Infinite genus surfaces and irrational polygonal billiards

We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end.     

On irrational lattice angles

The aim of this paper is to generalize the notions of ordinary and expanded lattice angles and their sums studied in the work  [7] by the author to the case of angles with lattice vertices but not necessarily with lattice rays. We find normal forms and extend the definition of lattice sums to a certain special case of such angles.     

On multiple coverings of irrational curves

The first named author was partially supported by MURST and GNSAGA of CNR (Italy).     

On the distribution of partial sums of irrational rotations

Periodica Mathematica Hungarica - For an irrational $$\alpha $$ , we investigate the sums $$\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) $$ and $$\sum _{i=1}^n \left\{ \left( \{i \alpha...     

On multiple ergodicity of affine cocycles over irrational rotations

Let \({T_\alpha }\) denote the rotation \({T_\alpha }x = x + \alpha \) (mod 1) by an irrational number α on the additive circle T = [0, 1). Let β ₁, …, β _d be d ≥ 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in ? ^d+1) generated over \({T_\alpha }\) by the vectorial function Ψ _d+1(x):= (φ(x), φ(x+β ₁), …, φ(x+β _d )), with φ(x) = {x}?½. It was already proved in [LeMeNa03] that Ψ₂ is regular for α with bounded partial quotients. In the present paper we show that Ψ₂ is regular for any irrational α. For higher dimensions, we give sufficient conditions for regularity. While the case d = 2 remains unsolved, for d = 3 we provide examples of non-regular cocycles Ψ₄ for certain values of the parameters β ₁, β ₂, β ₃. We also show that the problem of regularity for the cocycle Ψ _d+1 reduces to the regularity of the cocycles of the form \({\Phi _d} = {({1_{[0,{\beta _j}]}} - {\beta _j})_{j = 1, \ldots ,d}}\) (taking values in ? ^d ). Therefore, a large part of the paper is devoted to the classification problems of step functions with values in ? ^d .     

On values of hypergeometric functions with different irrational parameters

In this paper, we study the arithmetic nature of the values of generalized hypergeometric functions with different irrational parameters. Linear independence of such values is proved by means of an effective construction of the Pade approximation; corresponding quantitative results are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 65–72, 2005.     

On discrepancies of irrational rotations: an approach via rational rotation

Setokuchi and Takashima (Unif Distrib Theory (2) 9:31–57, 2014) and Setokuchi (Acta Math Hung, [11]) gave refinements of estimates for discrepancies by using Schoissengeier’s exact formula. Mori and Takashima (Period Math Hung, [7]) discussed the distribution of the leading digits of \(a^n\) by approximating irrational rotations by “rational rotations”. We apply here their methods to the estimation of discrepancies. We give much more accurate estimates for discrepancies by simple direct calculations, without using Schoissengeier’s formula. We show that the initial segment of the graph of discrepancies of irrational rotations with a single isolated large partial quotient is linearly decreasing, provided we observe the discrepancies on a linear scale with suitable step. We also prove that large hills, caused by single isolated large partial quotients, will appear infinitely often.     

Constructive irrational space

The Fréchet combination allows the construction of a complete metric on the set of irrational numbers. The constructive study of the resulting spaceM was begun by Errett Bishop. This paper studies the structure ofM in some detail. The constructive approach requires a strong form of the concept of irrational number and particular attention to the distinctions between the various notions of points exterior to a set. The main results are the characterization and construction of all compact and locally compact subspaces ofM.     

On the length of the period of a real quadratic irrational

We review some known and not so well known results on the length of the period of the continued fraction expansion of a quadratic irrational \(\sqrt D \) with D > 0: We also show that this length is o((DlogD)^1/2) for almost all D.     

On the Whitehead group of Novikov rings associated to irrational homomorphisms

Given a homomorphism ξ:G→R we show that the natural map from the Whitehead group of G to the Whitehead group of the Novikov ring is surjective. The group is of interest for the simple chain homotopy type of the Novikov complex. It also contains the Latour obstruction for the existence of a nonsingular closed 1-form within a fixed cohomology class ξ∈H¹(M;R), where M is a closed connected smooth manifold.