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**Zentralblatt MATH**

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

**Zbl.No: ** 811.11014

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

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

**Source: ** J. Number Theory 47, No.3, 329-347 (1994).

**Review: ** For a finite or infinite set A\subseteq **N** = **{** 1,2,...**}** let A(n) = |A\cap [1,n]| and 2A = **{**a+a' | a,a' in A**}**. A is called a Sidon set if all sums a+a' in 2A, a \leq a' are distinct.

Sum sets 2A of Sidon sets A cannot consist of ``few'' generalized arithmetic progressions of the same difference. To be more precise let B_{d} = **{**a in 2A | a-d\not in 2A**}** for d in **N**. There are absolute constants c_{1}, c_{2} > 0 such that for all d in **N** we have |B_{d}| > c_{1}|A|^{2} if A is a finite Sidon set and (*) **limsup**_{N ––> +oo} B_{d}(N) (A(N))^{-2} > c_{2} if A is an infinite Sidon set. For the proof in the case of infinite A the generating function f(z) = **sum**_{a in A} z^{a}, where z = e^{-1/N} e^{2\pi i\alpha} for large N in **N** and real \alpha is considered. Assuming the contrary of the proposition, ingenious estimates of I: = **int**_{0}^{1} |(1-z^{d})f^{2} (z)|^{2} d\alpha lead to contradicting lower and upper bounds for I. By example it is shown that (A(N))^{-2} in (*) cannot be replaced by (A(N))^{-2} log^{-1} N.

While these results in the case d = 1 deal with blocks of consecutive elements in 2A for Sidon sets A, the next theorems give information about gaps between consecutive elements of 2A. Let 2A = **{**s_{1},s_{2},...**}**, s_{1} < s_{2} < ... . For n in **N**, n > n_{0} there exists a Sidon set A\subseteq **{**1,2,..., n**}** such that s_{i+1}-s_{i} < 3\sqrt{n} for all s_{i+1} in 2A**\** **{**s_{1}**}**. The prime number theorem is used for constructing such sets A. For infinite Sidon sets the probabilistic method of Erd\H os and Rényi is adapted to prove the following result: For \epsilon > 0 there is a Sidon set A such that s_{i+1}-s_{i} < \sqrt{s_{i}} (log s_{i})^{(3/2)+\epsilon} for all i > i_{0} (\epsilon) and s_{i} in 2A. Also given are lower estimates for s_{i+1}- s_{i}. A catalog of unsolved problems concerning Sidon sets and B_{2}[g] sets closes this part I.

**Reviewer: ** J.Zöllner (Mainz)

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

**Keywords: ** addtive bases; B_{2}-sequences; sum sets of Sidon sets; infinite Sidon sets

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag