  Given a narrow signal over the real line, there is a limit to the
localisation of its Fourier transform. In spaces of prime dimensions, Tao
derived a sharp state-independent uncertainty relation which holds for the
support sizes of a pure qudit state in two bases related by a discrete Fourier
transform. We generalise Tao's uncertainty relation to complete sets of
mutually unbiased bases in spaces of prime dimensions. The bound we obtain
appears to be sharp for dimension three only. Analytic and numerical results
for prime dimensions up to nineteen suggest that the bound cannot be saturated
in general. For prime dimensions two to seven we construct sharp bounds on the
support sizes in $(d+1)$ mutually unbiased bases and identify some of the
states achieving them.
