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Eleven candidate damped Lya absorption systems were identified in 27 spectra of the quasars from the APM z>4 survey covering the redshift range 2.8<z(abs)<4.4 (8 with z(abs)>3.5). High resolution echelle spectra (0.8A FWHM) have been obtained for three quasars, including 2 of the highest redshift objects in the survey. Two damped systems have confirmed HI column densities of N(HI) >= 10^20.3 atoms cm^-2, with a third falling just below this threshold. We have discovered the highest redshift damped Lya absorber known at z=4.383 in QSO BR1202-0725. The APM QSOs provide a substantial increase in the redshift path available for damped surveys for z>3. We combine this high redshift sample with other quasar samples covering the redshift range 0.008 < z < 4.7 to study the redshift evolution and the column density distribution function for absorbers with log N(HI)>=17.2. In the HI column density distribution f(N)=kN^-beta we find evidence for breaks in the power law, flattening for 17.2< log N(HI)<21 and steepening for log N(HI)>21.2. The column density distribution function for the data with log N(HI)>=20.3 is better fit with the form f(N)=(f*/N*)(N/N*)^-beta exp(-N/N*). Significant redshift evolution in the number density per unit redshift is evident in the higher column density systems with an apparent decline in N(z) for z>3.5

APM z ≳ 4 survey: distribution and evolution of high column density H i absorbers