Semi-Progressions

Letg(n)⩾0 be a function. A sequence ofkpositive integers,a1<a2<…<ak, is called ak-term semi-progression forg(n) provided the diameter of the set of differences, diam{aj+1−aj|j=1, 2, …, k−1}, does not exceedg(k). A setAof integers is said to have property SP(g), if, for infinitely manyk,Acontains ak-term semi-progression forg(n). Ifg(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 functionsgthe 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.