Denmark goes through with the American offer to purchase Greenland in 1946. The United States fills the island with even more defensive and scientific installations, along with a significantly greater number of American soldiers and scientists, as well as the workers required to build and maintain these. Unlike Denmark, the United States was in possession of a number of large companies that were actually capable of massively ramping up of resource extraction. This would draw in a large amount of investment as well as workers to manage these mines. Along with miners and other blue collar workers, there would also be a need for accountants, managers, and various other middle class positions to maintain these mines and projects, as well as people to build and maintain the infrastructure required, from roads to railways to power stations, sewers, and telephone lines. A large number of prospectors, both in precious metals and real estate, also come in hope of a similar gold or oil rush as Alaska.
The current population of Greenland IOTL is roughly 56,000. ITTL, it'd likely be something like 150,000-250,000. A thing to consider is that, while Greenland is far less hospitable than Alaska, it's also far closer to the population centers of the American east coast and far more easily accessible. Essentially, a lot of the population that would've gone to Alaska would have instead gone to Greenland. Rather than having an Alaska with a population of 730,000 and a Greenland with a population of 56,000 as IOTL, it'd most likely be 630,000 in Alaska and 190,000 in Greenland, with the additional 40,000 having been people from the American east coast and parts of Canada and Europe who wouldn't have been able to go to Alaska or the Yukon, and who would have instead gone to Greenland. The Canadian Yukon would also have a smaller population than IOTL.