Optimal and typical discrepancy of 2-dimensional lattices

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.