scholarly journals A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

2007 ◽  
Vol 49 (2) ◽  
pp. 367-375 ◽  
Author(s):  
DZMITRY BADZIAHIN ◽  
JASON LEVESLEY
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
General Class ◽  
Simultaneous Approximation ◽  
Convergence Result ◽  
Planar Curves ◽  
Planar Curve ◽  
Induced Measure

AbstractLet $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

scholarly journals Hausdorff theory of dual approximation on planar curves

2018 ◽  
Vol 2018 (740) ◽  
pp. 63-76 ◽  
Author(s):  
Jing-Jing Huang
Keyword(s):  
Hausdorff Measure ◽  
General Class ◽  
Measure Theory ◽  
Complete Theory ◽  
Planar Curves ◽  
The Subject ◽  
Hausdorff Theory ◽  
Challenging Question

AbstractTen years ago, Beresnevich–Dickinson–Velani [Mem. Amer. Math. Soc. 179 (2006), no. 846] initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in{\mathbb{R}^{2}}, which represents the first complete theory of its kind for a general class of manifolds.

scholarly journals On weighted inhomogeneous Diophantine approximation on planar curves

2012 ◽  
Vol 154 (2) ◽  
pp. 225-241 ◽  
Author(s):  
MUMTAZ HUSSAIN ◽  
TATIANA YUSUPOVA
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Planar Curves ◽  
Planar Curve ◽  
Invent Math

AbstractThis paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdorff dimension results for the sets of simultaneously well-approximable points on planar curves, established in Badziahin and Levesley (Glasg. Math. J., 49(2):367–375, 2007), Beresnevich et al. (Ann. of Math. (2), 166(2):367–426, 2007), Beresnevich and Velani (Math. Ann., 337(4):769–796, 2007) and Vaughan and Velani (Invent. Math., 166(1):103–124, 2006).

scholarly journals Affine Subspaces of Curvature Functions from Closed Planar Curves

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Leonardo Alese
Keyword(s):  
Arc Length ◽  
Planar Curves ◽  
Affine Subspaces ◽  
Planar Curve ◽  
Real Functions ◽  
Discrete Counterpart ◽  
Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

scholarly journals A mass transference principle for systems of linear forms and its applications

2018 ◽  
Vol 154 (5) ◽  
pp. 1014-1047 ◽  
Author(s):  
Demi Allen ◽  
Victor Beresnevich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Linear Forms ◽  
Transference Principle ◽  
Integer Points ◽  
Additional Constraints

In this paper we establish a general form of the mass transference principle for systems of linear forms conjectured in 2009. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine–Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine–Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira.

The Generalized Baker–Schmidt Problem on Hypersurfaces

2019 ◽  
Author(s):  
Mumtaz Hussain ◽  
Johannes Schleischitz ◽  
David Simmons
Keyword(s):  
Hausdorff Measure ◽  
General Setting ◽  
Dimensional Hausdorff Measure ◽  
Planar Curves ◽  
Open Question

Abstract The generalized Baker–Schmidt problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $\psi $-approximable points on a nondegenerate manifold. There are two variants of this problem concerning simultaneous and dual approximation. Beresnevich–Dickinson–Velani (in 2006, for the homogeneous setting) and Badziahin–Beresnevich–Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The corresponding convergence counterpart represents a major challenging open question and the progress thus far has only been attained over planar curves. In this paper, we settle this problem for hypersurfaces in a more general setting, that is, for inhomogeneous approximations and with a non-monotonic multivariable approximating function.

scholarly journals METRICAL RESULTS ON SYSTEMS OF SMALL LINEAR FORMS

2013 ◽  
Vol 09 (03) ◽  
pp. 769-782 ◽  
Author(s):  
M. HUSSAIN ◽  
S. KRISTENSEN
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Linear Forms ◽  
Approximation Function

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine–Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximation function.

The simultaneous diophantine approximation of certain kth roots

1970 ◽  
Vol 67 (1) ◽  
pp. 75-86 ◽  
Author(s):  
Charles F. Osgood
Keyword(s):  
Diophantine Approximation ◽  
Simultaneous Approximation ◽  
Rational Numbers ◽  
Simultaneous Diophantine Approximation ◽  
Explicit Bounds ◽  
Positive Integers

In a recent paper in this journal(1) Baker obtained a result giving effectively computable bounds on the simultaneous approximation of certain numbers of the form (ab-1)nj/n by rational numbers of the formpιq-1 where a, b, n, the nι and q are positive integers while the pι are non-negative integers. In this paper we shall obtain explicit bounds on the simultaneous approximation of certain numbers of the form rβ1, …, rβn where β and each rι, 1 ≤ j ≤ n, are positive rational numbers.

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