Rational Points in Geometric Progression on the Unit Circle

TL;DR

This paper proves that there are infinitely many rational ratios for which the unit circle contains infinitely many rational points forming geometric progressions of length at least three.

Contribution

It establishes the existence of infinitely many rational ratios with infinitely many rational geometric progressions on the unit circle.

Findings

Infinitely many rational ratios $r$ exist.

For each such $r$, infinitely many $r$-geometric progressions are found.

These progressions have length at least 3.

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 $3$.