Abstract:
In connection with the two-dimensional model known as the
‘periodic Lorentz gas’, we study the asymptotic behaviour of
statistical characteristics of a free path interval of a point
particle before its first occurrence in an $h$-neighbourhood
(a circle of radius $h$) of a non-zero integer point as $h\to 0$
given that the particle starts from the $h$-neighbourhood
of the origin. We evaluate the limit distribution function
of the free path length and of the input aimed parameter
(the distance from the trajectory to the integer point
we are interested in) for a given value of the output aimed
parameter. This problem was studied earlier for a particle
starting from the origin (the homogeneous case).
Keywords:analytic number theory, dynamical systems, continued fractions, Kloosterman sums, billiards, geometry of numbers.
Fulltext:
PDF file (532 kB)
