Typical weighted random simplices Z µ , µ ∈ (−2, ∞), in a Poisson-Delaunay tessellation in R n are considered, where the weight is given by the (µ + 1)st power of the volume. As special cases this includes the typical (µ = −1) and the usual volume-weighted (µ = 0) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of Z µ satisfies a central limit theorem in high dimensions, that is, as n → ∞. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight µ = µ(n) to depend on the dimension n as well. A number of special cases are discussed separately. For fixed µ

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Let ϕ : R → R be a continuously differentiable function on an interval J ⊂ R and let α = (α 1 , α 2 ) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α 1 , α 2 is of degree ≤ n and height ≤ Q. Denote by M n ϕ (Q, γ, J) the set of such points α such that |ϕ(α 1 ) − α 2 | ≤ c 1 Q −γ . We show that for a real 0 < γ < 1 and any sufficiently large Q there exist positive valuesP (α 1 ) = P (α 2 ) = 0 is called the minimal polynomial of algebraic point α. Denote by deg(α) = deg P the degree of the algebraic point α and by H(α) = H(P ) the height of the algebraic point α. Define the following set of algebraic points:Problems related to calculating the number of integer points in shapes and bodies in R k can be naturally generalized to estimating the number of rational points in domains in Euclidean spaces. Let f : J 0 → R be a continuously differentiable function defined on a finite open interval J 0 in R. Define the following set:where J ⊂ J 0 and 0 ≤ γ < 2. In other words, the quantity #N f (Q, γ, J) denotes the number of rational points with bounded denominators lying within a certain neighborhood of the curve parametrized by f . The problem is to estimate the value #N f (Q, γ, J). In [7] Huxley proved that for functions f ∈ C 2 (J) such that 0 < c 4 := inf x∈J 0 |f ′′ (x)| ≤ c 5 := sup x∈J 0 |f ′′ (x)| < ∞ and an arbitrary constant ε > 0, the following upper bound holds:An estimate without using a quantity ε in the exponent has been obtained in 2006 in a paper by Vaughan and Velani [14]. One year later, Beresnevich, Dickinson and Velani [1] proved a lower estimate of the same order:This result was obtained using methods of metric theory introduced by Schmidt in [9].In this paper we consider a problem related to the distribution of algebraic points α ∈ A 2 n (Q) near smooth curves, which is a natural extension of the same problem formulated for rational points. Let ϕ : J 0 → R be a continuously differentiable function defined on a finite open interval J 0 in R satisfying the conditions:Define the following set:where c 1 = 1 2 + c 6 · c 8 and J ⊂ J 0 . This set contains algebraic points with a bounded degree and height lying within some neighborhood of the curve parametrized by ϕ. Our goal is to estimate the value #M n ϕ (Q, γ, J). The first advancement in solving this problem

In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals I 1 , . . . , I k ⊂ [0; π], 1 ≤ k ≤ m let Sal m,k (Q, I 1 , . . . , I k ) denote the set of ordered (k + 1)-tuples (α 0 , . . . , α k ) of conjugate algebraic integers, such that α 0 is a Salem numbers of degree 2m + 2 satisfying α ≤ Q for some positive real number Q and arg α i ∈ I i . We derive the following asymptotic approximationproviding explicit expressions for the constant ω m and the function ρ m,k (θ). Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by Q as Q → ∞.