Distribution of Farey Series and Free Path Lengths for a Certain Billiard in the Unit Square

Ledoan, Andrew Huyvu

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https://hdl.handle.net/2142/86878

Description

Title

Distribution of Farey Series and Free Path Lengths for a Certain Billiard in the Unit Square

Author(s)

Ledoan, Andrew Huyvu

Issue Date

2007

Doctoral Committee Chair(s)

Zaharescu, Alexandru

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

By Weyl's criterion the distribution of a given sequence is uniform if and only if certain exponential sums are small. For the Farey series FQ of fractions in their lowest terms with denominators not exceeding some bound Q ≥ 1, these exponential sums transform by way of Ramanujan sums into expressions involving the Mobius function. In 1924 Franel produced a quantitative form of this equivalence as an identity for the sum of the squares of the values of the local discrepancy function of the Mobius function and related this to the real parts of zeros of the Riemann zeta-function. In 1971 Huxley obtained a vast generalization of Franel's theorem for the zeros of a fixed Dirichlet L-function. In light of the difficulties surrounding the Riemann Hypothesis, in 1973 Niederreiter showed that it is, nevertheless, possible to obtain a best possible estimate for the absolute discrepancy of FQ . He proved that the discrepancy has order of magnitude 1/ Q. In 1999 Dress established the very remarkable result that the discrepancy is equal to 1/Q exactly, for every Q ≥ 1. In relation to these works, this dissertation has three distinct, but strongly connected, directions. First, it addresses the distribution of subsets of FQ with denominators belonging to a fixed arithmetic progression. Following Niederreiter, Dress, and Huxley, we establish a best possible estimate for the absolute discrepancy of these subsets in short intervals. Second, in relating the foregoing result to the distribution of visible points in the Cartesian plane, we study a problem in geometric probability on the free path lengths of a linear trajectory for a certain two-dimensional Euclidean billiard in the unit square, which conveys some of the spirit of the study of paths of particles. Third, we investigate the notion of Farey index introduced by Hall and Shiu in 2003, relate it to the position of consecutive visible points, and describe its asymptotic distribution as Q tends to infinity.

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