  We solve Chui's conjecture on the simplest fractions (i.e., sums of Cauchy
kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely,
for a wide class of weights, we prove that for every $N$, the simplest
fractions with $N$ poles on the unit circle have minimal norm if and only if
the poles are equispaced on the circle. We find sharp asymptotics of these
norms. Furthermore, we describe the closure of the simplest fractions in
weighted Bergman spaces, using an $L^2$ version of Thompson's theorem on
dominated approximation by simplest fractions.
