Small values of Weyl sums

We prove that the set of (x1,…,xd)∈[0,1)d, such thatlim_N→∞|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|=0, contains a dense Gδ set in [0,1)d and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums∑n=1Nexp⁡(2πixnd),x∈[0,1).