SEMI-PROGRESSIONS

Let g(n) greater than or equal to 0 be a function. A sequence of k pos itive integers, a(1) < a(2) < ... < a(k), is called a k-term semi-prog ression 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 i ntegers is said to have property SP(g), if, for infinitely many k, A c ontains a k-term semi-progression for g(n). If g(n) is a bounded funct ion, then this definition is similar to the earlier definition of havi ng property QP (containing arbitrarily long quasi-progression of bound ed diameter.) For unbounded functions g the property SP(g) is quite ne w and this paper examines its relation to several other properties eac h of which is generalization of the property AP of containing arbitrar ily long arithmetic progression. (C) 1996 Academic Press, Inc.