Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  588.10056
Autor:  Erdös, Paul; Sárközy, A.; Sós, V.T.
Title:  Problems and results on additive properties of general sequences. IV. (In English)
Source:  Number theory, Proc. 4th Matsci. Conf., Ootacamund/India 1984, Lect. Notes Math. 1122, 85-104 (1985).
Review:  [For the entire collection see Zbl 547.00014. - Part I, see the first and second author, Pac. J. Math. 118, 347-357 (1985; Zbl 569.10032).]
Let A = {a1 < a2 < ...} be an infinite sequence of positive integers and R1(n), R2(n), R3(n) denote the number of solutions of ax+ay = n,  ax in A,  ay in A in the cases: no restriction, x < y, x \leq y, respectively. It turns out that these functions behave quite different according to monotonicity.
The authors show that R1(n) is monotonous increasing iff A consists of all the integers from a point onwards. Denoting the number of elements of A up to n by A(n) they construct sequences A such that R2(n) is monotonous increasing and A(n) < n- cn1/3. There is no corresponding result for R3(n), however it is proved that R3(n) and R2(n) cannot be monotonous increasing when A(n) = o(n/ log n). The authors conjecture that this is true with A(n) = o(n).
Reviewer:  A.Balog
Classif.:  * 11P99 Additive number theory
                   11B13 Additive bases
                   05B10 Difference sets
                   00A07 Problem books
Keywords:  number of additive representations; infinite sequence of positive integers; monotonicity
Citations:  Zbl 547.00014; Zbl 569.10032

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