Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  161.04703
Autor:  Erdös, Pál; Hartman, S.
Title:  On sequences of distances of a sequence (In English)
Source:  Colloq. Math. 17, 191-193 (1967).
Review:  Let A = { a1 < a2 < ··· } be a sequence of positive integers and D(A) = {d1 < d2 < ··· } the sequence of integers of the form ai-aj, i > j. A subsequence B of D(A) will be called avoidable if there is an infinite subsequence A' of A such that D(A') contains no term of B. The authors prove:
(1) To every A there is a B \subset D(A) of density < \epsilon in D(A) which is not avoidable.
(2) If A has positive lower density in N = {1,2,...} and B has lower density 0 in N then B is avoidable.
The authors give an example of sequences A and B, such that B \subset D(A) and has lower density 0 in D(A) and is not avoidable and also give two sufficient conditions for avoidability.
Reviewer:  H.B.Mann
Classif.:  * 11B83 Special sequences of integers and polynomials
                   11B05 Topology etc. of sets of numbers
Index Words:  number theory

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