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I would not be surprised if the answer is the South Pole, because it is much colder and the vapor pressure drops very quickly with temperature.

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This is a fun question since the Sahara and Antarctica are not points but areas, and figuring out averages for areas in these cases are difficult because measurements are scarce. That said we can consider the following:

The Sahara is a desert region caused by the general circulation pattern on the earth. As such, measurements indicate less than 100 mm/y (Babaousmail et al., 2022). This number will, however, vary from year to year, and in addition it will also be highly dependent on climate variability; the Sahara during the ice ages was a wet region (Tierney et al., 2017).

Antarctica is also very dry but this is because its central surface largely sits at very high elevation 2000--4000 m a.s.l. and is also very distant from the moisture source of the surrounding ocean. The central parts of Antarctica is thus classified as a desert. The accumulation of snow is on the order of a mm to cm/year (e.g. Richardson et al 1997).

The problem comparing the two regions is that they differ not only in the obvious latitude but also in elevation and are subject to very different circulation patterns. Antarctica is largely isolated by both ocean circulation and wind patterns from the rest of the globe while the Sahara is part of the general and expected pattern of circulation caused by the Hadley cells. To be somewhat provocative, in some sense the driest point on Earth depends on where we place the stations measuring precipitation and indeed that true point may still be unknown.

References

Babaousmail, H., Hou, R., Ayugi, B., Sian, K. T. C. L. K., Ojara, M., Mumo, R., Chehbouni, A., & Ongoma, V. (2022). Future changes in mean and extreme precipitation over the Mediterranean and Sahara regions using bias-corrected CMIP6 models. International Journal of Climatology, 42(14), 7280–7297. https://doi.org/10.1002/joc.7644

Richardson, C., Aarholt, E., Hamran, S.-E., Holmlund, P., and Isaksson, E. (1997), Spatial distribution of snow in western Dronning Maud Land, East Antarctica, mapped by a ground-based snow radar, J. Geophys. Res., 102(B9), 20343–20353, doi:10.1029/97JB01441. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/97JB01441

Tierney, J.E., Pausata, F.S.R, deMenocal, P.B. (2017), Rainfall regimes of the Green Sahara.Sci. Adv. 3,e1601503. DOI:10.1126/sciadv.1601503

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