A Congruence Condition For The Four-Distance Problem

Abstract

Place the vertices of a rectangle at $\{(0, \pm 1/2), (a, \pm 1/2)\}$, where $a$ is rational. We show that if $v_3(a) = 0$, then any point $(x,y)$ that is rational distance from all four vertices of the rectangle has either $v_3(x) < 0$ or $v_3(y)<0$, where $v_3(\cdot)$ is the 3-adic valuation. The case of particular interest is the long-open four-distance problem, which asks whether such a rational distance point exists in the case $a=1$ of the unit square. For the four-distance problem, our result rules out one-fourth of all potential solutions with bounded height.