Absorbers seen in the spectrum of background quasars are a unique tool with which to select H i-rich galaxies at all redshifts. In turns, these galaxies allow us to determine the cosmological evolution of the H i gas ΩH i+He ii, which is a possible indicator of gas consumption as star formation proceeds. The damped Lyman α (Lyα) systems (DLAs with NH i≥ 1020.3 cm−2), in particular, are believed to contain a large fraction of the H i gas but there are also indications that lower column-density systems, called ‘sub-damped Lyα’ systems, play a role at high redshift. Here we present the discovery of high-redshift sub-DLAs based on 17 z > 4 quasar spectra observed with the Ultraviolet–Visual Echelle Spectrograph (UVES) on the Very Large Telescope (VLT). This sample is composed of 21 new sub-DLAs which, together with another 10 systems from previous European Southern Observatory archive studies, make up a homogeneous sample. The redshift evolution of the number density of several classes of absorbers is derived and shows that all systems seem to be evolving in the redshift range from z= 5 to z∼ 3. These results are further used to estimate the redshift evolution of the characteristic radius of these classes of absorbers, assuming a Holmberg relation and one unique underlying parent population. DLAs are found to have R*∼ 20 h−1100 kpc, while sub-DLAs have R*∼ 40 h−1100 kpc. The redshift evolution of the column density distribution, f(N,z), down to NH i= 1019 cm−2 is also presented. A departure from a power law due to a flattening of f(N,z) in the sub-DLA regime is present in the data. f(N,z) is further used to determine the H i gas mass contained in sub-DLAs at z > 2. The complete sample shows that sub-DLAs are important at all redshifts from z= 5 to z= 2. Finally, the possibility that sub-DLAs are less affected by the effects of dust obscuration than classical DLAs is discussed.