LpL^{p} and Weak-LpL^{p} estimates for the number of integer points in translated domains

Revisiting and extending a recent result of M.Huxley, we estimate the $L^{p}\left( \mathbb{T}^{d}\right)$ and Weak-$L^{p}\left( \mathbb{T}^{d}\right)$ norms of the discrepancy between the volume and the number of integer points in translated domains

Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R^2. The proof depends on simultaneous diophantine approximation and a general version of the Erdos-Turan inequality.Comment: 14 pages, 2 figure

Average decay estimates for Fourier transforms of measures supported on curves

We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.Comment: We have learned about an important reference on the subject matter, and revised the paper accordingl

Irregularities of distribution for bounded sets and half-spaces

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces