Context : The study of the dust extinction in high-redshift galaxies is fundamental to obtain an estimate of the corrected Star Formation Rate Density ( SFRD ) and to put constraints on galaxy evolution models .

Aims : We plan to analyse dust extinction in Lyman Break Galaxies ( LBGs ) by introducing a new and more reliable approach to their selection and to the characterization of their distribution of UV slopes \beta , using deep IR images from HST .
We fully discuss the methodology and the results focusing on a robust sample of z \sim 4 LBGs .

Methods : We exploit deep WFC3 IR observations of the ERS and HUDF fields over GOODS-South , combined with HST-ACS optical data , to select z \sim 4 LBGs through a new ( B-V ) vs. ( V-H ) colour diagram .
The UV slope of the selected galaxies is robustly determined by a linear fit over their observed I , Z , Y , J magnitudes , coherently with the original definition of \beta .
The same fit is used to determine their rest-frame UV magnitudes M _ { 1600 } through a simple interpolation .
We estimate the effect of observational uncertainties with detailed simulations that we also exploit , under a parametric maximum-likelihood approach , to constrain the probability density function of UV slopes PDF ( \beta ) as a function of rest-frame magnitude .

Results : We find 142 and 25 robust LBGs in the ERS and HUDF fields respectively , limiting our sample to S/N ( H ) > 10 objects .
Our newly defined criteria improve the selection of z \sim 4 LBGs and allow us to exclude red interlopers at lower redshift , especially z \sim 3-3.5 objects .
We show that the use of a linear fit to estimate \beta and an accurate characterization of observational effects are required in such analysis of flux limited samples .
We find that z \sim 4 LBGs are characterized by blue UV slopes , suggesting a low dust extinction : all L < L ^ { * } galaxies have an average UV slope \langle \beta \rangle \simeq - 2.1 , while brighter objects only are slightly redder ( \langle \beta \rangle \simeq - 1.9 ) .
We find an intrinsic dispersion \simeq 0.3 for PDF ( \beta ) at all magnitudes .
The SFRD at z \sim 4 corrected according to these estimates turns out to be lower than previously found : log ( SFRD ) \simeq - 1.09 ~ { } M _ { \odot } / yr / Mpc ^ { 3 } .
Finally , we discuss how the UV slope of z \sim 4 galaxies changes as a function of the dust-corrected UV magnitude ( i.e .
SFR ) .
We show that most galaxies with a high SFR ( \gtrsim 80 ~ { } M _ { \odot } / yr ) are highly extincted objects .
Among galaxies with lower SFR , we detect many with a much lower amount of reddening , although current observational limits prevent us from detecting those with high extinction , if they exist .

Conclusions :