Semi-Progressions

✍ Scribed by P. Ding; A.R. Freedman

• Book ID: 102588027
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 266 KB
• Volume: 76
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.0090

✦ Synopsis

Let g(n) 0 be a function. A sequence of k positive integers, a 1 <a 2 < } } } <a k , is called a k-term semi-progression for g(n) provided the diameter of the set of differences, diam[a j+1 &a j | j=1, 2, ..., k&1], does not exceed g(k). A set A of integers is said to have property SP( g), if, for infinitely many k, A contains a k-term semi-progression for g(n). If g(n) is a bounded function, then this definition is similar to the earlier definition of having property QP (containing arbitrarily long quasi-progressions of bounded diameter.) For unbounded functions g the property SP( g) is quite new and this paper examines its relation to several other properties each of which is a generalization of the property AP of containing arbitrarily long arithmetic progressions.