The CEO of Asustek said that over the course of the next three years, their notebooks will push them into the top five, then continuing on from there the three years after. They plan to do this by sticking to “mainstream” and high-end units, so likely they won't be engaged in price wars with companies like HP and Dell who offer sub-$600 notebooks. However, those low-cost units make up a significant portion of the market, so they must have some other plan.
Given that an Asustek laptop is something almost unheard of currently, they are definitely making some bold claims. I wouldn't mind seeing them try, however.