On the generalization of the Costas property in the continuum

We extend the definition of the Costas property to functions in thecontinuum, namely on intervals of the reals or the rationals, and argue thatsuch functions can be used in the same applications as discrete Costas arrays.We construct Costas bijections in the real continuum within the class ofpiecewise continuously differentiable functions, but our attempts to constructa fractal-like Costas bijection there are successful only under slight butnecessary deviations from the usual arithmetic laws. Furthermore, we are able,contingent on the validity of Artin's conjecture, to set up a limiting processaccording to which sequences of Welch Costas arrays converge to smooth Costasbijections over the reals. The situation over the rationals is different:there, we propose an algorithm of great generality and flexibility for theconstruction of a Costas fractal bijection. Its success, though, relies heavilyon the enumerability of the rationals, and therefore it cannot be generalizedover the reals in an obvious way.