For two very good, more detailed explanations of this problem for Canada, please follow the links below.
There are two major problems with Canada's claim to sovereignty over the Arctic. First, there are physical/geographic issues. Canada has to prove that the waters between its Arctic islands are "internal" waters (not open sea like, for example, the ocean between the US mainland and Hawaii). In addition, it has to show that those waters are not "used for international navigation." Both of these are not easy to do.
Second, there is also the issue of what Canada has and has not been doing to stake its claim. Canada has typically failed to patrol the waters that they claim. This is a problem because countries are supposed to patrol areas that belong to them. The issue here is whether Canada has done enough with the Arctic to uphold its claim.