Arithmetic Properties of Integers in Chains and Reflections of g-ary Expansions

During the preparation of this paper, the first author was partially supported by project MTM2014-55421-P from the Ministerio de Economia y Competitividad and the second author was partially supported by Australian Research Council Grant DP140100118

Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation

This dissertation explores relations between hyperbolic geometry and Diophantine approximation, with an emphasis on continued fractions over the Euclidean imaginary quadratic fields, Q(&radic;&minus;d), d = 1, 2, 3, 7, 11, and explicit examples of badly approximable numbers/vectors with an obvious geometric interpretation. The first three chapters are mostly expository. Chapter 1 briefly recalls the necessary hyperbolic geometry and a geometric discussion of binary quadratic and Hermitian forms. Chapter 2 briefly recalls the relation between badly approximable systems of linear forms and bounded trajectories in the space of unimodular lattices (the Dani correspondence). Chapter 3 is a survey of continued fractions from the point of view of hyperbolic geometry and homogeneous dynamics. The chapter discusses simple continued fractions, nearest integer continued fractions over the Euclidean imaginary quadratic fields, and includes a summary of A. L. Schmidt&rsquo;s continued fractions over Q(&radic;&minus;1). Chapters 4 and 5 contain the bulk of the original research. Chapter 4 discusses a class of dynamical systems on the complex plane associated to polyhedra whose faces are two-colorable (i.e. edge-adjacent faces do not share a color). To any such polyhedron, one can associated a right-angled hyperbolic Coxeter group generated by reflections in the faces of a (combinatorially equivalent) right-angled ideal polyhedron in hyperbolic 3-space. After some generalities, we discuss a simpler system, billiards in the ideal hyperbolic triangle. We then discuss continued fractions over Q(&radic;&minus;1) and Q(&radic;&minus;2) coming from the regular ideal right-angled octahedron and cubeoctahedron. Chapter 5 gives explicit examples of numbers/vectors in Rr&times;Cs that are badly approximable over number fields F of signature (r, s) with respect to the diagonal embedding. One should think of these examples as generalizations of real quadratic irrationalities, which we discuss first as our prototype. The examples are the zeros of (totally indefinite anisotropic F-rational) binary quadratic and Hermitian forms (the Hermitian case arises when F is CM). Such forms can be interpreted as compact totally geodesic subspaces in the relevant locally symmetric spaces SL2(OF )\SL2(F&otimes;R)/SO2(R)r &times; SU2(C)s. We discuss these examples from a few different angles: simple arguments stemming from Liouville&rsquo;s theorem on rational approximation to algebraic numbers, arguments using continued fractions (of the sorts considered in chapters 3 and 4) when they are available, and appealing to the Dani correspondence in the general case. Perhaps of special note are examples of badly approximable algebraic numbers and vectors, as noted in 5.10. Chapter 6 considers approximation in Rn (in the boundary &part;Hⁿ of hyperbolic n&minus;space) over &ldquo;weakly Euclidean&rdquo; orders in definite Clifford algebras. This includes a discussion of the relevant background on the &ldquo;SL₂&rdquo; model of hyperbolic isometries (with coefficients in a Clifford algbra) and a discription of the continued fraction algorithm. Some exploration in the case Z3 &sube; R3 is included, along with proofs that zeros of anisotropic rational Hermitian forms are &ldquo;badly approximable,&rdquo; and that the partial quotients of such zeros are bounded (conditional on increasing convergent denominators). Chapter 7 considers simultaneous approximation in Rr &times; Cs as a subset of the boundary of (H2)r &times; (H3)s over a diagonally embedding number field of signature (r, s). A continued fraction algorithm is proposed for norm-Euclidean number fields, but not even convergence is established. Some exploration and experimentation over the norm-Euclidean field Q(&radic;2) is included. Finally, chapter 8 includes some miscellaneous results&nbsp; related to the discrete Markoff spectrum. First, some identities for sums over Markoff numbers are proven (although they are closely related to Mcshane&rsquo;s identity). Secondly, transcendence of certain limits of roots of Markoff forms is established (a simple corollary to [1]). These transcendental numbers are badly approximable with only ones and twos in their continued fraction expansion, and can be written as infinite sums of ratios of Markoff numbers indexed by a path in the tree associated to solutions of the Markoff equation x2 + y2 + z2 = 3xyz. The geometry in this chapter can all be associated to a oncepunctured torus (with complete hyperbolic metric), going back to the observation of H. Cohn [23] v that the Markoff equation is a special case of Fricke&rsquo;s trace identity tr(A)2+tr(B)2+tr(AB)2 = tr(A)tr(B)tr(AB) + tr(ABA-1B-1) + 2, A, B &isin; SL2 in the case that A and B are hyperbolic with parabolic commutator of trace &minus;2 (in particular for the torus associated to the commutator subgroup of SL2(Z)).</p

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

Continued Fractions and Hyperbolic Geometry

This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. This intuitive approach gives insight into the theory of continued fractions that is not so easy to obtain from traditional algebraic methods. Using it, we provide a more extensive study of both Rosen continued fractions and even-integer continued fractions than in any previous works, yielding new results, and revisiting classical theorems. We also study two types of complex continued fractions, namely Gaussian integer continued fractions and Bianchi continued fractions. As well as providing a more elegant and simple theory of continued fractions, our approach leads to a natural generalisation of continued fractions that has not been explored before

TeV-scale gravity in Horava-Witten theory on a compact complex hyperbolic threefold

The field equations and boundary conditions of Horava-Witten theory, compactified on a smooth compact spin quotient of CH^3, where CH^3 denotes the hyperbolic cousin of CP^3, are studied in the presence of Casimir energy density terms. If the Casimir energy densities near one boundary result in a certain constant of integration taking a value greater than around 10^5 in units of the d = 11 gravitational length, a form of thick pipe geometry is found that realizes TeV-scale gravity by the ADD mechanism, with that boundary becoming the inner surface of the thick pipe, where we live. Three alternative ways in which the outer surface of the thick pipe might be stabilized consistent with the observed value of the effective d = 4 cosmological constant are considered. In the first alternative, the outer surface is stabilized in the classical region and the constant of integration is fixed at around 10^{13} in units of the d = 11 gravitational length for consistency with the observed cosmological constant. In the second alternative, the four observed dimensions have reduced in size down to the d = 11 gravitational length at the outer surface, and there are Casimir effects near the outer surface. In the third alternative, the outer surface is stabilized in the classical region by extra fluxes of the three-form gauge field, whose four-form field strength wraps three-cycles of the compact six-manifold times the radial dimension of the thick pipe. Some problems related to fitting the strong/electroweak Standard Model are considered.Comment: LaTeX2e, 315 pages. v2: corrections to subsections 5.1 and 5.3. Subsection 2.3.3 revised and extended, bibliography revised, other minor improvement