  In this paper, we exploit the relation between the regularity of refinable
functions with non-integer dilations and the distribution of powers of a fixed
number modulo 1, and show the nonexistence of a non-trivial {\bf C}^{\infty}
solution of the refinement equation with non-integer dilations. Using this, we
extend the results on the refinable splines with non-integer dilations and
construct a counterexample to some conjecture concerning the refinable splines
with non-integer dilations. Finally, we study the box splines satisfying the
refinement equation with non-integer dilation and translations. Our study
involves techniques from number theory and harmonic analysis.
