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Optimal and typical discrepancy of 2-dimensional lattices.

Bence Borda1

  • 1Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria.

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|September 16, 2024
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Summary
This summary is machine-generated.

This study analyzes the discrepancy in 2D Korobov lattices, characterizing optimal lattices using continued fractions. It provides precise asymptotics for known expansions and metric results for irrationals.

Keywords:
Continued fractionKorobov latticeLimit distributionLow discrepancyQuadratic irrationalSymmetrization

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Area of Science:

  • Number Theory
  • Discrepancy Theory
  • Geometric Discrepancy

Background:

  • Korobov lattices are essential in quasi-Monte Carlo methods.
  • Understanding lattice discrepancy is crucial for integration accuracy.
  • Symmetrization impacts lattice properties and discrepancy.

Purpose of the Study:

  • To fully characterize 2D Korobov lattices with optimal discrepancy.
  • To compute precise asymptotic formulas for the discrepancy.
  • To investigate the metric theory of discrepancy for rational and irrational lattices.

Main Methods:

  • Analysis of continued fraction partial quotients.
  • Asymptotic computations for specific irrational numbers (e.g., quadratic irrationals, Euler's number *e*).
  • Metric number theory techniques for almost all irrationals.

Main Results:

  • Complete characterization of 2D Korobov lattices with optimal discrepancy.
  • Explicit asymptotic formulas for discrepancy where continued fraction expansions are known.
  • Asymptotic behavior of discrepancy for almost all irrationals and limit distributions for random lattices.

Conclusions:

  • Continued fraction properties directly determine optimal discrepancy in Korobov lattices.
  • The study provides a comprehensive understanding of discrepancy for various lattice types.
  • Results advance the theory of quasi-Monte Carlo integration and related fields.