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BEGIN:VEVENT
SUMMARY:Osama Khalil (The University of Utah)
DTSTART;VALUE=DATE-TIME:20200603T140000Z
DTEND;VALUE=DATE-TIME:20200603T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/1
DESCRIPTION:Title: Random walks\, spectral gaps\, and Khintchine's theorem on fractals\nby Osama Khalil (The University of Utah) as part of Webinar on Diophan
tine approximation and homogeneous dynamics\n\n\nAbstract\nIn 1984\, Mahle
r asked how well typical points on Cantor’s set can be approximated by r
ational numbers. His question fits within a program\, set out by himself i
n the 1930s\, attempting to determine conditions under which subsets of $\
\mathbb{R}^d$ inherit the Diophantine properties of the ambient space. Sin
ce the approximability of typical points in Euclidean space by rational po
ints is governed by Khintchine’s classical theorem\, the ultimate form o
f Mahler’s question asks whether an analogous zero-one law holds for fra
ctal measures. Significant progress has been achieved in recent years\, al
beit\, almost all known results have been of “convergence type”.\nIn t
his talk\, we will discuss the first instances where a complete analogue o
f Khinchine’s theorem for fractal measures is obtained. The class of fra
ctals for which our results hold includes those generated by rational simi
larities of $\\mathbb{R}^d$ and having sufficiently small Hausdorff co-dim
ension. The main new ingredient is an effective equidistribution theorem f
or certain fractal measures on the space of unimodular lattices. The latte
r is established via a new technique involving the construction of $S$-ari
thmetic Markov operators possessing a spectral gap and encoding the arithm
etic structure of the maps generating the fractal. This is joint work in p
rogress with Manuel Luethi.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barak Weiss (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20200610T123000Z
DTEND;VALUE=DATE-TIME:20200610T140000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/2
DESCRIPTION:Title: New bounds on the covering density of a lattice\nby Barak Weiss (
Tel Aviv University) as part of Webinar on Diophantine approximation and h
omogeneous dynamics\n\n\nAbstract\nWe obtain new upper bounds on the minim
al density of lattice coverings of $\\mathbb{R}^n$ by dilates of a convex
body $K$. We also obtain bounds on the probability (with respect to the na
tural Haar-Siegel measure on the space of lattices) that a randomly chosen
lattice $L$ satisfies $L+K=\\mathbb{R}^n$. Joint work with Or Ordentlich
and Oded Regev.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Maynard (University of Oxford)
DTSTART;VALUE=DATE-TIME:20200617T130000Z
DTEND;VALUE=DATE-TIME:20200617T150000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/3
DESCRIPTION:Title: On the Duffin-Schaeffer Conjecture\nby James Maynard (University
of Oxford) as part of Webinar on Diophantine approximation and homogeneous
dynamics\n\n\nAbstract\nAlmost 80 years ago Duffin and Schaeffer conjectu
red a beautiful strengthening of Khinchin's classical result: Given a sequ
ence of possible forms of rational approximation\, either almost all reals
can be approximated in this manner or almost none can be\, and there is a
simple calculation to tell which case we are in.\nI'll talk about recent
work with D. Koukoulopoulos which establishes this conjecture. This relies
on a blend of different techniques\, recasting the problem as a structura
l question in additive combinatorics\, and then approaching this via study
ing a particular family of graphs to reduce it to a problem in analytic nu
mber theory.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lei Yang (Sichuan University)
DTSTART;VALUE=DATE-TIME:20200701T133000Z
DTEND;VALUE=DATE-TIME:20200701T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/4
DESCRIPTION:Title: Winning property of badly approximable points on curves\nby Lei Y
ang (Sichuan University) as part of Webinar on Diophantine approximation a
nd homogeneous dynamics\n\n\nAbstract\nWe will prove that badly approximab
le points (no matter weighted or unweighted) on any analytic non-degenerat
e curve in $\\mathbb{R}^n$ is an absolute winning set. This confirms a key
conjecture in the area stated by Badziahin and Velani (2014) which repres
ents a far-reaching generalisation of Davenport's problem from the 1960s.
Amongst various consequences of our main result is a solution to Bugeaud's
problem on real numbers badly approximable by algebraic numbers of arbitr
ary degree. This work is joint with Victor Beresnevich and Erez Nesharim.\
n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolay Moshchevitin (Moscow State University)
DTSTART;VALUE=DATE-TIME:20200708T133000Z
DTEND;VALUE=DATE-TIME:20200708T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/5
DESCRIPTION:Title: Singular vectors: from Khintchine to nowadays\nby Nikolay Moshche
vitin (Moscow State University) as part of Webinar on Diophantine approxim
ation and homogeneous dynamics\n\n\nAbstract\nMany concepts in Diophantine
Approximation have their origin in the \nfamous paper "Über eine Klasse
linearer diophantischer Approximationen" by A. Khintchine (1926). The resu
lts of this paper were rediscovered many times by different mathematicians
. In particular\, Khintchine was the first who observed the phenomenon of
singularity in higher-dimensional Diophantine Approximation. In our lectur
e we discuss several problems related to singular vectors and best approxi
mation (minimal points) as well as some related topics dealing with Diopha
ntine exponents and approximation on algebraic and analytic surfaces which
were considered recently in author's joint papers with D. Kleinbock and B
. Weiss. Also we suppose to discuss some related open problems.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Baowei Wang (Huazhong University of Sci. & Tech.)
DTSTART;VALUE=DATE-TIME:20200722T133000Z
DTEND;VALUE=DATE-TIME:20200722T143000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/6
DESCRIPTION:Title: Mass transference principle from rectangles to rectangles in Diophant
ine approximation\nby Baowei Wang (Huazhong University of Sci. & Tech.
) as part of Webinar on Diophantine approximation and homogeneous dynamics
\n\n\nAbstract\nAs is well known\, Dirichlet's theorem and Minkowski's the
orem are two fundamental results in Diophantine approximation. One says th
at all points in R^d will fall into infinitely many balls centered at rati
onals with specific radius\; while the other says that all points will fal
l into infinitely many rectangles centered at rationals with specific side
lengths. This motives a further study on the metric theory of limsup sets
defined by a sequence of balls or rectangles. Since the landmark works of
Beresnevich & Velani (2006) and Beresnevich\, Dickinson & Velani (2006) w
here the mass transference principle was found\, the metric theory for lim
sup sets defined by a sequence balls or isotropic thicken of general sets
has been sufficiently well established. While\, the metric theory for lims
up sets defined by a sequence of rectangles are not as rich as the ball ca
se. In this talk\, I will talk about some progess on the metric theory of
the latter case by modifying the settings in the above mentioned impressin
g works.\n\nThis talk will only take one hour.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengyu Yang (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20200729T133000Z
DTEND;VALUE=DATE-TIME:20200729T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/7
DESCRIPTION:Title: Dirichlet Improvability\, Equidistribution\, and Grassmannians\nb
y Pengyu Yang (ETH Zurich) as part of Webinar on Diophantine approximation
and homogeneous dynamics\n\n\nAbstract\nAs a natural generalisation of Di
richlet's approximation theorem on real numbers\, Dirichlet's approximatio
n theorem on $m\\times n$ real matrices tells us the following: given $m$
real linear forms in $n$ variables\, we can find an integral vector such t
hat the evaluations of all the linear forms at this integral vector are si
multaneous small. In the 1960s Davenport and Schmidt showed that Dirichlet
’s theorem is non-improvable for almost all matrices\, and they asked if
the analogous result holds for a submanifold of the space of $m\\times n$
matrices. This problem is related to an equidistribution problem in the s
pace of unimodular lattices in $\\mathbb{R}^n$. In this talk I will presen
t some recent progress on this problem\, and I will explain its connection
s to the geometry of Grassmannian manifolds.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anurag Rao (Brandeis University)
DTSTART;VALUE=DATE-TIME:20200805T130000Z
DTEND;VALUE=DATE-TIME:20200805T140000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/8
DESCRIPTION:Title: Some problems in uniform Diophantine approximation\nby Anurag Rao
(Brandeis University) as part of Webinar on Diophantine approximation and
homogeneous dynamics\n\n\nAbstract\nWe study a norm sensitive Diophantine
approximation problem arising from the work of Davenport and Schmidt on t
he improvement of Dirichlet's theorem. Its supremum norm case was recently
considered by the Kleinbock and Wadleigh\, and here we extend the set-up
by replacing the supremum norm with an arbitrary norm. This gives rise to
a class of shrinking target problems for one-parameter diagonal flows on t
he space of lattices\, with the targets being neighborhoods of the critica
l locus of a suitably scaled norm ball. We use methods from geometry of nu
mbers and dynamics to generalize a result due to Andersen and Duke on meas
ure zero and uncountability of the set of numbers for which Minkowski appr
oximation theorem can be improved. The choice of the Euclidean norm on $\\
mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface whi
ch visit a decreasing family of balls. An application of a dynamical Borel
-Cantelli lemma of Maucourant produces a zero-one law for improvement of D
irichlet's theorem in Euclidean norm. Based on joint works with Dmitry Kle
inbock and Srinivasan Sathiamurthy.\n\nThe starting time is earlier than u
sual. The talk will only take one hour.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anish Ghosh (Tata Institute of Fundamental Research)
DTSTART;VALUE=DATE-TIME:20200812T133000Z
DTEND;VALUE=DATE-TIME:20200812T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/9
DESCRIPTION:Title: Diophantine approximations\, large intersections and geodesics in neg
ative curvature\nby Anish Ghosh (Tata Institute of Fundamental Researc
h) as part of Webinar on Diophantine approximation and homogeneous dynamic
s\n\n\nAbstract\nI will discuss new results on the `shrinking target pro
blem' including a logarithm law for approximation by geodesics in negati
vely curved manifolds and Hausdorff dimension estimates for finer spiralin
g phenomena of geodesics. I will also discuss the large intersection prope
rty of Falconer in the context of negative curvature and some applications
to Diophantine approximation and to hyperbolic geometry. This is joint wo
rk with Debanjan Nandi.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Denis Koleda (Institute of mathematics\, Minsk)
DTSTART;VALUE=DATE-TIME:20200819T133000Z
DTEND;VALUE=DATE-TIME:20200819T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/10
DESCRIPTION:Title: The distribution of conjugate algebraic numbers: a random polynomial
approach\nby Denis Koleda (Institute of mathematics\, Minsk) as part
of Webinar on Diophantine approximation and homogeneous dynamics\n\n\nAbst
ract\nIn the talk we consider the spatial distribution of points that have
algebraic (Galois) conjugate coordinates of fixed degree and bounded heig
ht. We give an asymptotic formula for counting such points in a wide class
of regions of Euclidean space (as the parameter that bounds heights grows
to infinity). We explain connection of this formula to random polynomials
with i.i.d. coefficients. We also discuss some corollaries and applicatio
ns of the formula. The talk is based on a joint work with F. Götze and D.
Zaporozhets.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20200826T133000Z
DTEND;VALUE=DATE-TIME:20200826T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/11
DESCRIPTION:Title: Higher order uniformity of the Liouville function\nby Kaisa Mato
mäki (University of Turku) as part of Webinar on Diophantine approximatio
n and homogeneous dynamics\n\n\nAbstract\nThe Liouville function takes a v
alue +1 or -1 at a natural number $n$ depending on whether $n$ has an even
or an odd number of prime factors. The Liouville function is believed to
behave more or less randomly. In particular a famous conjecture of Sarnak
says that the Liouville function does not correlate with any sequence of "
low complexity" whereas a longstanding conjecture of Chowla says that the
Liouville function has negligible correlations with its own shifts.\nI wil
l discuss conjectures of Sarnak and Chowla and my very recent work with Ra
dziwiłł\, Tao\, Teräväinen\, and Ziegler\, where we show that\, in alm
ost all intervals of length $X^\\varepsilon$\, the Liouville function does
not correlate with polynomial phases or more generally with nilsequences.
I will also discuss applications to superpolynomial word complexity for t
he Liouville sequence and to a new averaged version of Chowla's conjecture
.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damien Roy (University of Ottawa)
DTSTART;VALUE=DATE-TIME:20200923T133000Z
DTEND;VALUE=DATE-TIME:20200923T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/12
DESCRIPTION:Title: Simultaneous rational approximation to exponentials of algebraic num
bers\nby Damien Roy (University of Ottawa) as part of Webinar on Dioph
antine approximation and homogeneous dynamics\n\n\nAbstract\nThe theorem o
f Lindemann-Weierstrass asserts that the exponentials of distinct algebrai
c numbers are linearly independent over the field of rational numbers. The
proof uses a construction of simultaneous rational approximations to such
exponentials values\, which goes back to Hermite. In this talk\, we sho
w that\, from an adelic perspective\, these approximations are essentially
best possible. This point of view partly explains the nature of the alg
ebraic numbers whose exponentials have a structured continuous fraction ex
pansion. We also propose few specific conjectures regarding simultaneo
us approximations to such values in adèle rings.\n\nThe proof of our main
result requires a separate analysis for each place of the associated numb
er field. For the Archimedean places\, it relies on the structure of the
graph drawn in the complex plane by the paths of fastest descent for the
norm of a general univariate complex polynomial starting from the roots of
its derivative and ending in the roots of the polynomial. It happens
that this graph is a tree and that the lengths of those paths can be estim
ated from above in terms of the degree of the given polynomial and the dia
meter of its set of zeros. We will mention some instances where these up
per bounds can be greatly improved and state as an open problem whether or
not such improvements hold in general.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART;VALUE=DATE-TIME:20200916T133000Z
DTEND;VALUE=DATE-TIME:20200916T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/13
DESCRIPTION:Title: A fully-inhomogeneous version of Gallagher's theorem\nby Sam Cho
w (University of Warwick) as part of Webinar on Diophantine approximation
and homogeneous dynamics\n\n\nAbstract\nGallagher's theorem describes the
multiplicative diophantine \napproximation rate of a typical vector. We es
tablish a fully-inhomogeneous \nversion of Gallagher's theorem\, a diophan
tine fibre refinement\, and a \nsharp and unexpected threshold for Liouvil
le fibres. Along the way\, we \nprove an inhomogeneous version of the Duff
in--Schaeffer conjecture for a \nclass of non-monotonic approximation func
tions. Joint with Niclas Technau.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ralf Spatzier (University of Michigan)
DTSTART;VALUE=DATE-TIME:20200930T133000Z
DTEND;VALUE=DATE-TIME:20200930T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/14
DESCRIPTION:Title: Hyperbolic Actions of Higher Rank Abelian Groups\nby Ralf Spatzi
er (University of Michigan) as part of Webinar on Diophantine approximatio
n and homogeneous dynamics\n\n\nAbstract\nWe study transitive $\\mathbb{R}
^k \\times \\mathbb{Z}^\\ell$ actions on arbitrary compact manifolds with
a projectively dense set of Anosov elements and 1-dimensional coarse Lyap
unov foliations. Such actions are called totally Cartan actions. We comple
tely classify such actions as built from low-dimensional Anosov flows and
diffeomorphisms and affine actions\, verifying the Katok-Spatzier conjectu
re for this class. This is achieved by introducing a new tool\, the action
of a dynamically defined topological group describing paths in coarse Lya
punov foliations\, and understanding its generators and relations. We obta
in applications to the Zimmer program. This talk is based on joint work wi
th Kurt Vinhage.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mishel Skenderi (Brandeis University)
DTSTART;VALUE=DATE-TIME:20201014T133000Z
DTEND;VALUE=DATE-TIME:20201014T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/15
DESCRIPTION:Title: Small values at integer points of generic subhomogeneous functions\nby Mishel Skenderi (Brandeis University) as part of Webinar on Diophan
tine approximation and homogeneous dynamics\n\n\nAbstract\nThis talk will
be based on joint work with Dmitry Kleinbock that has been motivated by se
veral recent papers (among them\, those of Athreya-Margulis\, Bourgain\, G
hosh-Gorodnik-Nevo\, Kelmer-Yu). Given a certain sort of group $G$ and cer
tain sorts of functions $f: \\mathbb{R}^n \\to \\mathbb{R}$ and $\\psi : \
\mathbb{R}^n \\to \\mathbb{R}_{>0}$\, we obtain necessary and sufficient c
onditions so that for Haar-almost every $g \\in G$\, there exist infinitel
y many (respectively\, finitely many) $v \\in \\mathbb{Z}^n$ for which $|(
f \\circ g)(v)| \\leq \\psi(\\|v\\|)$\, where $\\|\\cdot\\|$ is an arbitra
ry norm on $\\mathbb{R}^n$. We also give a sufficient condition in the set
ting of uniform approximation. As a consequence of our methods\, we obtain
generalizations to the case of vector-valued (simultaneous) approximation
with no additional effort. In our work\, we use probabilistic results in
the geometry of numbers that go back several decades to the work of Siegel
\, Rogers\, and W. Schmidt\; these results have recently found new life th
anks to a 2009 paper of Athreya-Margulis.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jialun Li (University of Zurich)
DTSTART;VALUE=DATE-TIME:20201021T133000Z
DTEND;VALUE=DATE-TIME:20201021T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/16
DESCRIPTION:Title: Decay of Fourier transforms of fractal measures\nby Jialun Li (U
niversity of Zurich) as part of Webinar on Diophantine approximation and h
omogeneous dynamics\n\n\nAbstract\nWe will talk about some of the recent w
orks on estimating the decay of Fourier transforms of fractal measures\, s
uch as self-similar measures and Furstenberg measures. The proof is based
on renewal theorems for stopping times of random walks on $\\mathbb{R}$.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro Pezzoni (University of York)
DTSTART;VALUE=DATE-TIME:20201028T143000Z
DTEND;VALUE=DATE-TIME:20201028T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/17
DESCRIPTION:Title: Simultaneous Diophantine approximation on manifolds by algebraic num
bers\nby Alessandro Pezzoni (University of York) as part of Webinar on
Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nSimulta
neous Diophantine approximation on manifolds is notoriously\ncomplicated\,
since it requires to take into account the arithmetic\nproperties of a ma
nifold\, as well as the analytic ones. In this talk\nwe will make some pro
gress towards a metric theory of approximation on\nmanifolds by algebraic
numbers with algebraic conjugate coordinates\,\ngeneralising a conjecture
of Sprindžuk's.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Marnat (TU Graz)
DTSTART;VALUE=DATE-TIME:20201104T143000Z
DTEND;VALUE=DATE-TIME:20201104T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/18
DESCRIPTION:Title: Dirichlet is not just Bad and Sing\nby Antoine Marnat (TU Graz)
as part of Webinar on Diophantine approximation and homogeneous dynamics\n
\n\nAbstract\nIt is well known that in dimension one\, the set of Dirichle
t \nimprovable real numbers consists precisely of badly approximable and \
nsingular numbers. We show that in higher dimensions this disjoint union \
nis not the full set of Dirichlet improvable vectors: we prove that there
\nexist uncountably many Dirichlet improvable vectors that are neither \nb
adly approximable nor singular. We construct these numbers using the \npar
ametric geometry of numbers. Furthermore\, by doing so we can choose \nthe
exponent of Diophantine approximation by a rational subspace of \ndimensi
on exactly $d$\, for any d between $0$ and $n-1$.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Haynes (University of Houston)
DTSTART;VALUE=DATE-TIME:20201111T133000Z
DTEND;VALUE=DATE-TIME:20201111T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/19
DESCRIPTION:Title: Higher dimensional gap theorems in Diophantine approximation\nby
Alan Haynes (University of Houston) as part of Webinar on Diophantine app
roximation and homogeneous dynamics\n\n\nAbstract\nThe three distance theo
rem states that\, if $x$ is any real number and $N$ is any positive intege
r\, the points $x\, 2x\, … \, Nx \\mod 1$ partition the unit interval in
to component intervals having at most $3$ distinct lengths. There are many
higher dimensional analogues of this theorem\, and in this talk we will d
iscuss two of them. In the first we consider points of the form $mx+ny \\m
od 1$\, where $x$ and $y$ are real numbers and $m$ and $n$ are integers ta
ken from an expanding set in the plane. This version of the problem was pr
eviously studied by Geelen and Simpson\, Chevallier\, Erdős\, and many ot
her people\, and it is closely related to the Littlewood conjecture in Dio
phantine approximation. The second version of the problem is a straightfor
ward generalization to rotations on higher dimensional tori which\, surpri
singly\, has been largely overlooked in the literature. For the two dimens
ional torus\, we are able to prove a five distance theorem\, which is best
possible. In higher dimensions we also have bounds\, but establishing opt
imal bounds is an open problem. The first hour of this talk will be exposi
tory\, and the second half will focus on proofs. The new results presented
in this talk are joint work with Jens Marklof and with Roland Roeder.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bersudsky (IIT Technion)
DTSTART;VALUE=DATE-TIME:20201118T143000Z
DTEND;VALUE=DATE-TIME:20201118T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/20
DESCRIPTION:Title: On the image in the torus of sparse points on expanding analytic cur
ves\nby Michael Bersudsky (IIT Technion) as part of Webinar on Diophan
tine approximation and homogeneous dynamics\n\n\nAbstract\nIt is known tha
t the projection to the $2$-torus of the normalised parameter measure on a
circle of radius $R$ in the plane becomes uniformly distributed as $R$ gr
ows to infinity. I will discuss the following natural discrete analogue fo
r this problem. Starting from an angle and a sequence of radii $\\{R_n\\}$
which diverges to infinity\, I will consider the projection to the 2-toru
s of the $n$'th roots of unity rotated by this angle and dilated by a fact
or of $R_n$. The interesting regime in this problem is when $R_n$ is much
larger than $n$ so that the dilated roots of unity appear sparsely on the
dilated circle. I will discuss 3 types of results:\n\n1. Validity of equid
istribution for all angles when the sparsity is polynomial.\n\n2. Failure
of equidistribution for some super polynomial dilations.\n\n3. Equidistrib
ution for almost all angles for arbitrary dilations.\n\nI will then pass t
o discuss more general results on the projection to the $d$-torus of dilat
ions of varying analytic curves in $d$-space.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Han Yu (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20201125T143000Z
DTEND;VALUE=DATE-TIME:20201125T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/21
DESCRIPTION:Title: Rational points near self-similar sets\nby Han Yu (University of
Cambridge) as part of Webinar on Diophantine approximation and homogeneou
s dynamics\n\n\nAbstract\nWe show that the rational points are quite well\
n'equidistributed' near the middle 15th Cantor set $K$. As a consequence\,
\nit is possible to show that the set of well-approximable numbers has\nfu
ll Hausdorff dimension inside $K$. This answers a question of\nLevesley-Sa
lp-Velani for $K$. In fact\, it is possible to prove a\nslightly stronger
result which partially answers a question of\nBugeaud-Durand. The results
also hold for some self-similar sets other\nthan $K$. We will provide a su
fficient condition and some other\nexamples. We suspect that the above res
ults hold for all self-similar\nsets with Hausdorff dimension bigger than
$1/2$ and with the open set\ncondition. We will see some heuristics in the
talk.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreyasi Datta (University of Michigan)
DTSTART;VALUE=DATE-TIME:20201202T143000Z
DTEND;VALUE=DATE-TIME:20201202T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/22
DESCRIPTION:Title: Recent progress in p-adic Diophantine approximation\nby Shreyasi
Datta (University of Michigan) as part of Webinar on Diophantine approxim
ation and homogeneous dynamics\n\n\nAbstract\nStudying the $p$-adic analog
ue of Mahler's conjecture was initiated by Sprind zuk in 1969. Subsequentl
y\, there were several partial results culminating in the work of Kleinboc
k and Tomanov\, where the $S$-adic case of the Baker-Sprindzuk conjectures
were settled in full generality. We provide a complete $p$-adic analogue
of the results of D. Kleinbock on Diophantine exponents of affine subspace
s. This answers a conjecture of Kleinbock and Tomanov. Recently\, we prove
d $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondeg
enerate manifolds. For $S$ consisting of more than one valuation\, the div
ergence results are new even in the homogeneous setting. This aformentione
d result answers questions posed by Badziahin\, Beresnevich and Velani and
also it generalizes the work of Golsefidy and Mohammadi. In the first hal
f of the talk\, I will go over these results and in the seocnd half\, I wi
ll try to concentrate on some of the technical details of the proofs. The
new results presented in this talk are joint work with Anish Ghosh.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahriar Mirzadeh (Michigan State University)
DTSTART;VALUE=DATE-TIME:20201209T143000Z
DTEND;VALUE=DATE-TIME:20201209T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/23
DESCRIPTION:Title: Upper bound for the Hausdorff dimension of exceptional orbits in hom
ogeneous spaces\nby Shahriar Mirzadeh (Michigan State University) as p
art of Webinar on Diophantine approximation and homogeneous dynamics\n\n\n
Abstract\nConsider the set of points in a homogeneous space $G/\\Gamma$\nw
hose $g_t$-orbit misses a fixed open set. It has measure zero\nif the flow
is ergodic. It has been conjectured that this set has\nHausdorff dimensio
n strictly smaller than the dimension of whole space. This\nconjecture is
proved when $G/\\Gamma$ is compact or when has real rank. In\nthis talk we
will prove the conjecture for probably the most important\nexample of the
higher rank case namely: $\\SL(m+n\, \\R)/\\SL(m+n\, \\Z)$ and $g_t = \\m
athrm{diag}\\{e^{t/m}\, \\dots \, e^{t/m}\, e^{-t/n}\, \\dots\, e^{-t/n}\\
}$. This\nhomogeneous space has many applications in Diophantine\napproxim
ation that will be discussed in the talk if time permits. This\nproject is
joint work with Dmitry Kleinbock.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetlana Jitomirskaya (UC Irvine)
DTSTART;VALUE=DATE-TIME:20201216T160000Z
DTEND;VALUE=DATE-TIME:20201216T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/24
DESCRIPTION:Title: Inhomogeneous Diophantine approximation in the coprime setting\n
by Svetlana Jitomirskaya (UC Irvine) as part of Webinar on Diophantine app
roximation and homogeneous dynamics\n\n\nAbstract\nGiven $n\\in\n$ and $x\
\in\\R$\, let \n $$||nx||^\\prime=\\min\\{|nx-m|:m\\in\\Z\, gcd(n\,m)=1\
\}.$$\nTwo conjectures in the coprime inhomogeneous Diophantine approximat
ion stated\, by analogy with the classical Diophantine approximation\, tha
t for any irrational number $\\alpha$ and almost every $\\gamma\\in \\R$\,
\n $$\\liminf_{n\\to \\infty}n||\\gamma -n\\alpha||^{\\prime}=0\,$$\nand t
hat there exists $C$ such that for all $\\gamma\\in \\R$\,\n\n$$\\liminf_{
n\\to \\infty}n||\\gamma -n\\alpha||^{\\prime} < C.$$\n\nWe will present o
ur joint work with W. Liu that proves one of those and disproves the other
.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manos Zafeiropoulos (TU Graz)
DTSTART;VALUE=DATE-TIME:20210210T143000Z
DTEND;VALUE=DATE-TIME:20210210T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/25
DESCRIPTION:Title: Inhomogeneous Diophantine Approximation on $M_0$ Sets with restricte
d denominators\nby Manos Zafeiropoulos (TU Graz) as part of Webinar on
Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nLet $\\
mu$ be a probability measure with $\\widehat{\\mu}(t)\\ll (\\log |t|)^{-A}
$ for some $A>0$\, supported on a set $F\\subseteq [0\,1]$. Let $\\mathcal
{A}=(q_n)_{n=1}^{\\infty}$ be an increasing \nsequence of integers. We
establish a quantitative inhomogeneous Khintchine-type theorem in which t
he points of interest lie in $F$ and the "denominators" of the approximant
s belong to $\\mathcal{A}$ in the following cases: \n(i) $(q_n)_{n=1}^{
\\infty}$ is lacunary and $A>2$.\n(ii)The prime divisors of $(q_n)_{n=1}^{
\\infty}$ are restricted in a set of $k$ prime numbers and $A>2k$.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (Claude Bernard University Lyon 1)
DTSTART;VALUE=DATE-TIME:20210217T143000Z
DTEND;VALUE=DATE-TIME:20210217T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/26
DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite automata
\nby Boris Adamczewski (Claude Bernard University Lyon 1) as part of Webin
ar on Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nIt
is commonly expected that expansions of numbers in multiplicatively indep
endent bases\, such as 2 and 10\, should have no common structure. However
\, it seems extraordinarily difficult to confirm this naive heuristic prin
ciple in some way or another. In the late 1960s\, Furstenberg suggested a
series of conjectures\, which became famous and aim to capture this heuris
tic. The work I will discuss in this talk is motivated by one of these con
jectures. Despite recent remarkable progress by Shmerkin and Wu\, it remai
ns totally out of reach of the current methods. While Furstenberg’s conj
ectures take place in a dynamical setting\, I will use instead the languag
e of automata theory to formulate some related conjectures that formalize
and express in a different way the same general heuristic. I will explain
how the latter follow from some recent advances in Mahler's method\; a met
hod in transcendental number theory initiated by Mahler in the end of the
1920s. This a joint work with Colin Faverjon.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Badziahin (University of Sydney)
DTSTART;VALUE=DATE-TIME:20210224T100000Z
DTEND;VALUE=DATE-TIME:20210224T120000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/27
DESCRIPTION:Title: Diophantine approximation on the Veronese curve\nby Dmitry Badzi
ahin (University of Sydney) as part of Webinar on Diophantine approximatio
n and homogeneous dynamics\n\n\nAbstract\nIn the talk we discuss the set $
S_n(\\lambda)$ of simultaneously $\\lambda$-well approximable points in $\
\mathbb{R}^n$. These are the points $x$ such that the inequality\n$$\\| x
- p/q\\|_\\infty < q^{-\\lambda - \\epsilon}$$\nhas infinitely many soluti
ons in rational points $p/q$. Investigating the intersection of this set w
ith a suitable manifold comprises one of the most challenging problems in
Diophantine approximation. It is known that the structure of this set\, es
pecially for large $\\lambda$\, depends on the manifold. For some manifold
s it may be empty\, while for others it may have relatively large Hausdorf
f dimension.\n\nWe will concentrate on the case of the Veronese curve $V_
n$. We discuss\, what is known about the Hausdorff dimension of the set $S
_n(\\lambda) \\cap V_n$ and will talk about the recent results of the spea
ker and Bugeaud which impose new bounds on that dimension.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minju Lee (Yale University)
DTSTART;VALUE=DATE-TIME:20210317T143000Z
DTEND;VALUE=DATE-TIME:20210317T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/28
DESCRIPTION:Title: Orbit closures of unipotent flows for hyperbolic manifolds with Fuch
sian ends.\nby Minju Lee (Yale University) as part of Webinar on Dioph
antine approximation and homogeneous dynamics\n\n\nAbstract\nThis is joint
work with Hee Oh. We establish an analogue of Ratner's orbit closure theo
rem for any connected closed subgroup generated by unipotent elements in
$\\mathrm{SO}(d\,1)$ acting on the space $\\Gamma\\backslash\\mathrm{SO}(d
\,1)$\, assuming that the associated hyperbolic manifold $M=\\Gamma\\backs
lash\\mathbb{H}^d$ is a convex cocompact manifold with\nFuchsian ends. For
$d = 3$\, this was proved earlier by McMullen\, Mohammadi and Oh. In a hi
gher dimensional case\, the possibility of accumulation on closed orbits o
f intermediate subgroups causes serious issues\, but in the end\, all orbi
t closures of unipotent flows are relatively\nhomogeneous. Our results imp
ly the following: for any $k\\geq 1$\,\n\n(1) the closure of any $k$-horos
phere in $M$ is a properly immersed submanifold\;\n\n(2) the closure of an
y geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold\;\n\n(3
) an infinite sequence of maximal properly immersed geodesic $(k+1)$-plane
s intersecting $\\mathrm{core} M$ becomes dense in $M$.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pablo Shmerkin (Torcuato Di Tella University)
DTSTART;VALUE=DATE-TIME:20210310T163000Z
DTEND;VALUE=DATE-TIME:20210310T183000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/29
DESCRIPTION:Title: Beyond Furstenberg's intersection conjecture\nby Pablo Shmerkin
(Torcuato Di Tella University) as part of Webinar on Diophantine approxima
tion and homogeneous dynamics\n\n\nAbstract\nHillel Furstenberg conjecture
d in the 1960s that the intersections of closed $\\times 2$ and $\\times 3
$-invariant Cantor sets have "small" Hausdorff dimension. This conjecture
was proved independently by Meng Wu and by myself\; recently\, Tim Austin
found a simple proof. I will present some generalizations of the intersect
ion conjecture and other related results.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lifan Guan (Gottingen)
DTSTART;VALUE=DATE-TIME:20210414T133000Z
DTEND;VALUE=DATE-TIME:20210414T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/30
DESCRIPTION:Title: Divergent trajectories on products of homogenous spaces\nby Lifa
n Guan (Gottingen) as part of Webinar on Diophantine approximation and hom
ogeneous dynamics\n\n\nAbstract\nThanks to Dani correspondence\, it is now
well-known that the set of singular vectors is closely related to the set
of points with divergent trajectories in certain homogeneous dynamical sy
stems. Since Yitwah Cheung's breakthrough work on the Hausdorff dimension
of the set of 2-dim singular vectors\, there have been lots of progress in
singular vectors and divergent trajectories in the so-called "unweighted"
cases. Otherwise\, our understanding is quite limited. In this talk\, I w
ill mainly discuss the dimension formula for the set of divergent trajecto
ries in products of "unweighted" homogeneous dynamical systems. This is a
joint work with Jinpeng An\, Antoine Marnat and Ronggang Shi.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalia Jurga (University of St Andrews)
DTSTART;VALUE=DATE-TIME:20210421T133000Z
DTEND;VALUE=DATE-TIME:20210421T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/31
DESCRIPTION:Title: Random matrix products and self-projective sets\nby Natalia Jurg
a (University of St Andrews) as part of Webinar on Diophantine approximati
on and homogeneous dynamics\n\n\nAbstract\nA finite set of matrices $A \\s
ubset SL(2\,R)$ acts on one-dimensional real projective space $RP^1$ throu
gh its linear action on $R^2$. In this talk we will be interested in the s
mallest closed subset of $RP^1$ which contains all attracting and neutral
fixed points of matrices in $A$ and which is invariant under the projectiv
e action of $A$. Recently\, Solomyak and Takahashi proved that if $A$ is u
niformly hyperbolic and satisfies a Diophantine property\, then the invari
ant set has Hausdorff dimension equal to the minimum of 1 and the critical
exponent. In this talk we will discuss an extension of their result beyon
d the uniformly hyperbolic setting. This is based on joint work with Argyr
ios Christodoulou.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pratyush Sarkar (Yale University)
DTSTART;VALUE=DATE-TIME:20210428T133000Z
DTEND;VALUE=DATE-TIME:20210428T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/32
DESCRIPTION:Title: Generalization of Selberg's 3⁄16 theorem for convex cocompact thin
subgroups of SO(n\, 1)\nby Pratyush Sarkar (Yale University) as part
of Webinar on Diophantine approximation and homogeneous dynamics\n\n\nAbst
ract\nSelberg’s 3/16 theorem for congruence covers of the modular surfac
e is a beautiful theorem which has a natural dynamical interpretation as u
niform exponential mixing. Bourgain-Gamburd-Sarnak's breakthrough works in
itiated many recent developments to generalize Selberg's theorem for infin
ite volume hyperbolic manifolds. One such result is by Oh-Winter establish
ing uniform exponential mixing for convex cocompact hyperbolic surfaces. T
hese are not only interesting in and of itself but can also be used for a
wide range of applications including uniform resonance free regions for th
e resolvent of the Laplacian\, affine sieve\, and prime geodesic theorems.
I will present a further generalization to higher dimensions and some of
these immediate consequences.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henna Koivusalo (University of Bristol)
DTSTART;VALUE=DATE-TIME:20210505T133000Z
DTEND;VALUE=DATE-TIME:20210505T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/33
DESCRIPTION:Title: Linear repetitivity in polytopal cut and project sets\nby Henna
Koivusalo (University of Bristol) as part of Webinar on Diophantine approx
imation and homogeneous dynamics\n\n\nAbstract\nCut and project sets are a
periodic point patterns obtained by projecting an irrational slice of the
integer lattice to a subspace. One way of classifying aperiodic sets is to
study the number and repetition of finite patterns. From this perspective
\, sets with patterns repeating linearly often\, called linearly repetitiv
e sets\, can be viewed as the most ordered aperiodic sets. Repetitivity of
a cut and project set depends on the slope and shape of the irrational sl
ice. The cross-section of the slice is known as the window. In earlier wor
ks\, joint with subsets of {Haynes\, Julien\, Sadun\, Walton}\, we showed
that many properties of cut and project sets with a cube window can be stu
died in the language of Diophantine approximation. For example\, linear re
petitivity holds if and only if the following two conditions are satisfied
: (i) the cut and project set has minimal number of different finite patte
rns (minimal complexity)\, and (ii) the irrational slope satisfies a badly
approximable condition. In a new joint work with Jamie Walton\, we give a
generalisation of this result to all polytopal windows satisfying a mild
geometric condition. A key step in the proof is a decomposition of the cut
and project scheme\, which allows us to make sense of condition (ii) for
general polytopal windows.\nThe talk will cover motivation and history of
studying cut and project sets\, showcase a series of results on their repe
titivity properties highlighting the number theory connections\, and finis
h with the new results which move beyond Diophantine approximation.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (University of Göttingen)
DTSTART;VALUE=DATE-TIME:20210512T133000Z
DTEND;VALUE=DATE-TIME:20210512T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/34
DESCRIPTION:Title: Density of rational points near/on compact manifolds with certain cu
rvature conditions\nby Damaris Schindler (University of Göttingen) as
part of Webinar on Diophantine approximation and homogeneous dynamics\n\n
\nAbstract\nIn this talk I will discuss joint work with Shuntaro Yamagishi
where we establish an asymptotic formula for the number of rational point
s\, with bounded denominators\, within a given distance to a compact subma
nifold $M$ of $\\mathbb{R}^n$ with a certain curvature condition. Technica
lly we build on work of Huang on the density of rational points near hyper
surfaces. One of our goals is to explore generalisations to higher codimen
sion. In particular we show that assuming certain curvature conditions in
codimension at least two\, leads to upper bounds for the number of rationa
l points on $M$ which are even stronger than what would be predicted by th
e analogue of Serre's dimension growth conjecture.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nattalie Tamam (UC San Diego)
DTSTART;VALUE=DATE-TIME:20210519T153000Z
DTEND;VALUE=DATE-TIME:20210519T173000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/35
DESCRIPTION:Title: When can we find a simple description of the divergence of trajector
ies?\nby Nattalie Tamam (UC San Diego) as part of Webinar on Diophanti
ne approximation and homogeneous dynamics\n\n\nAbstract\nIt is well known
that the only singular numbers are rational numbers. Dani's correspondence
ties this property to a simple algebraic description of divergent traject
ories in $\\mathrm{SL}_2(\\mathbb{R})$ under the action of the diagonal gr
oup. Similar principles can be utilised to define obvious divergent trajec
tories in a more general setting. We will discuss the existence of non-obv
ious divergent trajectories under the action of different diagonal subgrou
ps\, and the diophantine meaning of their existence (or lack thereof). Thi
s is a joint work with Omri Nisan Solan.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Scoones (University of York)
DTSTART;VALUE=DATE-TIME:20210526T133000Z
DTEND;VALUE=DATE-TIME:20210526T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/36
DESCRIPTION:Title: On the abc Conjecture in Algebraic Number Fields\nby Andrew Scoo
nes (University of York) as part of Webinar on Diophantine approximation a
nd homogeneous dynamics\n\n\nAbstract\nWhile the abc Conjecture remains op
en\, much work has been done on weaker versions\, and on generalising the
conjecture to number fields. Stewart and Yu were able to give an exponenti
al bound for $\\max\\{a\, b\, c\\}$ in terms of the radical over the integ
ers\, while Györy was able to give an exponential bound for the projectiv
e height $H(a\, b\, c)$ in terms of the radical for algebraic integers. We
generalise Stewart and Yu's method to give an improvement on Györy's bou
nd for algebraic integers.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Poels and Damien Roy (University of Ottawa)
DTSTART;VALUE=DATE-TIME:20211008T140000Z
DTEND;VALUE=DATE-TIME:20211008T160000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/37
DESCRIPTION:Title: Simultaneous rational approximation to successive powers of a real n
umber\nby Anthony Poels and Damien Roy (University of Ottawa) as part
of Webinar on Diophantine approximation and homogeneous dynamics\n\n\nAbst
ract\nLet $\\xi\\in\\mathbb{R}\\setminus\\bar{\\mathbb{Q}}$ be a real tran
scendental number and let $n$ be a positive integer. By pioneer work of D
avenport \nand Schmidt from 1969\, we know that the exponent $\\tau_{n+1}(
\\xi)$ \nof best approximation to $\\xi$ by algebraic integers of degree\n
at most $n+1$ is at least equal to $1+1/\\lambda_n(\\xi)$\, where \n$\\lam
bda_n(\\xi)$ stands for the uniform exponent of rational\napproximation to
the successive powers $1\,\\xi\,\\dots\,\\xi^n$ \nof $\\xi$. So any uppe
r bound on $\\lambda_n(\\xi)$ which holds \nfor any $\\xi\\in\\mathbb{R}\\
setminus\\bar{\\mathbb{Q}}$ provides a lower bound on \n$\\tau_{n+1}(\\xi)
$ which is also independent of $\\xi$. In this talk\, \nwe present new to
ols which yield\, for each integer $n\\ge 4$\, a\nsignificantly improved u
pper bound on $\\lambda_n(\\xi)$ and thus\na refined lower bound on $\\tau
_{n+1}(\\xi)$. The new lower bound is\n$n/2+a\\sqrt{n}+4/3$ with $a=(1-\\
log(2))/2\\simeq 0.153$\, instead of the\ncurrent $n/2+\\mathcal{O}(1)$.\n
\nAs usual\, the starting point is the sequence of so-called \nminimal poi
nts $\\mathbf{x}_1\,\\mathbf{x}_2\,\\mathbf{x}_3\,\\ldots$ in $\\mathbb{Z}
^{n+1}$ defined initially \nby Davenport and Schmidt. Our strategy consist
s in estimating from above \nthe height of the subspaces of $\\mathbb{R}^{
n-\\ell+1}$ generated by $n-\\ell+1$ \nconsecutive coordinates from each p
oint among $\\mathbf{x}_i\,\\mathbf{x}_{i+1}\,\\dots\,\\mathbf{x}_q$\nfor
given $i\\le q$. To this end\, we first need a lower bound for the \ndime
nsion of such spaces.\n\nIn the first part of the talk\, we present the re
quired background with some historical perspective and our key algebraic r
esult concerning the dimension of the above mentioned spaces. In the secon
d part\, we look at their heights from different \nperspectives and outlin
e the general strategy of the proof.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Gorodnik (University of Zurich)
DTSTART;VALUE=DATE-TIME:20211022T140000Z
DTEND;VALUE=DATE-TIME:20211022T160000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/38
DESCRIPTION:Title: Products of linear forms and distribution of translated measures
\nby Alex Gorodnik (University of Zurich) as part of Webinar on Diophantin
e approximation and homogeneous dynamics\n\n\nAbstract\nWe explore the beh
avior of the counting function which represents the number of solutions\no
f a multiplicative Diophantine problem. The argument is based on analysis
of measures translated under a group action on homogeneous spaces. Ultimat
ely we explain how estimates on correlations of translated measures lead t
o a quantitative asymptotic formula for the counting function. This is a j
oint work with Björklund and Fregolli.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jimmy Tseng (University of Exeter)
DTSTART;VALUE=DATE-TIME:20211105T143000Z
DTEND;VALUE=DATE-TIME:20211105T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/39
DESCRIPTION:Title: Shrinking target and horocycle equidistribution\nby Jimmy Tseng
(University of Exeter) as part of Webinar on Diophantine approximation and
homogeneous dynamics\n\n\nAbstract\nConsider a shrinking neighborhood of
a cusp of the unit tangent bundle of a noncompact hyperbolic surface of fi
nite area. We discuss how a closed horocycle whose length goes to infin
ity can become equidistributed on this shrinking neighborhood\, giving a s
harp criterion in a natural case. This setup is closely related to number
theory\, and\, as an example\, our method yields a number-theoretic identi
ty.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yitwah Cheung (Tsinghua University)
DTSTART;VALUE=DATE-TIME:20211119T143000Z
DTEND;VALUE=DATE-TIME:20211119T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/40
DESCRIPTION:Title: Invariants of Diophantine Approximation\nby Yitwah Cheung (Tsing
hua University) as part of Webinar on Diophantine approximation and homoge
neous dynamics\n\n\nAbstract\nThere is a natural generalization of the con
cept of convergents of the continued fraction to higher dimensions that do
es not involve any specific choice of norm. In this talk\, I will motivat
e this concept from several different angles\, within the framework of sta
ircases\, which is a rectilinear version of Kleinian sails. I will descri
be some results about dual convergents and illustrate the method of our ap
proach towards constructing slowly unbounded A-orbits by sketching the pro
of of dichotomy of Hausdorff dimension phenomenon obtained in joint work w
ith P. Hubert and H. Masur.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiyoung Han (Tata Institute)
DTSTART;VALUE=DATE-TIME:20211203T143000Z
DTEND;VALUE=DATE-TIME:20211203T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/41
DESCRIPTION:Title: Higher moment formulas for Siegel transforms and applications to lim
it distributions of functions of counting lattice points\nby Jiyoung H
an (Tata Institute) as part of Webinar on Diophantine approximation and ho
mogeneous dynamics\n\n\nAbstract\nThe Siegel transform is one of the main
tools when we consider\nproblems related to counting lattice points using
homogeneous dynamics. It\nis revealed by many mathematicians that Siegel
’s integral formula and\nRogers’ second moment formula are very useful
to solve various\nquantitative and effective variants of classical proble
ms in the geometry\nof numbers\, such as the Gauss circle problem (general
ized to convex sets)\nand Oppenheim conjecture. Furthermore\, Rogers’ hi
gher moment formulas\,\ntogether with the method of moments\, give us info
rmation about limit\ndistributions related to these problems.\nIn this tal
k\, we revisit Rogers’ higher moment formulas with a new\napproach\, and
introduce higher moment formulas for Siegel transforms on\nthe space of a
ffine unimodular lattices and the space of unimodular\nlattices with a con
gruence condition. Using these formulas\, we obtain the\nresults of limit
distributions\, which are generalizations of the work of\nRogers (1956)\,
Södergren (2011)\, and Strömbergsson and Södergren (2019).\nThis is joi
nt work with Mahbub Alam and Anish Ghosh.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Demi Allen (University of Warwick)
DTSTART;VALUE=DATE-TIME:20211217T143000Z
DTEND;VALUE=DATE-TIME:20211217T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T081420Z
UID:DAHD-webinar/42
DESCRIPTION:Title: An inhomogeneous Khintchine-Groshev Theorem without monotonicity
\nby Demi Allen (University of Warwick) as part of Webinar on Diophantine
approximation and homogeneous dynamics\n\n\nAbstract\nThe classical (inhom
ogeneous) Khintchine-Groshev Theorem tells us that for a monotonic approxi
mating function $\\psi: \\mathbb{N} \\to [0\,\\infty)$ the Lebesgue measur
e of the set of (inhomogeneously) $\\psi$-well-approximable points in $\\R
^{nm}$ is zero or full depending on\, respectively\, the convergence or di
vergence of $\\sum_{q=1}^{\\infty}{q^{n-1}\\psi(q)^m}$. In the homogeneous
case\, it is now known that the monotonicity condition on $\\psi$ can be
removed whenever $nm>1$\, and cannot be removed when $nm=1$. In this talk
I will discuss recent work with Felipe A. Ramírez (Wesleyan\, US) in whic
h we show that the inhomogeneous Khintchine-Groshev Theorem is true withou
t the monotonicity assumption on $\\psi$ whenever $nm>2$. This result brin
gs the inhomogeneous theory almost in line with the completed homogeneous
theory. I will survey previous results towards removing monotonicity from
the homogeneous and inhomogeneous Khintchine-Groshev Theorem before discus
sing the main ideas behind the proof our recent result.\n
LOCATION:https://researchseminars.org/talk/DAHD-webinar/42/
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