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Diophantine Inequalities for Forms

, 1991
A form F(λ) of degree k can be written as $$ F\left( \lambda \right) = \mathop{\sum }\limits_{{1 \leqslant {{i}_{1}}, \ldots ,{{i}_{k}} \leqslant s}} a\left( {{{i}_{1}}, \ldots ,{{i}_{k}}} \right){{\lambda }_{{{{i}_{l}}}}} \cdots {{\lambda }_{{{{i}_{k}}}}} $$ we associate the multilinear form $$ \hat F\left( \lambda \right) = \sum\limits_{1 \
Wang Yuan
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Asymptotic lower bounds for Diophantine inequalities

Mathematika, 2000
Let \(F({\mathbf x})=\lambda_1 x_1^k+ \cdots +\lambda_s x_s^k\) be a diagonal form with non-zero real coefficients, whose ratios are not all rational, and such that, if \(k\) is even, then not all coefficients have the same sign. In this paper the author proves that there is an absolute real positive constant \(C\), such that for every \(\epsilon >0 ...
exaly   +2 more sources

Cubic Diophantine Inequalities

Acta Mathematica Sinica, English Series, 2001
In this paper, it is proved that for any real numbers \(\lambda_1\), \(\lambda_2,\ldots,\lambda_7\) with \(\lambda_i\geq 1\) \((1\leq i\leq 7)\), the Diophantine inequality \[ |\lambda_1x_1^3+\lambda_2x_2^3+\cdots+\lambda_7x_7^3|
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On a Diophantine Inequality with Reciprocals

Proceedings of the Steklov Institute of Mathematics, 2017
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Proportionally modular diophantine inequalities and their multiplicity

Acta Mathematica Sinica, English Series, 2010
A proportionally modular Diophantine inequality has the form \(ax \, \text{mod} \, b \leq cx\), where \(a,b,c\) are positive integers. The set of integer solutions is a numerical semigroup: a subset of the natural numbers which contains zero, is closed under addition, and has finite complement. The authors present properties of such semigroups.
Rosales, J.C., Branco, M.B., Vasco, P
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The Solubility of Certain Diophantine Inequalities

Proceedings of the London Mathematical Society, 1958
The author proves the following theorem: Let \(\lambda_1, \ldots, \lambda_{14}\) be non-zero real numbers, not all of the same sign, and suppose that \(\lambda_1/\lambda_2\) is irrational. Then, for any real \(\gamma\), and any \(\varepsilon > 0\), the inequality \[ \vert \lambda_1x_1^4 + \ldots + \lambda_{14}x_{14}^4 < \varepsilon \] has infinitely ...
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Systems of proportionally modular Diophantine inequalities

Semigroup Forum, 2008
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Delgado, M., García-Sánchez, P. A., Rosales, J. C., Urbano-Blanco, J. M.   +3 more
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Dual Diophantine Systems of Linear Inequalities

Journal of Mathematical Sciences, 2020
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On a ternary Diophantine inequality

The Ramanujan Journal, 2016
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On a Diophantine Inequality Over Primes (II)

Monatshefte für Mathematik, 2006
Suppose that \(c > 1\) is not an integer. Let \(H(c)\) denote the least integer \(r\) such that for any fixed \(\varepsilon > 0\) and for any \(N > N_0(c, \varepsilon) > 0\), the inequality \[ \left| p_1^c + p_2^c + \dots + p_r^c - N \right| < \varepsilon \] has solutions in prime numbers \(p_1, p_2, \dots, p_r\).
Zhai, Wenguang, Cao, Xiaodong
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