ON THE NON-MONOTONICITY OF (jIm(zn)j)

We show that the sequence ? jIm(z n)j is never monotone when z is a non-real complex number, and ? jRe(z n)j is never monotone, unless z = re ii , where = k m for some odd m, and r is suuciently large or suuciently small. Similar results hold for the length of the projection of z n onto any line through the origin. The purpose of this note is to prove a simple, amusing and apparently new application of some familiar number theory. The most interesting special case is the following: if z is a non-real complex number, then the sequence ? jIm(z n)j is not monotone; in fact, it increases and decreases innnitely often. This oscillation contrasts with the familiar fact that ? jx n j is monotone for real x. Recall that if`denotes the line in C through the origin with angle to the real axis, then the distance from a point w tò is equal to jIm(e ?ii w)j. This distance is jIm(w)j when = 0 and jRe(w)j when = 1 2. Theorem. Let z = a + bi = re ii , be a non-real complex number (b 6 = 0), and let u(n) = jIm(e ?ii z n)j = jr n sin (nn ?)j. Then the sequence ? u(n) increases and decreases innnitely often, (indeed, each event occurs with positive probability), unless the following conditions are met: = k m , gcd(k; m) = 1, mm 6 2 Z and r is suuciently large or suuciently small. In these cases ? u(n) is monotone increasing or monotone decreasing. Proof. Note rst that u(n) = 0 if and only if nn ? 2 Z; under our hypotheses, = 2 Z, so (n 1) ? = 2 Z, hence u(n 1) > 0.