# Browsing by Subject "Diophantine approximation"

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Item Dirichlet's Theorem in projective general linear groups and the Absolute Siegel's Lemma(2006) Pekker, Alexander; Vaaler, Jeffrey D.Show more This dissertation addresses two problems in diophantine number theory: (1) an analogue of classical Dirichlet’s Theorem in a projective general linear group over a local field and (2) a sharp bound on the conjugate products of successive minima in the geometry of numbers over the adeles. For the first problem, we show that if k is a local field and P GL(N, k) is endowed with a natural norm Φ, then unless A ∈ P GL(N, k) is conjugate to an isometry, the orbit of A, {A, A2 , . . .} is bounded away from the identity element 1N ; otherwise, we quantify min1≤m≤M Φ(Am) in terms of M. We use classical techniques for the case k = C and group-theoretic techniques for k nonarchimedean; as a side fact, we show that the maximal order of an element in P GL(N, Fq) is (q N − 1)/(q − 1). We also discuss a more general vii group-theoretic approach due to Vaaler that is suitable for any compact abelian group and apply it to P GL(N, k). For the second problem, let k be a number field, let A ∈ GL(N, kA) be an automorphism of the adeles kA, and let µn be the n-th absolute successive minimum of A. We show that for 1 ≤ n ≤ N, the conjugate proudct µn(A)µN−n+1(A∗ ) is bounded above by e (N−1)/2 where A∗ = (AT ) −1 is the dual of A. As a corollary, we deduce the (already known) Absolute Siegel’s Lemma, i.e., the fact that QN n=1 µn(A) ≤ e N(N−1)/4 . We use induction, a symmetric algebra argument due to Roy and Thunder, and new duality arguments to derive our results.Show more Item On some distribution problems in Analytic Number Theory(2009-12) Homma, Kosuke; Vaaler, Jeffrey D.; Voloch, Felipe; Ciperiani, Mirela; Hadani, Ronny; Haynes, Alan K.Show more This dissertation consists of three parts. In the first part we consider the equidistribution of roots of quadratic congruences. The roots of quadratic congruences are known to be equidistributed. However,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms, especially Kuznetsov's formula, together with an Erdős-Turán inequality. Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers. In the second and third part of this dissertation we consider some aspects of Farey fractions. The set of Farey fractions of order at most [mathematical formula] is, of course, a classical object in Analytic Number Theory. Our interest here is in certain sumsets of Farey fractions. Also, in this dissertation we study Farey fractions by working in the quotient group Q/Z, which is the modern point of view. We first derive an identity which involves the structure of Farey fractions in the group ring of Q/Z. Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula]. Our method uses results about divisors in short intervals due to K. Ford. We also prove a new form of the Erdős-Turán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L²(R/Z).Show more Item Tools and techniques in diophantine approximation(2006) Haynes, Alan Kaan; Vaaler, Jeffrey D.Show more In the study of any branch of mathematics it is useful to be able to identify a central body of tools and techniques which can together be used to establish a wide variety of lemmas and theorems. Although the historical development of a mathematical problem or an area of mathematics does not always follow this kind of pedagogy, in hindsight the crucial results and central themes which pervade a subject are many times more obvious. In this work we develop the theory of a collection of functions with an abundance of applications to Diophantine approximation. These functions satisfy a wealth of identities which lead to elegant proofs of many known results and of many new ones. In the course of our study we will demonstrate how these functions play a central role in problems about the continued fraction expansions of real numbers. Perhaps the most interesting property possessed by the functions is that they form a system of martingale differences. We will show how this fact can be used to advantage by proving several nontrivial results in metric number theory. The techniques which we use do not appear to have been previously exploited to prove theorems in number theory. It is our hope that the introduction of martingales will provide the tools necessary for further advancement in this subject.Show more