  We study uncertainty relations for pairs of conjugate variables like number
and angle, of which one takes integer values and the other takes values on the
unit circle. The translation symmetry of the problem in either variable implies
that measurement uncertainty and preparation uncertainty coincide
quantitatively, and the bounds depend only on the choice of two metrics used to
quantify the difference of number and angle outputs, respectively. For each
type of observable we discuss two natural choices of metric, and discuss the
resulting optimal bounds with both numerical and analytic methods. We also
develop some simple and explicit (albeit not sharp) lower bounds, using an
apparently new method for obtaining certified lower bounds to ground state
problems.
