In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332-4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128-3272$. We also construct some dense lattices of dimensions in the range $4098-8232$. Finally we also obtain some new lattices of moderate dimensions such as $68,84,85,86$, which are denser than the...

The set $𝒰{𝒟}_r$ of point sets of ${ℝ}^n,n\ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r\>0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $𝒰{𝒟}_{r,f}\subset 𝒰{𝒟}_r$ of the finite point sets is compatible with the restriction of this topology to $𝒰{𝒟}_{r,f}$. We show that its subsets of Delone sets of given constants in ${ℝ}^n,n\ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature,...

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r{(1-x)}^r$, r ≥ 0, and let $P_{n,r}^{\mathbb{Z}}\subset P_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^{\mathbb{Z}}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (P_{n,r}^{\mathbb{Z}};L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ for p ≠ 2. It follows that $\mu (P_{n,r}^{\mathbb{Z}};L_p)\asymp n^{-2r-2/p}$ as n → ∞. The results partially...