  Precise thermometry is of wide importance in science and technology in
general and in quantum systems in particular. Here, we investigate fundamental
precision limits for thermometry on cold quantum systems, taking into account
constraints due to finite measurement resolution. We derive a tight bound on
the optimal precision scaling with temperature, as the temperature approaches
zero. The bound can be saturated by monitoring the non-equilibrium dynamics of
a single-qubit probe. We support this finding by accurate numerical simulations
of a spin-boson model. Our results are relevant both fundamentally, as they
illuminate the ultimate limits to quantum thermometry, and practically, in
guiding the development of sensitive thermometric techniques applicable at
ultracold temperatures.
