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On probability measures arising from lattice points on circles.

Pär Kurlberg1, Igor Wigman2

  • 11Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden.

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Attainable probability measures from lattice points on circles include all extreme points of symmetric measures. However, not all symmetric measures are attainable, revealing a fractal structure in attainable measures, especially those derived from prime powers.

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Area of Science:

  • Number Theory
  • Probability Theory
  • Geometric Measure Theory

Background:

  • Probability measures on the unit circle can be derived from lattice points on circles.
  • These 'attainable' measures and their limits are central to the study.
  • Understanding their properties is key to exploring lattice point distributions.

Purpose of the Study:

  • To investigate the set of attainable probability measures derived from lattice points on circles.
  • To determine if this set includes all extreme points of symmetric invariant measures.
  • To characterize the structure and limitations of attainable measures.

Main Methods:

  • Convex geometry to analyze extreme points of measure sets.
  • Convolution operations to study closure properties of attainable measures.
  • Analysis of projections onto Fourier coefficients to reveal singularities.

Main Results:

  • The set of attainable measures contains all extreme points of symmetric invariant probability measures.
  • Attainable measures are closed under convolution.
  • Symmetric probability measures exist that are not attainable, exhibiting fractal singularities related to prime powers.

Conclusions:

  • Attainable measures from lattice points on circles form a rich set with complex geometric properties.
  • The presence of fractal singularities highlights the intricate relationship between number theory and probability.
  • Square-free radii avoid these singularities, offering a simpler structure.