On small fractional parts of polynomials

TextWe prove that for any real polynomial f(x)∈R[x]f(x)∈R[x] the set{α∈R:lim infn→∞nlogn‖αf(n)‖>0} has positive Hausdorff dimension. Here ‖ξ‖‖ξ‖ means the distance from ξ to the nearest integer. Our result is based on an original method due to Y. Peres and W. Schlag.VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=GNWDrfQnV2c.