Discrete Gaussian measures and new bounds of the smoothing parameter for lattices In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that (varepsilon) is suitably small, we obtain two estimations of the smoothing parameter:  :67,displaystyle eta _varepsilon (mathbb Z) le sqrtfracln big (fracvarepsilon 44frac2varepsilon big )pi .  is a :81,practically useful case. For this case, our upper bound is very close to the exact value of (eta _varepsilon (mathbb Z)) in that (sqrtfracln big (fracvarepsilon 44frac2varepsilon big )pi eta _varepsilon (mathbb Z)le fracvarepsilon  a :118,lattice (mathcal Lsubset mathbb Rn) of dimension n,  :67,displaystyle eta _varepsilon :128,(mathcal L) le sqrtfracln big (n1frac2nvarepsilon big )pi tildebl(mathcal L). endaligned

New Weak Error bounds and expansions for Optimal Quantization We propose new weak error bounds and expansion in dimension one for optimal quantizationbased cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewisedefined locally Lipschitz or xcexb1Holder derivatives. This new results rest on the local behaviors of optimal quantizers, the L rL s distribution mismatch problem and Zadoru0027s Theorem. This new expansion supports the definition of a RichardsonRomberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.