##
**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 = **{** a_{1} < a_{2} < ··· **}** be a sequence of positive integers and D(A) = **{**d_{1} < d_{2} < ··· **}** the sequence of integers of the form a_{i}-a_{j}, 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag