It goes back to Lagrange that a real quadratic irrational has always a periodic continued fraction. Starting from decades ago, several authors proposed different definitions of a $p$-adic continued fraction, and the definition depends on the chosen system of residues mod $p$. It turns out that the theory of p-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagranges theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known until now. In recent work with F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the $p$-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion, somewhat surprisingly, depends on the ‘real' value of the $p$-adic continued fraction.[-]

The course will explore several related topics in number theory with dynamical and/or geometric facets: continued fractions, Diophantine approximation, and Apollonian circle packings. We will focus on both theoretical and experimental tools, a parallel goal will be to experience the role of visualization and illustration in mathematical research. In covering background material, the approach will emphasize the visual and dynamical:
(1) Continued fractions, quadratic forms, and Diophantine approximation.
(2) Hyperbolic geometry, Minkowski space, and Kleinian groups.
With these tools at hand, we will study some areas of current research:
(1) The geometry of Diophantine approximation and continued fractions in the complex plane,including algebraic starscapes and Schmidt arrangements.
(2) Apollonian circle packings, with an emphasis on their surprising relationships to the preceding topics.[-]

Optimal results on the improvements to Dirichlet's Theorem are obtained in the one-dimensional case. For simultaneous approximation the problem is open. I will describe reduction of the problem to dynamics both in one-dimensional case (via continued fractions) and for higher dimensions (via diagonal flows on the space of lattices). If time allows I'll mention an inhomogeneous version which is easier than the homogeneous one. Joint work with Nick Wadleigh.[-]

Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded discrepancy provides particularly strong convergence properties of ergodic sums. It is also closely related to the notions of balance in word combinatorics.[-]

We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared to the one-dimensional case our criterion is rather unusual. We apply this criterion to the analysis of Dirichlet spectrum for simultaneous Diophantine approximation.[-]

Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a < q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.[-]

The course will explore several related topics in number theory with dynamical and/or geometric facets: continued fractions, Diophantine approximation, and Apollonian circle packings. We will focus on both theoretical and experimental tools, a parallel goal will be to experience the role of visualization and illustration in mathematical research. In covering background material, the approach will emphasize the visual and dynamical:
(1) Continued fractions, quadratic forms, and Diophantine approximation.
(2) Hyperbolic geometry, Minkowski space, and Kleinian groups.
With these tools at hand, we will study some areas of current research:
(1) The geometry of Diophantine approximation and continued fractions in the complex plane,including algebraic starscapes and Schmidt arrangements.
(2) Apollonian circle packings, with an emphasis on their surprising relationships to the preceding topics.[-]

The course will explore several related topics in number theory with dynamical and/or geometric facets: continued fractions, Diophantine approximation, and Apollonian circle packings. We will focus on both theoretical and experimental tools, a parallel goal will be to experience the role of visualization and illustration in mathematical research. In covering background material, the approach will emphasize the visual and dynamical:
(1) Continued fractions, quadratic forms, and Diophantine approximation.
(2) Hyperbolic geometry, Minkowski space, and Kleinian groups.
With these tools at hand, we will study some areas of current research:
(1) The geometry of Diophantine approximation and continued fractions in the complex plane,including algebraic starscapes and Schmidt arrangements.
(2) Apollonian circle packings, with an emphasis on their surprising relationships to the preceding topics.[-]