11Jxx Diophantine approximation
This subtopic studies Diophantine approximation, focusing on how well real or algebraic quantities can be approximated by rationals or algebraic numbers.
Specific topics
11J04 Homogeneous approximation to one number
Overview
11J04 studies homogeneous approximation to one number within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Homogeneous approximation to one number (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for homogeneous approximation to one number
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J06 Markov and Lagrange spectra
Overview
11J06 studies markov and lagrange spectra within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Markov and Lagrange spectra (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for markov and lagrange spectra
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J13 Simultaneous homogeneous approximation
Overview
11J13 studies simultaneous homogeneous approximation within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Simultaneous homogeneous approximation (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for simultaneous homogeneous approximation
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J20 Inhomogeneous linear forms
Overview
11J20 studies inhomogeneous linear forms within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Inhomogeneous linear forms (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for inhomogeneous linear forms
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J25 Diophantine inequalities
Overview
11J25 studies diophantine inequalities within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Diophantine inequalities (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for diophantine inequalities
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J54 Small fractional parts of polynomials
Overview
11J54 studies small fractional parts of polynomials within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Small fractional parts of polynomials (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for small fractional parts of polynomials
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J68 Approximation to algebraic numbers
Overview
11J68 studies approximation to algebraic numbers within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Approximation to algebraic numbers (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for approximation to algebraic numbers
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J81 Transcendence (general theory)
Overview
11J81 studies transcendence (general theory) within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Transcendence (general theory) (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for transcendence (general theory)
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J83 Metric theory
Overview
11J83 studies metric theory within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Metric theory (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for metric theory
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11J86 Linear forms in logarithms; Baker's method
Overview
11J86 studies linear forms in logarithms; baker's method within Diophantine approximation. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Linear forms in logarithms; Baker's method (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for linear forms in logarithms; baker's method
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks