On weakly arithmetic progressions

✍ Scribed by Egbert Harzheim

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 230 KB
• Volume: 138
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00207-y

✦ Synopsis

A set of real numbers a~ < a 2 <... < cl L is called a weakly arithmetic progression of length L, if there exist L consecutive intervals I i = [x i\_ ~, xl), i = 1 ..... L, of equal length with a~El i. Here we consider conditions from which the existence of weakly arithmetic progressions can (resp. cannot) be deduced of a given length.
In the following JV (resp. W'o) denotes the set of positive (resp. nonnegative) integers.
Generalizing the notion of arithmetic progression the following was defined in [1]. Definition. Let L~JV'. A set of real numbers aa <..-< aL is called a weakly arithmetic' progression of length L (shortly a WAPL) if there exist real numbers Xo < '.-(L--1) FKq (case 1) or [AI>FLK-k-].(L-1) k (case 2) holds, then A contains a WAPL.
Proof. We assume that A has no WAPL and that case 1 holds. Then we subdivide [0, L r) into L consecutive subintervals (left-closed, right-open) of equal length L r-1