scholarly journals On two lattice points problems about the parabola

2019 ◽  
Vol 16 (04) ◽  
pp. 719-729
Author(s):  
Jing-Jing Huang ◽  
Huixi Li
Keyword(s):  
Fourier Analysis ◽  
Character Sums ◽  
Upper Bounds ◽  
Gauss Sums ◽  
Lattice Points ◽  
Square Root ◽  
Optimal Error ◽  
Asymptotic Formulae ◽  
Error Terms

We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the square root cancellation in the context of the parabola, whereas its analogues are wide open conjectures for the circle and the hyperbola. We also obtain essentially sharp upper bounds for the latter lattice points problem associated with the parabola. Our proofs utilize techniques in Fourier analysis, quadratic Gauss sums and character sums.

Estimates of lattice points in the discriminant aspect over abelian extension fields

2018 ◽  
Vol 30 (3) ◽  
pp. 767-773 ◽  
Author(s):  
Wataru Takeda ◽  
Shin-ya Koyama
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Abelian Extension ◽  
Lattice Points ◽  
Abelian Extensions ◽  
Lindelöf Hypothesis ◽  
Error Terms

AbstractWe estimate the number of relatively r-prime lattice points in {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

scholarly journals The Mean Values of Character Sums and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 318
Author(s):  
Jiafan Zhang ◽  
Yuanyuan Meng
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Mean Values ◽  
Special Polynomials ◽  
Elementary Methods ◽  
The Mean

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

scholarly journals On the Mean Values of Certain Character Sums

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Zhefeng Xu ◽  
Huaning Liu
Keyword(s):  
Character Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Mean Values ◽  
Asymptotic Formulae ◽  
Fourth Power Mean ◽  
The Mean

Letq≥5be an odd number. In this paper, we study the fourth power mean of certain character sums∑χmodq,χ-1=-1*∑1≤a≤q/4aχa4and∑χmodq,χ-1=1*∑1≤a≤q/4aχa4, where∑‍*denotes the summation over primitive characters moduloq, and give some asymptotic formulae.

scholarly journals Simultaneous Additive Equations: Repeated and Differing Degrees

2017 ◽  
Vol 69 (02) ◽  
pp. 258-283 ◽  
Author(s):  
Julia Brandes ◽  
Scott T. Parsell
Keyword(s):  
Hybrid Systems ◽  
Existence Of Solutions ◽  
Square Root ◽  
Multiple Forms ◽  
Quadratic Equations ◽  
Asymptotic Formulae ◽  
Cubic Forms

Abstract We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning–Heath–Brown, which give weak results when specialized to the diagonal situation, this is the first result on such “hybrid” systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.

Markoff’s Theorem for Approximations

2018 ◽  
pp. 55-62
Author(s):  
Christophe Reutenauer
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Golden Ratio ◽  
Periodic Pattern ◽  
Square Root ◽  
Error Terms

This chapter provesMarkoff’s theorem for approximations: if x is an irrational real number such that its Lagrange number L(x) is <3, then the continued fraction of x is ultimately periodic and has as periodic pattern a Christoffel word written on the alphabet 11, 22. Moreover, the bound is attained: this means that there are indeed convergents whose error terms are correctly bounded. For this latter result, one needs a lot of technical results, which use the notion of good and bad approximation of a real number x satisfying L(x) <3: the ranks of the good and bad convergents are precisely given. These results are illustrated by the golden ratio and the number 1 + square root of 2.

Asymptotic Formulae for the Lattice Point Enumerator

1999 ◽  
Vol 51 (2) ◽  
pp. 225-249 ◽  
Author(s):  
U. Betke ◽  
K. Böröczky
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Positive Curvature ◽  
Convex Bodies ◽  
Mixed Volume ◽  
Lattice Points ◽  
Asymptotic Formulae ◽  
General Convex ◽  
Lattice Surface ◽  
Small Deformations

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.

scholarly journals On a Counting Theorem for Weakly Admissible Lattices

2020 ◽  
Author(s):  
Reynold Fregoli
Keyword(s):  
Diophantine Approximation ◽  
General Setting ◽  
Upper Bounds ◽  
Lattice Points ◽  
Precise Estimate ◽  
Linear Subspaces ◽  
Minimal Structure ◽  
Partition Method

Abstract We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve “hyperbolic spikes” and occur naturally in multiplicative Diophantine approximation. We use Wilkie’s o-minimal structure $\mathbb{R}_{\exp }$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The 1st one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The 2nd application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result, we develop a sophisticated partition method that is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

scholarly journals On Classical Gauss Sums and Some of Their Properties

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 625 ◽  
Author(s):  
Li Chen
Keyword(s):  
Recursive Formula ◽  
Character Sums ◽  
Gauss Sums ◽  
Computational Problem ◽  
Rational Polynomials

The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it.

scholarly journals Lifting, restricting and sifting integral points on affine homogeneous varieties

2012 ◽  
Vol 148 (6) ◽  
pp. 1695-1716 ◽  
Author(s):  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Homogeneous Spaces ◽  
Acta Math ◽  
Upper Bounds ◽  
Lattice Points ◽  
List Type ◽  
Integral Points ◽  
Counting Problem ◽  
Symmetric Varieties ◽  
Almost Prime ◽  
Homogeneous Varieties

AbstractIn [Gorodnik and Nevo,Counting lattice points, J. Reine Angew. Math.663(2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak,Prime and almost prime integral points on principal homogeneous spaces, Acta Math.205(2010), 361–402] and use them to establish several useful consequences of this property, including:(1)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

scholarly journals A hybrid mean value of the inversion of L-functions and general quadratic Gauss sums

2002 ◽  
Vol 167 ◽  
pp. 1-15 ◽  
Author(s):  
Wenpeng Zhang ◽  
Yuping Deng
Keyword(s):  
Character Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Analytic Method ◽  
Power Mean ◽  
Hybrid Mean Value

AbstractThe main purpose of this paper is, using the estimates for character sums and the analytic method, to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and give two interesting asymptotic formulas.

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