On Schmidtʹs game and the set of points with non-dense orbits under a class of expanding maps

Given an expanding interval map, we consider sets of points whose forward orbits are bounded away from a given point. It is shown that such sets are 1/2-winning in the sense of Schmidtʹs game for a class of expanding interval maps including the Gauss transformation and all beta transformations. This improves upon several results dating back to a work of W. Schmidt.