Diophantine Type Research Articles

Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

<p style='text-indent:20px;'>We prove quantitative statistical stability results for a large class of small <inline-formula><tex-math id="M1">\begin{document}$ C^{0} $\end{document}</tex-math></inline-formula> perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.</p>

A genus 4 origami with minimal hitting time and an intersection property

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami, the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.

The Distribution of Spacings Between the Fractional Parts of ndα

Abstract We study the distribution of spacings between the fractional parts of $n^d\alpha $. For $\alpha $ of high enough Diophantine type we prove a necessary and sufficient condition for $n^d\alpha \mod 1, 1\leq n\leq N,$ to be Poissonian as $N\to \infty $ along a suitable subsequence.

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $\Gamma\backslash G$ which project to pieces of closed horocycles in SL$(2,Z)\backslash$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-\alpha)^2+(m_2-\beta)^2-(m_3-\alpha)^2-(m_4-\beta)^2$, for $(\alpha,\beta)$ of Diophantine type, following the approach by Marklof [24] using theta sums.

Linear forms of a given Diophantine type and lattice exponents

In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.

Simultaneous Approximation on Affine Subspaces

Abstract We solve the convergence case of the generalized Baker–Schmidt problem for simultaneous approximation on affine subspaces, under natural diophantine type conditions. In one of our theorems, we do not require monotonicity on the approximation function. In order to prove these results, we establish asymptotic formulae for the number of rational points close to an affine subspace. One key ingredient is a sharp upper bound on a certain sum of reciprocals of fractional parts associated with the matrix defining the affine subspace.

Long hitting time for translation flows and L-shaped billiards

We consider the flow in direction $ \theta $ on a translation surface and we study the asymptotic behavior for $ r\to 0 $ of the time needed by orbits to hit the $ r $-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction $ \theta $. In higher genus, we consider a generalized geometric notion of Diophantine type of a direction $ \theta $ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.

On the discrepancy of random subsequences of $\{n\alpha\}$

For irrational $\alpha$, $\{n\alpha\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences $\{n_k \alpha\}$, with the exception of metric results for exponentially growing $(n_k)$. It is therefore natural to consider random $(n_k)$, and in this paper we give nearly optimal bounds for the discrepancy of $\{n_k \alpha\}$ in the case when the gaps $n_{k+1}-n_k$ are independent, identically distributed, integer-valued random variables. As we will see, the discrepancy behavior is determined by a delicate interplay between the distribution of the gaps $n_{k+1}-n_k$ and the rational approximation properties of $\alpha$. We also point out an interesting critical phenomenon, a sudden change of the order of magnitude of the discrepancy of $\{n_k \alpha\}$ as the Diophantine type of $\alpha$ passes through a certain critical value.

Basics on the theory of heights and its applications to certain diophantine problems

The present article offers an overview of the basic theory of heights of algebraic points on varieties, illustrating some applications, mainly of diophantine type. The exposition is informal and with little detail, modeled on three lectures delivered by the author at the Fields Institute in Toronto in February 2017.

Rational approximation of affine coordinate subspaces of Euclidean space

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the basepoint of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence.

Asymptotic diophantine approximation: the multiplicative case

Let \(\alpha \) and \(\beta \) be irrational real numbers and \(0<\varepsilon <1/30\). We prove a precise estimate for the number of positive integers \(q\le Q\) that satisfy \(\Vert q\alpha \Vert \cdot \Vert q\beta \Vert <\varepsilon \). If we choose \(\varepsilon \) as a function of Q, we get asymptotics as Q gets large, provided \(\varepsilon Q\) grows quickly enough in terms of the (multiplicative) Diophantine type of \((\alpha ,\beta )\), e.g., if \((\alpha ,\beta )\) is a counterexample to Littlewood’s conjecture, then we only need that \(\varepsilon Q\) tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Le and Vaaler.

Mean value estimates for odd cubic Weyl sums

We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $\alpha x^3+\beta x$. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown's variant of Weyl's inequality, and other applications of Diophantine type.

Equivalence classes of codimension-one cut-and-project nets

We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement equivalent to lattices. Our proof demonstrates that for any irrational ${\it\alpha}$, regardless of Diophantine type, there is a collection of intervals in $\mathbb{R}/\mathbb{Z}$ which is closed under translation, contains intervals of arbitrarily small length, and along which the discrepancy of the sequence $\{n{\it\alpha}\}$ is bounded above uniformly by a constant.

Diophantine type of interval exchange maps

AbstractRoth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two non-equivalent Roth type conditions, the first (condition (Z)) by means of the Zorich cocycle [Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier 46(2) (1996), 325–370], the second (condition (A)) by means of a further acceleration of the continued fraction algorithm by Marmi–Moussa–Yoccoz introduced in [The cohomological equation for Roth type interval exchange maps, J. Amer. Math. Soc. 18 (2005), 823–872]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or with respect to uniform convergence (respectively conditions (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R), which are equivalent among them. For maps of four intervals or more we prove several results; the only relation that we cannot decide is whether (Z) implies (R) or not.

Inhomogeneous Diophantine approximation with general error functions

Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_{\varphi}(\al):={y\in \R: |n\al -y| < \varphi(n) \text{for infinitely many} n},] where $|\cdot|$ denotes the distance to the nearest integer.

Optimal bound for the discrepancies of lacunary sequences

The law of the iterated logarithm for discrepancies of lacunary sequences is studied. An optimal bound is given under a very mild Diophantine type condition.

Information theory: Sources, Dirichlet series, and realistic analyses of data structures

Most of the text algorithms build data structures on words, mainly trees, as digital trees (tries) or binary search trees (bst). The mechanism which produces symbols of the words (one symbol at each unit time) is called a source, in information theory contexts. The probabilistic behaviour of the trees built on words emitted by the same source depends on two factors: the algorithmic properties of the tree, together with the information-theoretic properties of the source. Very often, these two factors are considered in a too simplified way: from the algorithmic point of view, the cost of the Bst is only measured in terms of the number of comparisons between words --from the information theoretic point of view, only simple sources (memoryless sources or Markov chains) are studied. We wish to perform here a realistic analysis, and we choose to deal together with a general source and a realistic cost for data structures: we take into account comparisons between symbols, and we consider a general model of source, related to a dynamical system, which is called a dynamical source. Our methods are close to analytic combinatorics, and our main object of interest is the generating function of the source Lambda(s), which is here of Dirichlet type. Such an object transforms probabilistic properties of the source into analytic properties. The tameness of the source, which is defined through analytic properties of Lambda(s), appears to be central in the analysis, and is precisely studied for the class of dynamical sources. We focus here on arithmetical conditions, of diophantine type, which are sufficient to imply tameness on a domain with hyperbolic shape.

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series \({\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}}\) for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λn = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series \({\sum_{n=1}^{\infty} g(n\alpha) z^n}\). We prove that a Dirichlet series \({\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s}\) has an abscissa of convergence σ0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ0 satisfies σ0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζg,α(s) has an analytic continuation to the entire complex plane.

Linear forms of a given Diophantine type

Nous demontrons un resultat sur l'existence des formes lineaires de type Diophantien donne. .

CONSTRUCTING HERMAN RINGS BY TWISTING ANNULUS HOMEOMORPHISMS

AbstractLet F(z) be a rational map with degree at least three. Suppose that there exists an annulus $H \subset \widehat {\mathbb {C}}$ such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.