Close Lattice Points on Circles

Abstract

AbstractWe classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length tR1/3 on a circle of radius R, for any given t > 0. In particular we prove that any arc of length on a circle of radius R, with , contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points, (ν1, n, ν2, n, ν3, n, ν4, n), each of which lies on an arc of length on a circle of radius Rn.