_{m} (a) is close to |A|^{2} we have the opposite situation. For all finite Sidon sets A it is shown that D_{m}(A) > 2^{-m-1} |A|^{2}. On the other hand for all m in **N** there exists a finite Sidon set A such that D_{m}(A) \leq ^{1}/_{2} | A|^{2}. These two theorems are proved in a more general form for B_{2}[g] sets.

**Reviewer: ** J.Zöllner (Mainz)

**Classif.: ** * 11B13 Additive bases

11B25 Arithmetic progressions

**Keywords: ** sum sets of Sidon sets; additive bases; B_{2}-sequences; B_{2}[g] sets; arithmetic progressions

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag