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On some applications of diophantine approximations
1Department of Mathematics, Columbia University, New York, NY 10027.
Summary
Researchers refined E-function value analysis, showing their irrationality measures match typical numbers. This provides optimal bounds for diophantine approximations of E-function values at rational points.
Area of Science:
- Number Theory
- Analytic Number Theory
- Diophantine Approximation
Background:
- Siegel's foundational work (1929) established results on the transcendence and algebraic independence of E-function values.
- Understanding the diophantine approximation properties of E-function values is crucial in number theory.
Purpose of the Study:
- To refine Siegel's results by establishing the best possible bounds for measures of irrationality and linear independence of E-function values.
- To investigate the nature of diophantine approximations for E-function values at rational points.
Main Methods:
- Utilizing graded Padé approximations for systems of functions.
- Applying methods to functions satisfying linear differential equations with rational function coefficients.
Main Results:
- Established optimal bounds for measures of irrationality and linear independence of E-function values.
- Demonstrated that E-function values at rational points exhibit diophantine approximation properties typical of "almost all" numbers.
- Proved that such numbers possess an "epsilon + 2" exponent of irrationality.
Conclusions:
- The findings provide significant advancements in understanding the arithmetic properties of E-functions.
- The results answer open problems posed by Lang regarding the diophantine approximation of E-function values.
- The employed techniques offer a robust framework for analyzing similar problems in number theory.
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