Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions

By Ññøöö Øøøóöý, Óó Óôôôòøøòò, Ôôöóüüññøøóòº Óö Ü Ý ¾ Ê Ò Û Ððø Ü ¡ Ý. 2002

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Notation. The main objects of this paper are n-tuples y = (y1, . . . , yn) of real numbers viewed as linear forms, i.e. as row vectors. In what follows, y will always mean a row vector, and we will be interested in values of a linear form given by y at integer points q = (q1, . . . , qn)T , the latter being a column vector. Thus yq will stand for y1q1 + · · ·+ ynqn. Hopefully it will cause no confusion. We will study differentiable maps f = (f1, . . . , fn) from open subsets U of R to R; again, f will be interpreted as a row vector, so that f(x)q stands for q1f1(x) + · · · + qnfn(x). In contrast, the elements of the “parameter set” U will be denoted by x = (x1, . . . , xd) without boldfacing, since the linear structure of the parameter space is not significant. For f as above we will denote by ∂if : U 7→ R, i = 1, . . . , d, its partial derivative (also a row vector) with respect to xi. If F is a scalar function on U , we will denote by∇F the column vector consisting of partial derivatives of F . With some abuse of notation, the same way we will treat vector functions f : namely, ∇f will stand for the matrix function U 7→ Md×n(R) with rows given by partial derivatives ∂if . We will also need higher order differentiation: for a multiindex β = (i1, . . . , id), ij ∈ Z+, we let |β| = ∑d j=1 ij and ∂β = ∂ i1 1 ◦ · · · ◦ ∂ id d . Unless otherwise indicated, the norm ‖x‖ of a vector x ∈ R (either row or column vector) will stand for ‖x‖ = max1≤i≤k |xi|. In some cases however we will work with the Euclidean norm ‖x‖ = ‖x‖e = √∑k i=1 x 2 i , keeping the same notation. This distinction will be clearly emphasized to avoid confusion. We will denote by R1 the set of unit vectors in R (with respect to the Euclidean norm). We will use the notation |〈x〉| for the distance between x ∈ R and the closest integer, |〈x〉| def = mink∈Z |x− k|. (It is quite customary to use ‖x‖ instead, but we are not going to do this in order to save the latter notation for norms in vector spaces.) If B ⊂ R, we let |B| stand for the Lebesgue measure of B.

Subfields: Geometry and Topology, Mathematical Physics

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Citations in the corpus

Rational points near manifolds and metric Diophantine approximation (2009, background): “1 has been fully settled [3], [6], [19] and very deep answers regarding the dual version of Problem 2.”
DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS (2006, background): “We remark that the constant C1 from Theorem 3.2, and hence C2 from Theorem 3.5 and"0 from Theorem 1.5, can be explicitly extimated in terms of the input data of those theorems, see [KM2, BKM , KT, Kl2].”
Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory (2002, methodology): “14, proved in [21] by a modification of the method described above:”
Real Interference Alignment: Exploiting the Potential of Single Antenna Systems (2009, background): “Theorem 5 ( [24] and [25]): and g 1 , g 2 , . . . , g m be functions from R n to R with the following conditions: 1) g i for i ∈ { 1 , 2 , . . . , . . . , , m is analytic, 2) 1 , g 1 , g 2 g m are linearly independent over R .”
Badly approximable vectors on fractals (2005, background): “Here is one of them, a modied version of [ BKM , Lemma 3.3]: Lemma 2.3.”
Interference Alignment: A New Look at Signal Dimensions in a Communication Network (2011, methodology): “The separability of rational dimensions is a consequence of the (extensions of) Khintchine–Groshev theorem [10, 12] in Diophantine approximation theory.”
Badly approximable points on manifolds (2013, background): “4 from [11] that refines the results of [33].”