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Hausdorff Discretizations of Algebraic Sets and Diophantine Sets
[chapter]

2000
*
Lecture Notes in Computer Science
*

Actually we give some decidable

doi:10.1007/3-540-44438-6_9
fatcat:5ub74splyvdb5kirz67b3om4w4
*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*. ...##
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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. ...##
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Metric Diophantine approximation on homogeneous varieties

2014
*
Compositio Mathematica
*

AbstractWe develop the metric theory

doi:10.1112/s0010437x13007859
fatcat:3lxi6ehbn5dqnpsosghmdcuryy
*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*...##
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Ergodic Theory on Homogeneous Spaces and Metric Number Theory
[chapter]

2009
*
Encyclopedia of Complexity and Systems Science
*

Article outline This article gives a brief overview

doi:10.1007/978-0-387-30440-3_180
fatcat:a6y5ud7wm5fonkuhkjdchi2opa
*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 ...##
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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. ...##
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MTK volume 32 issue 2 Cover and Front matter

1985
*
Mathematika
*

Classes

doi:10.1112/s0025579300010974
fatcat:uupcz3lt5zg23ez7m5yjvbztta
*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. ...##
###
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

2015
*
Compositio Mathematica
*

We give a characterization

doi:10.1112/s0010437x14007854
fatcat:t3b5qhdhtzfbznds6pcu2saclq
*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. ...##
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SUBGROUPS OF FRACTIONAL DIMENSION IN NILPOTENT OR SOLVABLE LIE GROUPS

2013
*
Mathematika
*

In

doi:10.1112/s0025579313000077
fatcat:mps622zhejaqbd5ic5zfmdqh3y
*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. ...##
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Metrical Diophantine approximation for quaternions

2014
*
Mathematical proceedings of the Cambridge Philosophical Society (Print)
*

Analogues

doi:10.1017/s0305004114000462
fatcat:qrta3oacmjbqnjeuue4ehoynxy
*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] ). ...##
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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}. ...##
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How smooth is quantum complexity?
[article]

2021
*
arXiv
*
pre-print

We use ideas from

arXiv:2106.08324v2
fatcat:jvotpx6fkrapvplsu6lwsdjydm
*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. ...##
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Diophantine approximation exponents on homogeneous varieties
[article]

2014
*
arXiv
*
pre-print

We concentrate specifically on the problem

arXiv:1401.6581v1
fatcat:gzjtwufewff37iahec2tv622le
*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. ...##
###
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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