Hardy-Littlewood Problems on the Uniform Distribution of Arithmetic Progressions

Abstract

We apply canonical Poisson-Lie T-duality transformations to bosonic open string worldsheet boundary conditions, showing that the form of these conditions is invariant at the classical level, and therefore they are compatible with Poisson-Lie T-duality. In particular the conditions for conformal invariance are automatically preserved, rendering also the dual model conformal. The boundary conditions are defined in terms of a gluing matrix which encodes the properties of D-branes, and we derive the duality map for this matrix. We demonstrate explicitly the implications of this map for D-branes in two non-Abelian Drinfel'd doubles.

The Size of (q; q)nforqon the Unit Circle

“…Thus Using estimates for quadrature sums and results from the theory of uniform distribution, one can then show [17] that for almost all {, (See also [18].) It turns out that this is a rather crude estimate, and one can show much more: for x # R, we let [x] denote the greatest integer and let a.…”

Torsion homology growth beyond asymptotics

“…Inspired by work of Hardy and Littlewood, Oskolkov introduced his notion of functions of Class H and identified conditions for Equation 5.1 to hold, [Osk90], [Osk94]. We shall work with a stronger version due to Baxa and Schoißengeier.…”

“…Remark 5.10. If one can ensure stronger conditions on f , a precursor of this result is given in [Osk90]. If f is a function of Oskolkov's Class H and θ is badly approximable, its partial quotients are bounded, so {nθ} is regularly distributed by [Osk94, Theorem 3].…”

Calculation of improper integrals using uniformly distributed sequences

“…K. A. Driver, D. S. Lubinsky, G. Petruska and P. Sarnak [3] constructed functions for which (1.2) does not hold to study the radius of convergence of hypergeometric functions. V. A. Oskolkov [15] proved that (1. where f satisfies the same conditions as in [6] and q m denotes the denominator of the mth convergent of the continued fraction expansion of α. (In a follow-up paper [16] he proved a similar result for sequences satisfying a certain technical condition.…”