## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  818.11009
Autor:  Erdös, Paul; Sárközy, A.; Sós, V.T.
Title:  On additive properties of general sequences. (In English)
Source:  Discrete Math. 136, No.1-3, 75-99 (1994).
Review:  The authors give a survey of their papers on additive properties of general sequences and they prove several further results on the range of additive representation functions and on difference sets. Many related unsolved problems are discussed.
Let N = {1, 2,...} and N0 = N\cup {0}. For A\subseteq N0 and n in N the number of solutions of n = a+a'; a,a' in A is denoted by r1 (A,n). For the additional conditions a \leq a' or a < a' this number is denoted by r2 (A,n) or r3 (A,n) resp. For given ranges of the representation function r1 two theorems are proved:
Theorem 11. Let B\subseteq N0. There exists a set A\subseteq N0 such that B equals {r1 (A,n) | n in N} if and only if either B = {0,1} or {0,1, 2}\subseteq B.
Theorem 12. Let B\subseteq N0. There exists a set A\subseteq N0 such that B equals {m in N | m = r1 (A,n) for infinitely many n in N}. Corresponding results are also stated for i = 2 and 3.
For A\subseteq N0 let D(A) = {a- a' | a in A, a' in A, a > a'}. Generalizing a theorem by O. Grosek and R. Jajcay [J. Comb. Math. Comb. Comput. 13, 167-174 (1993; Zbl 777.05025)], the authors show that if a set B\subseteq N contains arbitrary long sequences of consecutive integers then there exists a set A\subseteq N0 such that D(A) = B. In contrast to sum sets it is possible that a difference set D(A) is dense' while A is extremely thin' because small differences d = a-a' can be formed using large elements a,a' in A. Two related results are given. The question if for a given infinite set B\subseteq N the equation D(A) = B with 0 in A can have a unique solution is answered positively in the case that B = D(A1) for an infinite B3 sequence A1 with 0 in A1.
Apart from using Lemma 1 (to prove theorem 12) all proofs use elementary combinatorial methods only.
Reviewer:  J.Zöllner (Mainz)
Classif.:  * 11B13 Additive bases
11B34 Representation functions
05B10 Difference sets
11B75 Combinatorial number theory
11B83 Special sequences of integers and polynomials
Keywords:  additive bases; difference bases; Bk sequences; survey; additive representation functions; difference sets; unsolved problems
Citations:  Zbl 777.05025

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