Shigeki Akiyama (University of Tsukuba)
Title: A criterion for almost periodicity of sequences
Abstract: Given a subshift, we wish to decide almost periodicity of the system. There are many known criteria. We wish to add another geometric sufficient condition, which stems from construction of Rauzy fractal. For the subshift generated by irreducible Pisot substitution, the condition is shown to be necessary as well. One good feature of this method is that this is checkable by automata computation. We apply several interesting examples including an S-adic system. This is a joint work with Paul Mercat.
Yann Bugeaud (Université de Strasbourg)
Title: On simultaneous rational approximation to a real number and its integral powers
Abstract: For a positive integer $n$ and a real number $\xi$, let $\lambda_n (\xi)$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||, \ldots , ||q \xi^n||$ are all less than $q^{-\lambda}$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We present new results on the Hausdorff dimension of the set of real numbers $\xi$ such that $\lambda_n (\xi)$ is equal (or greater than or equal) to a given value. This is a joint work with Dmitry Badziahin.
Byungchul Cha (Muhlenberg College)
Title: Intrinsic Diophantine Approximation of Spheres
Abstract: In Diophantine approximation, we would like to know how well (or badly) a real number can be approximated by rational numbers. Associated to each real number, we can define its approximation constant, which measures its approximability. The collection of all approximation constants is called the Lagrange spectrum. A classical result of Markoff, which had first appeared in 1879 and 1880, gives a complete description of the discrete part of the Lagrange spectrum. This characterizes all the badly approximable real numbers.
In this talk, we study the intrinsic Diophantine approximation of spheres of low dimensions, especially, a circle and a 2-sphere. In particular, we give complete descriptions of discrete parts of the Lagrange spectra, arising from intrinsically approximating points on a circle by the rational points on it. We also present some relevant results for a 2-sphere.
This is joint work with Dong Han Kim.
Arnaud Durand (Université Paris-Sud)
Title: Metric Diophantine approximation on fractals: large intersection properties and probabilistic aspects
Abstract: Given $\mu\geq 2$, let $M(\mu)$ denote the set of real numbers approximable at order at least $\mu$ by rational numbers. Jarn\'ik and Besicovitch established that the Hausdorff dimension of $M(\mu)$ is equal to $2/\mu$. In a recent joint paper with Yann Bugeaud, we were interested in the size of the intersection of $M(\mu)$ with compact subsets of the interval $[0,1]$. In particular, we proposed a conjecture for the exact value of the dimension of $M(\mu)$ intersected with the middle-third Cantor set. Among other supporting ideas, we showed that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The purpose of the talk is: (1) to recall the above work; (2) to put it into perspective using a new potential theoretical framework which connects hitting probabilities for random sets (in particular, sets approximated by systems of random points), and the size and large intersection properties of their intersection with arbitrary compacts.
Anish Ghosh (Tata Institute of Fundamental Research)
Title: On Inhomogeneous quadratic forms
Abstract: I will discuss some qualitative and quantitative versions of the Oppenheim conjecture for inhomogeneous quadratic forms.
Jiyoung Han (Seoul National University)
Title: The quantitative Oppenheim conjecture for S-arithmetic quadratic forms of low rank
Abstract: The quantitative Oppenheim conjecture, proved by Eskin, Margulis and Mozes in 1987, is an equidistribution problem of the image set of integral lattice vectors under a given quadratic form. It is known that it is hard to solve the problem completely when the signature of a quadratic form is (2,1) or (2,2). On the other hand, for the case of quadratic forms of low rank, one can involve more geometic intuitions of the homogeneous space of their special orthogonal group in the proof. In this talk, we will provide the S-arithmetic analogy of equidistribution problem for quadratic forms of rank 3 and 4, following the work of Eskin-Margulis-Mozes. In the case of rank 3, the homogeneous spaces of their special orthogonal groups are the product of the hyperbolic plane and (p+1)-regular trees, for a finite number of odd primes p.
Michihiro Hirayama (University of Tsukuba)
Title: On Möbius disjointness for infinite measure preserving systems
Abstract: Sarnak formulated the "randomness" of the Möbius function in terms of disjointness of the function from any topological dynamics on a compact metric space (Sarnak’s conjecture). In this talk, we will discuss such a disjointness in infinite measure preserving systems. This is a joint work with Davit Karagulyan.
Taehyeong Kim (Seoul National University)
Title: Hausdorff dimension in weighted inhomogeneous Diophantine approximation
Abstract: In 2018, Yann Bugeaud, Dong Han Kim, Seonhee Lim, and Michal Rarms showed that the set of ε-badly approximable numbers for α has full Hausdorff dimension if and only if α is singular on average. They also gave the equivalent conditon for singular on average property using the denominator of its convergent. Wooyeon Kim and Seonhee Lim recently partially generalized this theorem to the matrix case. In this talk, we will give a generalization of the remaining parts with weights. This is a joint work with Seonhee Lim.
Sanghoon Kwon (Catholic Kwandong University)
Title: Rate of equidistribution of long closed horocycle
Abstract: We review the Zagier's horocycle criterion for RH and equidistribution of positive characteristic analogue. We discuss some dynamical results as well as an ongoing work about the rate of equidistribution. The main strategy is constructing Young's tower for discrete geodesic flow.
Jeong-Yup Lee (Catholic Kwandong University)
Title: Regular model set and pure discrete spectrum on substitution tilings in R^d
Abstract: It has been known that every regular model set has pure point spectrum, but the converse is not true in general. The relation between regular model set and pure point spectrum is well studied in Baake-Moody '04, Baake-Lenz-Moody '07, Strungaru '17 in quite a general setting. When we restrict to substitution tilings, it has been shown Lee '07 that pure point spectrum and inter model set are equivalent. However the inter model set is a projected point set in a cut-and-project scheme(CPS) with an abstract internal space. It was not easy to extract information from the internal space. Here we show that the internal space can be an Euclidean space under some additional assumption. This result generalizes a remark in Barge-Kwapisz '06 which shows the equivalence between regular model set and pure point spectrum in the case of 1-dimension substitution tilings.
Seulbee Lee (Seoul National University)
Title: Odd-odd continued fraction
Abstract: It is known that regular continued fraction gives the best approximation of an irrational number with rationals. We investigate a continued fraction, say odd-odd continued fraction, which gives the best approximation of an irrational with rationals whose numerator and denominator are odd. This is joint work with Dong Han Kim and Lingmin Liao.
Lingmin Liao (University Paris-East Créteil)
Title: Normal sequences with given limits of multiple ergodic averages
Abstract: We study the set of normal sequences in the space {0,1}^N with a given frequency of the pattern "11" in the positions k, 2k. The topological entropy of such sets is determined. This is a joint work with Michal Rams.
Stefano Marmi (Scuola Normale Superiore)
Title: Birkhoff sums for diophantine interval exchange maps
Luca Marchese (Université Paris 13)
Title: Dimension of bad sets for non-uniform Fuchsian lattices
Abstract: In classical diophantine approximations it is natural to consider the set "Bad" of those real numbers which are badly approximable by rationals: it is a set of zero Lebesgue measure and full dimension. Finer metric properties have been investigated in depth, both for the classical case and for several generalizations, which arise from the relation between diophantine approximations and the dynamics on homogeneous spaces (or other moduli spaces). The set Bad admits a natural exhaustion by sub-sets Bad(c), in terms a positive parameter c>0, and the dimension of Bad(c) converges to 1 as c goes to 0. D. Hensley computed the asymptotic for the dimension up to the first order in c, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded. I will prove a generalization of Hensley's asymptotic formula in the context of Fuchsian groups, considering the set of points in the boundary of the hyperbolic space which are badly approximable by the orbits of a non-uniform lattice G in PSL(2,R), and an exhaustion of such set by subsets Bad(G,c), in terms of a parameter c>0. Bowen and Series introduced a "boundary expansion" which enables to approximate any set Bad(G,c) by a dynamically defined Cantor set, whose dimension can be estimated with great precision by thermodynamic techniques introduced by Ruelle and Bowen. A perturbative analysis of the spectral radius of the transfer operator gives the dimension of Bad(G,c) up to the first order in c.
Rene Rühr (Technion - Israel Institute of Technology)
Title: Classification of cut and project schemes
Abstract: We will report on work in prgress about classification of the Ratner groups defining the moduli space of cut and project sets introduced by Marklof-Strömbergsson (Communications in Mathematical Physics 2013).
Nicolas de Saxcé (CNRS - Université Paris 13)
Title: Diophantine approximation on quadrics
Abstract: Following work of Kleinbock, Merrill, Fishman and Simmons, we shall study the diophantine exponent of a point x on a quadric hypersurface. In particular, we shall explain how the classical theorems of Roth on algebraic numbers, and of Kleinbock and Margulis on extremality of non-degenerate submanifolds can be adapted to our setting.
Tom Schmidt (Oregon State University)
Title: Planar extensions and relaxing the Markov condition
Abstract: We give easily verified conditions implying that an interval map f is eventually expanding and is ergodic with respect to an absolutely continuous invariant measure. The conditions are in terms of a planar extension system. Expansiveness follows from the planar domain having finite nonzero Lebesgue measure and being such that the fibers above some full cylinder of the interval map f are mapped to a proper fraction of the receiving fibers (with respect to Lebesgue measure). In addition, ergodicity follows from what we call the "non-full range" condition: there is a full cylinder of f that is avoided by the f-orbits of the endpoints of all non-full cylinders. We give examples of families of interval maps where the conditions apply. This is joint work with Cor Kraaikamp.
Younghwan Son (Pohang University of Science and Technology)
Title: Joint ergodicity
Abstract: Berend and Bergelson introduced the notion of joint ergodicity and investigated some properties. Joint ergodicity of commuting invertible measure preserving transformations is a natural extension of ergodicity of a measure preserving transformation. In this talk, we will present some extension of the previous results including characterization of joint ergodicity along generalized sequences, which is a joint work with Vitaly Bergelson and Alexander Leibman.