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?
2026-03-26 22:13:35.1774563215
Meaning of "non-trivial three-term arithmetic progressions"
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"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").