ABOUT THE MULTIFRACTAL NATURE OF CANTOR’S BIJECTION: BOUNDS FOR THE HÖLDER EXPONENT AT ALMOST EVERY IRRATIONAL POINT

Abstract

In this note, we investigate the regularity of Cantor’s one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the fractal nature of this map by showing that its Hölder regularity lies between 0.35 and 0.72 almost everywhere (with respect to the Lebesgue measure).