Approximation in the mean and quasianalyticity

Let E be a compact subset of the complex plane with positive dx dy measure. For each p, 1 ≤ p < ∞, let R^p(E dx dy) denote the closure in L^p(E, dx dy) of the rational functions having no poles on E. We give an example where E has empty interior and the functions in R^p(E, dx dy), p ≥ 2, are uniquely determined by their values on any set of positive dx dy measure in E. Earlier, Sinanjan (Sibirsk Mat. Ž. 6 (1965), 1365-1381) obtained a somewhat weaker result along these lines and he also proved that this phenomenon can never occur when 1 ≤ p < 2.