  Recall that the Hilbert (Riemann-Hilbert) boundary value problem was recently
solved in \cite{R1} for arbitrary measurable coefficients and for arbitrary
measurable boundary data in terms of nontangential limits and principal
asymptotic values. Here it is developed a new approach making possible to
obtain new results on tangential limits. It is shown that the spaces of the
found solutions have the infinite dimension for prescribed collections of
Jordan arcs terminating in almost every boundary point. Similar results are
proved for the Riemann problem.
