In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on $\mathbb{R}^2$ prove that, each such norm, set improvable pairs contains badly approximable pairs, hence is hyperplane absolute winning. To this we make careful study some classical results in geometry numbers...

Abstract We prove that the discrete spherical maximal functions (in spirit of Magyar, Stein, and Wainger) corresponding to Euclidean spheres in $\mathbb {Z}^{d}$ with dyadic radii have $\ell ^{p}(\mathbb {Z}^{d})$ bounds for all $p\in [2, \infty ]$ independent dimensions $d\ge 5$. An important part our argument is asymptotic formula Waring problem squares a dimension-free multiplicative error t...

We compute the number of summands in q-averages of norms needed to approximate an Euclidean ball. Surprisingly, for a large range of q-s, these numbers depend essentially only on the maximal ratio of the norm and the Euclidean norm. Some properties of a family of projective caps on the sphere are central in this study. Moyennes de normes sur la sphère et comportement d’une famille de calottes p...

This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset X ⊂ Rn, associated with the Euclidean metric, with points in the cube {±1}m and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in X . Our methods rely on qua...