# Meaning of “non-trivial three-term arithmetic progressions”

In this paper "ON ROTH’S THEOREM ON PROGRESSIONS" by Tom Sanders, the author gives a bound related to "non-trivial three-term arithmetic progressions", but what exactly means "non-trivial" in this context? Or what is an example of a trivial three-term arithmetic progression?

"Non-trivial" just means "non-constant". That is, a non-trivial three-term arithmetic progression is a triple of elements of the form $$(a,a+b,a+2b)$$ where $$b\neq 0$$ (the $$b\neq 0$$ condition being what makes it "non-trivial").