Dusty star-forming galaxies are among the most prodigious systems at high redshift (z>1), characterized by high star formation rates and huge dust reservoirs. The bright end of this population has been well characterized in recent years, but considerable uncertainties remain for fainter dusty star-forming galaxies, which are responsible for the bulk of star formation at high redshift and thus play a key role in galaxy growth and evolution. In this first paper of our series, we describe our methods for finding high redshift faint dusty galaxies using millimeter observations with ALMA. We obtained ALMA 1.1 mm mosaic images for three strong-lensing galaxy clusters from the Frontier Fields survey. The 2x2 mosaics overlap with the deep HST WFC3/IR footprints and encompass the high magnification regions of each cluster. The combination of extremely high ALMA sensitivity and the magnification power of these clusters allows us to systematically probe the sub-mJy population of dusty star-forming galaxies over a large surveyed area. We present a description of the reduction and analysis of the ALMA continuum observations for the galaxy clusters Abell 2744 (z=0.308), MACSJ0416.1-2403 (z=0.396) and MACSJ1149.5+2223 (z=0.543), for which we reach observed rms sensitivities of 55, 59 and 71 $mu$Jy beam$^{-1}$ respectively. We detect 12 dusty star-forming galaxies at S/N>5.0 across the three clusters, all of them presenting coincidence with NIR detected counterparts in the HST images. None of the sources fall close to the lensing caustics, thus they are not strongly lensed. The observed 1.1 mm flux densities for the total sample of galaxies range from 0.41 to 2.82 mJy, with observed effective radii spanning <0.05 to 0.37$pm$0.21. The lensing-corrected sizes of the detected sources appear to be in the same range as those measured in brighter samples, albeit with possibly larger dispersion.
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