  Positron binding to silver and gold atoms was studied using a fully {\it ab
initio} relativistic method, which combines the configuration interaction
method with many-body perturbation theory. It was found that the silver atom
forms a bound state with a positron with binding energy 123 ($\pm$ 20%) meV,
while the gold atom cannot bind a positron. Our calculations reveal the
importance of the relativistic effects for positron binding to heavy atoms. The
role of these effects was studied by varying the value of the fine structure
constant $\alpha$. In the non-relativistic limit, $\alpha = 0$, both systems
$e^+$Ag and $e^+$Au are bound with binding energies of about 200 meV for
$e^+$Ag and 220 meV for $e^+$Au. Relativistic corrections for a negative ion
are essentially different from that for a positron interacting with an atom.
Therefore, the calculation of electron affinities cannot serve as a test of the
method used for positron binding in the non-relativistic case. However, it is
still a good test of the relativistic calculations. Our calculated electron
affinities for silver (1.327 eV) and gold (2.307 eV) atoms are in very good
agreement with corresponding experimental values (1.303 eV and 2.309 eV
respectively).
