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There was a significant population bottleneck after the europeans reached north/south america. Nearly all native americans died, and then their population rebounded again (but not fully).

A lot of research like this makes an assumption that the genetics of modern native americans are close to pre-columbus genetics but that is almost certainly not the case.

When I was in graduate school I got a smithsonian fellowship to study the feasibility of amplifying dna from pre-columbus incan bones (they left their dead above ground on a plateau and so no contamination from tannins from the soil). I showed the dna could successfully be amplified.

I then left science and I'm not sure where the research went after that.

It should also be noted that Polynesians very likely had pre-Columbian contact with South America; it's the most plausible explanation for how the sweet potato (a South American indigenous crop) became a Polynesian staple, even if there is a stunning lack of cultural or archaeological evidence for the theory.
The linked article indeed provides genetic evidence for pre-Colombian Polynesian migrations to S. America.
"Archaic" Austronesians, Malays, Polyesians, etc, were remarkable sea-goers. Not only did they populate Madagascar from Borneo, it now appears that they populated South America as well... mind boggling.

Previous data on the matter:

https://en.wikipedia.org/wiki/File:Migraciones_austronesias....

True. And they somehow managed to get to the Easter Island. And from there, the nearest continental point is in Chile. So, seafaring seems like a plausible way for Austronesians to reach South America. Going over Siberia appears unlikely - you got seafaring people used to tropical climates (they never saw snow or ice) suddenly dropping the seafaring stuff and becoming winter-hardy and walking all the way across Siberia and then down the North America... quite a stretch.

https://en.wikipedia.org/wiki/Easter_Island