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Hausdorff Discretizations of Algebraic Sets and Diophantine Sets [chapter]

Mohamed Tajine, Christian Ronse
2000 Lecture Notes in Computer Science  
Actually we give some decidable and undecidable properties concerning Hausdorff discretizations of algebraic sets and we prove that some Hausdorff discretizations of algebraic sets are diophantine sets  ...  We study the properties of Hausdorff discretizations of algebraic sets.  ...  Hausdorff discretizations of algebraic sets and we prove that some Hausdorff discretizations of algebraic sets are diophantine sets.  ... 
doi:10.1007/3-540-44438-6_9 fatcat:5ub74splyvdb5kirz67b3om4w4

Page 6773 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
This paper is concerned with analogues of the classical questions in Diophantine approximation (Hausdorff dimension and Khinchin type theorems) in the field of Laurent series.  ...  (F-IHES; Bures-sur- Yvette) ; Pérez-Marco, R. (1- UCLA; Los Angeles, CA) Diophantine conditions in small divisors and transcendental number theory. (English summary) Discrete Contin. Dyn.  ... 

Metric Diophantine approximation on homogeneous varieties

Anish Ghosh, Alexander Gorodnik, Amos Nevo
2014 Compositio Mathematica  
AbstractWe develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems.  ...  In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using$\def \xmlpi #1{}\def  ...  In fact, we consider, more generally, • the problem of simultaneous Diophantine approximation with respect to several completions, • the problem of Diophantine approximation over number fields by S-algebraic  ... 
doi:10.1112/s0010437x13007859 fatcat:3lxi6ehbn5dqnpsosghmdcuryy

Ergodic Theory on Homogeneous Spaces and Metric Number Theory [chapter]

Dmitry Kleinbock
2009 Encyclopedia of Complexity and Systems Science  
Article outline This article gives a brief overview of recent developments in metric number theory, in particular, Diophantine approximation on manifolds, obtained by applying ideas and methods coming  ...  When G is a Lie group and Γ is a discrete subgroup, the space G/Γ is a smooth manifold and locally looks like G itself. lattice; unimodular lattice A lattice in a Lie group is a discrete subgroup of finite  ...  Hausdorff dimension A nonnegative number attached to a metric space and extending the notion of topological dimension of "sufficiently regular" sets, such as smooth submanifolds of real Euclidean spaces  ... 
doi:10.1007/978-0-387-30440-3_180 fatcat:a6y5ud7wm5fonkuhkjdchi2opa

Page 851 of Mathematical Reviews Vol. , Issue 81C [page]

1981 Mathematical Reviews  
V. 81c: 10070 +1006 Hausdorff dimension in Diophantine approximations of p-adic om numbers. (Russian) Ukrain. Mat. Z. 32 (1980), no. 1, 118-124, 144.  ...  850 851 2 and Melniéuk, Ju. V. 81c: 10069 1eorem | —-«dDiophantine approximations on the circle and Hausdorff iber of | dimension. (Russian) rem 3; | Mat.  ... 

Page 97 of Mathematical Reviews Vol. , Issue 2002A [page]

2002 Mathematical Reviews  
M. (4-YORK; York Simultaneous Diophantine approximation on the circle and Hausdorff dimension. Math. Proc. Cambridge Philos.  ...  (NL-LEID-MI; Leiden Algebraic aspects of discrete tomography. (English summary) J. Reine Angew. Math. 534 (2001), 119-128.  ... 

MTK volume 32 issue 2 Cover and Front matter

1985 Mathematika  
Classes of sets with large intersection 191 K. J. FALCONER On the Hausdorff dimensions of distance sets 206 P. MATTILA On the Hausdorff dimensions and capacities of intersections 213 G.  ...  JAYNE and C. A. ROGERS Piece-wise closed functions and almost discretely ^-decomposable families 229 R.  ...  SMITH MATHEMATIKA is published by the Department of Mathematics, University College London. It contains original notes and memoirs on mathematics and its applications.  ... 
doi:10.1112/s0025579300010974 fatcat:uupcz3lt5zg23ez7m5yjvbztta

Page 7882 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
The set N with the discrete topology is embedded in a compact space /N, the Stone- Cech compactification of N.  ...  In the present note the author shows that the set of those x’s for which (t,()),>o is not uniformly distributed has Hausdorff dimension | too.  ... 

Diophantine properties of nilpotent Lie groups

Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, Nicolas de Saxcé
2015 Compositio Mathematica  
We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra.  ...  We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.  ...  of ISEF and Advanced Research Grant 228304 from the ERC.  ... 
doi:10.1112/s0010437x14007854 fatcat:t3b5qhdhtzfbznds6pcu2saclq

SUBGROUPS OF FRACTIONAL DIMENSION IN NILPOTENT OR SOLVABLE LIE GROUPS

Nicolas de Saxcé
2013 Mathematika  
In algebraic groups defined over a finite extension of the rationals, using diophantine properties of algebraic numbers, we are also able to construct dense subgroups of arbitrary dimension, but the general  ...  We construct dense Borel measurable subgroups of Lie groups of intermediate Hausdorff dimension.  ...  Acknowledgements: This article is part of the research I did for my doctoral thesis, under the supervision of Emmanuel Breuillard.  ... 
doi:10.1112/s0025579313000077 fatcat:mps622zhejaqbd5ic5zfmdqh3y

Metrical Diophantine approximation for quaternions

MAURICE DODSON, BRENT EVERITT
2014 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of  ...  general 'lim sup' sets.  ...  and a family R = {R j : j ∈ J} of resonant sets, where J is a countable discrete index set (see [51] ).  ... 
doi:10.1017/s0305004114000462 fatcat:qrta3oacmjbqnjeuue4ehoynxy

Page 91 of Mathematical Reviews Vol. , Issue 94a [page]

1994 Mathematical Reviews  
M. (4-YORK; York) Hausdorff dimension, lower order and Khintchine’s theorem in metric Diophantine approximation. J. Reine Angew. Math. 432 (1992), 69-76. A little-known result of A. Groshev [Dokl.  ...  A subset A of N is intersective if for each subset S of N with b(S) > 0, the set AN(S—S) is nonempty. Let y be a polynomial with integer coefficients and let P, = {y(p): pa rational prime}.  ... 

How smooth is quantum complexity? [article]

Vir B. Bulchandani, S. L. Sondhi
2021 arXiv   pre-print
We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness.  ...  The "quantum complexity" of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates.  ...  Moessner and especially A. Brown for helpful discussions. We are grateful to A. Harrow for comments on the manuscript.  ... 
arXiv:2106.08324v2 fatcat:jvotpx6fkrapvplsu6lwsdjydm

Diophantine approximation exponents on homogeneous varieties [article]

Anish Ghosh, Alexander Gorodnik, Amos Nevo
2014 arXiv   pre-print
We concentrate specifically on the problem of estimating exponents of Diophantine approximation by arithmetic lattices acting on algebraic varieties.  ...  Recent years have seen very important developments at the interface of Diophantine approximation and homogeneous dynamics.  ...  The first author acknowledges support of the Royal Society. The second author acknowledges support of EPSRC, ERC and RCUK. The third author acknowledges support of ISF.  ... 
arXiv:1401.6581v1 fatcat:gzjtwufewff37iahec2tv622le

Page 4 of Mathematical Reviews Vol. , Issue 89I [page]

1989 Mathematical Reviews  
Then the limit of this sequence lies in A;. The final section of the paper gives a proof of the result that for any positive d, the set C — Az has d-dimensional Hausdorff measure zero.  ...  Math. 78 (1984), no. 3, 445-490; MR 86e:11053] to obtain a new and sharp formulation of Roth’s theorem on the approxi- mation of algebraic numbers by algebraic numbers.  ... 
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