scholarly journals FULLY INHOMOGENEOUS MULTIPLICATIVE DIOPHANTINE APPROXIMATION OF BADLY APPROXIMABLE NUMBERS

Mathematika ◽  
2021 ◽  
Vol 67 (3) ◽  
pp. 639-646
Author(s):  
Sam Chow ◽  
Agamemnon Zafeiropoulos
Keyword(s):  
Diophantine Approximation ◽  
Badly Approximable

scholarly journals A NOTE ON WEIGHTED BADLY APPROXIMABLE LINEAR FORMS

2016 ◽  
Vol 59 (2) ◽  
pp. 349-357 ◽  
Author(s):  
STEPHEN HARRAP ◽  
NIKOLAY MOSHCHEVITIN
Keyword(s):  
Diophantine Approximation ◽  
Affirmative Answer ◽  
Famous Conjecture ◽  
Linear Forms ◽  
Badly Approximable

AbstractWe prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

scholarly journals Badly approximable points on self-affine sponges and the lower Assouad dimension

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Assouad Dimension ◽  
Badly Approximable Points ◽  
Bounded Below ◽  
Badly Approximable ◽  
Sierpinski Sponges

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

scholarly journals Diophantine approximation and badly approximable sets

2006 ◽  
Vol 203 (1) ◽  
pp. 132-169 ◽  
Author(s):  
Simon Kristensen ◽  
Rebecca Thorn ◽  
Sanju Velani
Keyword(s):  
Diophantine Approximation ◽  
Badly Approximable

scholarly journals A Hausdorff measure version of the Jarník–Schmidt theorem in Diophantine approximation

2017 ◽  
Vol 164 (3) ◽  
pp. 413-459 ◽  
Author(s):  
DAVID SIMMONS
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Higher Dimensions ◽  
Dimension Function ◽  
Hausdorff Dimensions ◽  
Asymptotic Bounds ◽  
Badly Approximable

AbstractWe solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalising to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorfff-measure of the set of matrices which are not ψ-approximable, given a dimension functionfand a function ψ : (0, ∞) → (0, ∞). This complements earlier work by Dickinson and Velani ('97) who found the Hausdorfff-measure of the set of matrices which are ψ-approximable.

scholarly journals Badly approximable vectors, curves and number fields

2015 ◽  
Vol 36 (6) ◽  
pp. 1851-1864 ◽  
Author(s):  
MANFRED EINSIEDLER ◽  
ANISH GHOSH ◽  
BEVERLY LYTLE
Keyword(s):  
Number Field ◽  
Diophantine Approximation ◽  
Number Fields ◽  
Winning Set ◽  
Field Form ◽  
Set Of Points ◽  
Badly Approximable

We show that the set of points on $C^{1}$ curves which are badly approximable by rationals in a number field form a winning set in the sense of Schmidt. As a consequence, we obtain a number field version of Schmidt’s conjecture in Diophantine approximation.

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