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Unconventional height functions in simultaneous Diophantine approximation.

Lior Fishman1, David Simmons2

  • 11Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017 USA.

Monatshefte Fur Mathematik
|April 10, 2020
PubMed
Summary
This summary is machine-generated.

This study explores simultaneous Diophantine approximation by using nonstandard height functions instead of the standard one. This leads to significant differences from classical theory, requiring new mathematical methods and yielding precise results.

Keywords:
Continued fractionsDiophantine approximationDirichlet’s theoremHardy L-functionsHeight functions

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Area of Science:

  • Number Theory
  • Diophantine Approximation

Background:

  • Simultaneous Diophantine approximation traditionally uses a standard height function to approximate points.
  • Understanding the impact of alternative height functions is crucial for advancing the field.

Purpose of the Study:

  • To investigate the effects of replacing the standard height function with nonstandard ones in simultaneous Diophantine approximation.
  • To develop new mathematical methods necessitated by these changes.

Main Methods:

  • Analysis of three distinct nonstandard height functions.
  • Computation of exponents of irrationality for these functions.

Main Results:

  • Demonstration of dramatic differences compared to classical Diophantine approximation theory.
  • Derivation of more precise results using nonstandard height functions.

Conclusions:

  • Nonstandard height functions significantly alter simultaneous Diophantine approximation.
  • New theoretical frameworks and methods are required for these variations.