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diophantine inequality involving binary forms

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Frontiers of Mathematics in China 2014年第3期9卷 641-657页

作者： Boqing XUE Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China

Let r =2^d-1 ＋ 1. We investigate the diophantine inequality｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.

关键词： diophantine inequality binary form

One Additive diophantine inequality with Mixed Powers 2 and 4

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Chinese Quarterly Journal of Mathematics 2008年第1期23卷 1-7页

作者：龚克李伟平 Department of Mathematics Tongji University College of Information Henan University of Finance and Economics

In this paper, it is shown that： if λ1 ,……λs axe nonzero real numbers, not all of the same sign, such that A1/A2 is irrational, then for any real number η and ε 〉 0 the inequality ｜λ1x1^2 ＋ λ2x2^2＋ λ3x3^... 详细信息

关键词： diophantine inequality mixed powers circle method

On the diophantine inequality Problem

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Chinese Quarterly Journal of Mathematics 2021年第1期36卷 79-89页

作者： SONG Yan-bo Department of Mathematics Xian Jiaotong UniversityXian 710049China

In this paper,we deal with a diophantine inequality involving a prime,two squares of primes and one k-th power of a prime which give an improvement of the result given by Alessandro Gambini.

关键词： diophantine inequality Hardy-littlewood method Davenport-heilbronn method

Cubic diophantine Inequalities

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Acta Mathematica Sinica,English Series 2001年第1期17卷 153-160页

作者： Hong Ze LI Department of Mathematics Shandong University

Let λ1, λ2,...,λ7 be real numbers satisfying λi≥1. In this paper, we prove there are integers x1,...,x7 such that the inequalities |λ1x13+λ2x23+...+λ7x73|i|xi]3 （λ1λ2…λ7）89814) hold simultaneously.

关键词： diophantine inequality Cubic equation Non-trivial solution

Results of diophantine approximation by unlike powers of primes

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Frontiers of Mathematics in China 2018年第4期13卷 797-808页

作者： Gaiyun GAO Zhixin LIU School of Mathematics Tianjin University Tianjin 300350 China

Let k be an integer with k ≥ 6. Suppose that λ1, λ2, ... , λ5 be nonzero real numbers not all of the same sign, satisfying that λ1/λ2 is irrational, and suppose that η is a real number. In this paper, for any ε 〉 0, we consider the inequality ｜λ1p1 ＋ λ2p2^2 ＋λ3p3^3＋λ4p4^4 ＋ λ5p5^k ＋ η｜〈（maxpj）^-σ（k）＋ε has infinitely many solutions in prime variables P1, P2,...,P5, where σ（k） depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on diophantine approximation by unlike powers of primes, and get the related problem.

关键词： Waring-Goldbach problem diophantine inequality

Proportionally Modular diophantine Inequalities and Their Multiplicity

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Acta Mathematica Sinica,English Series 2010年第11期26卷 2059-2070页

作者： José Carlos ROSALES Manuel Batista BRANCO Paulo VASCO Departamento de lgebra Universidad de Granada Departamento de Matemática Departamento de Matemática Universidade de Trás-os-Montes e Alto Douro

Let I be an interval of positive rational numbers. Then the set S （I） = T ∩ N, where T is the submonoid of （Q0＋, ＋） generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a diophantine inequality of the form ax rood b 〈 cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S （I） has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals.

关键词： Numerical semigroup diophantine inequality multiplicity Frobenius number

Values of binary linear forms at prime arguments

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Frontiers of Mathematics in China 2015年第6期10卷 1449-1459页

作者： Yuchao WANG Department of Mathematics Shanghai University Shanghai 200444 China

We investigate the exceptional set of real numbers not close to some value of a given binary linear form at prime arguments. Let λ1 and λ2 be positive real numbers such that λ1/λ2 is irrational and algebraic. For any （C, c） well-spaced sequence v and δ 〉 0, let E（v, X,δ ） denote the number of ν∈v with v ≤ X for which the inequality ｜λ1P1 ＋ λ2P2 - v｜〈 v-δ has no solution in primes P1,P2. It is shown that for any ε 〉 0, we have E（V, X, δ）〈〈 max（X3/5＋2δ＋ε, X1/3＋4/3δ＋ε）.

关键词： Circle method diophantine inequality

diophantine Inequalities with Mixed Powers

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Chinese Quarterly Journal of Mathematics 2015年第4期30卷 545-554页

作者：牟全武吕晓东 College of Science Xi'an Polytechnic University Department of Mathematics Tongji University

It is proved that if λ1,λ2,…,λ7 are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 〈 σ 〈 1/16, the inequality ｜λ1x1^2＋λ2x2^2＋∑i=3 7λixi^4＋η｜〈（max1≤i≤7｜xi｜-σhas infinitely many solutions in positive integers xl, x2,... , x7. Similax result is proved for ｜λ1x1^2＋λ2x2^2＋λ3x3^2＋λ4x4^4＋λ5x5^4＋λ6x6^4＋η｜〈（max1≤i≤7｜xi｜-σ.These results constitute an improvement upon those of Shi and Li.

关键词： diophantine inequality mixed power the Davenport-Heilbronn method