Fractal properties of spacing distributions

The paper reports a link between the Hausdorff dimension of a number theoretically based set, and certain arithmetic properties of the spacing distribution of the two-dimensional harmonic oscillator. It is shown that the set of points omega is an element of [0, 1], with continued fraction [a(1), a(2),...], such that log Pi(i=1)(n) a(i)(1/n) diverges, has Hausdorff dimension 1/2. The set of convergents p(n)/q(n) = [a(1), ..., a(n)], such that the series q(n)(1/n) diverge, is also shown to have a Hausdorff dimension 1/2. Although this result can be seen as a purely number-theoretic result, it is related to level spacing distributions in the following manner. For the two-dimensional harmonic oscillator with frequency ratio, omega, that has a continued fraction satisfying the above condition, the level spacing distribution is delta(s). Thus, the non-ergodic behaviour of the two-dimensional oscillator has Hausdorff dimension 1/2. Similar results are found for the system of a particle trapped in a box, using a number-theoretic result of Ramanujan.