  Assume that in a fundamental theory of quantum gravity spatial information is
encoded through elements x_i of an associative, complex and possibly
noncommutative algebra in which the involution acts as x^*_i = x_i. Without
further assumptions it can be shown that such x_i can describe only three
different types of short distance structures: I. a lattice, II. a continuum or
III. a finite lower bound on the uncertainty in positions, as e.g. described in
a stringy uncertainty relation. All other cases are mixtures of the three. We
briefly review recent results on the case III short distance structure, in
particular its ultraviolet regularity and a possible new mechanism that turns
the external degrees of freedom lost through the UV-cutoff into internal
degrees of freedom.
