On algebraic points in the plane near smooth curves∗

Bernik, V. I.; Götze, Friedrich; Kukso, Olga

doi:10.1007/s10986-014-9241-0

Cited by 5 publications

(11 citation statements)

References 10 publications

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“…Furthermore, the techniques used to derive such estimates can be of interest in and of themselves; for example, in the case of rational points they have been adapted to derive an efficient algorithm to compute the rational points with bounded denominator on a given manifold, see [25], or [35,Section 11] for a nice overview. This problem was first considered for planar curves by Bernik, Götze and Kukso in [16]. In other words, let B ⊂ R be a bounded open interval and let f 1 : B → R be a C 1 function; also, define the sets…”

Section: Introductionmentioning

confidence: 99%

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Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds

Pezzoni

2020

Preprint

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Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds by Bernik, Götze et al. for the number of points with algebraic conjugate coordinates close to a given manifold. In the process, we also improve on a Khinchin-Groshev-type theorem for a problem of constrained approximation by polynomials.

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Section: Introductionmentioning

confidence: 99%

“…A lower bound for #M n f1 (Q, γ, B) was provided in [16] for 0 < γ < 1 2 . This was soon extended in [13], where Bernik, Götze and Gusakova also provided an upper bound.…”

Section: Introductionmentioning

confidence: 99%

Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds

Pezzoni

2020

Preprint

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“…However, it should be noted that this result is not best possible since for the quantity #M n ϕ (Q, γ, J) an upper bound of order Q n+1−γ can be proved for γ < 1. In this paper we are going to fill this gap in the result of [5] by obtaining lower and upper bounds of the same order for 0 < γ < 1. Our main result is as follows:…”

Section: Introductionmentioning

confidence: 99%

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

2017

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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

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“…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.…”

Section: Introductionmentioning

confidence: 99%

On algebraic integers in short intervals and near smooth curves

Götze¹,

Gusakova²

2017

Acta Arith.

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In 1970 A. Baker and W. Schmidt introduced regular systems of numbers and vectors, showing that the set of real algebraic numbers forms a regular system on any fixed interval. This fact was used to prove several important results in the metric theory of transcendental numbers. In this paper the concept of a regular system is applied to the set of algebraic integers α of height ≤ Q in intervals of length depending on Q.

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On algebraic points in the plane near smooth curves∗

Cited by 5 publications

References 10 publications

Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds

Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds

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

On algebraic integers in short intervals and near smooth curves

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