Rational Points in Geometric Progression on the Unit Circle

Abstract

A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we prove the existence of infinitely many rational numbers $r$ such that for each $r$ there exist infinitely many $r$-geometric progression sequences on the unit circle $x^2+y^2=1$ of length at least $3$.

7 pages, accepted for publication in Publicationes Mathematicae Debrecen