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 Bd = {a in 2A | a-d\not in 2A} for d in N. There are absolute constants c1, c2 > 0 such that for all d in N we have |Bd| > c1|A|2 if A is a finite Sidon set and (*) limsupN ––> +oo Bd(N) (A(N))-2 > c2 if A is an infinite Sidon set. For the proof in the case of infinite A the generating function f(z) = suma in A za, where z = e-1/N e2\pi i\alpha for large N in N and real \alpha is considered. Assuming the contrary of the proposition, ingenious estimates of I: = int01 |(1-zd)f2 (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 = {s1,s2,...}, s1 < s2 < ... . For n in N, n > n0 there exists a Sidon set A\subseteq {1,2,..., n} such that si+1-si < 3\sqrt{n} for all si+1 in 2A\ {s1}. 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 si+1-si < \sqrt{si} (log si)(3/2)+\epsilon for all i > i0 (\epsilon) and si in 2A. Also given are lower estimates for si+1- si. A catalog of unsolved problems concerning Sidon sets and B2[g] sets closes this part I.
Reviewer: J.Zöllner (Mainz)
Classif.: * 11B13 Additive bases
Keywords: addtive bases; B2-sequences; sum sets of Sidon sets; infinite Sidon sets
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