On the Distribution of Points with Algebraically Conjugate Coordinates in a Neighborhood of Smooth Curves

Abstract

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

On algebraic integers in short intervals and near smooth curves

“…Hence the values v 1 and v 2 lie between −1 and n. Remark 6. It it easily seen (for example from Lemma 2.4) that for a fixed polynomial P the set of points (x, y) ∈ R 2 satisfying the system (2.2) is contained in a rectangle σ P = J 1 × J 2 of measure µ 2 σ P ≤ 1 4 µ 2 E (see [8]). If I 1 ⊂ J 1 or I 2 ⊂ J 2 , we consider the rectangle I 1 × J 2 or J 1 × I 2 instead of the rectangle σ P to estimate the measure of L 2 n .…”

“…In the second part of our paper we proceed with the study of twodimensional analogue of Theorem 1.2. An interesting result related to the distribution of points with algebraically conjugate coordinates in the Euclidean plane was obtained in the papers [7,8]. Let us consider a rectangle E = I 1 × I 2 , where I 1 , I 2 are intervals of lengths |I 1 | = Q −s 1 , |I 2 | = Q −s 2 for 0 < s 1 + s 2 < 1.…”

“…We call a point (α, β) an algebraic point if α and β are algebraically conjugate numbers, and an algebraic integer point if α and β are algebraically conjugate integers. In the paper [8] it is shown that for Q > Q 0 any rectangle E of size…”

On distribution of points with conjugate algebraic integer coordinates close to planar curves

“…A first attempt to solve this problem for 0 < γ ≤ 1 2 has been made in [14]. This result was complemented by lower bound of the right order for 0 < γ < 3 4 [21] and finally by lower and upper bounds of the same order for 0 < γ < 1 [22]. We are going to state the final result in the following form: for any smooth function ϕ with conditions (1.1) we have #M n ϕ (Q, γ, J) ≍ Q n+1−γ for Q > Q 0 (n, J, ϕ, γ) and 0 < γ < 1.…”

“…2 + c 4 (the full description of this scheme is given in [22]). Thus, in order to prove Theorem 1 we need to estimate the number of integer algebraic points lying in such a square Π.…”