scholarly journals Estimating the critical determinants of a class of three-dimensional star bodies

2017 ◽  
Vol 25 (2) ◽  
pp. 149-157
Author(s):  
Werner Georg Nowak
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Positive Constant ◽  
Three Dimensional ◽  
Double Cone ◽  
Critical Determinant ◽  
Simultaneous Diophantine Approximation ◽  
Hurwitz’S Theorem ◽  
Star Bodies

Abstract In the problem of (simultaneous) Diophantine approximation in ℝ3 (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body K2 : (y2 + z2)(x2 + y2 + z2) ≤ 1 play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant ∆ (Kc) of more general star bodies Kc : (y2 + z2)c/2(x2 + y2 + z2) ≤ 1 ; where c is any positive constant. These are obtained by inscribing into Kc either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of c.

scholarly journals On the critical determinants of certain star bodies

2017 ◽  
Vol 25 (1) ◽  
pp. 5-11 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Calculus Of Variations ◽  
Diophantine Approximation ◽  
Star Body ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Critical Determinant ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies ◽  
Classic Paper

Abstract In a classic paper [14], W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body|x1|(|x1|3 + |x2|3 + |x3|3 ≤ 1.In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body|x1|(|x1|3 + |x22 + x32)3/2≤ 1.

scholarly journals KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES

2006 ◽  
Vol 02 (03) ◽  
pp. 431-453
Author(s):  
M. M. DODSON ◽  
S. KRISTENSEN
Keyword(s):  
Distance Function ◽  
Diophantine Approximation ◽  
Direct Proof ◽  
Distance Functions ◽  
Transference Principle ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

DIOPHANTINE APPROXIMATION IN IMAGINARY QUADRATIC FIELDS

2010 ◽  
Vol 06 (04) ◽  
pp. 731-766 ◽  
Author(s):  
L. YA. VULAKH
Keyword(s):  
Half Space ◽  
Hyperbolic Space ◽  
Lower Bounds ◽  
Diophantine Approximation ◽  
Three Dimensional ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Space Model ◽  
Limit Points ◽  
Group B

Let H3 be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups Γ of an extended Bianchi group Bd, the extremality of the axis of hyperbolic F ∈ Γ in H3 with respect to Γ implies its extremality with respect to Bd. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of Bd for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields with discriminants -120 and -132 are given. The second minima are also indicated for these fields.

Complete caps in AG(3, q) from elliptic curves

2014 ◽  
Vol 13 (08) ◽  
pp. 1450050 ◽  
Author(s):  
Irene Platoni
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Complete Caps ◽  
Odd Order ◽  
Galois Space

In a three-dimensional Galois space of odd order q, the known infinite families of complete caps have size far from the theoretical lower bounds. In this paper, we investigate some caps defined from elliptic curves. In particular, we show that for each q between 100 and 350 they can be extended to complete caps, which turn out to be the smallest complete caps known in the literature.

scholarly journals Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

2014 ◽  
Vol 58 (1) ◽  
pp. 1-26
Author(s):  
Faustin Adiceam
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Rational Approximations ◽  
Vertical Shift ◽  
Polynomial Curves ◽  
Integer Coefficients ◽  
Simultaneous Diophantine Approximation ◽  
First Results

AbstractThe Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well-approximable points lying on a curve that is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.

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