Badly Approximable Unimodular Functions in Weighted L p Spaces

2005 ◽  
Vol 4 (2) ◽  
pp. 315-326 ◽  
Author(s):  
Alexei B. Aleksandrov
Keyword(s):  
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 Very badly approximable matrix functions

2005 ◽  
Vol 11 (1) ◽  
pp. 127-154 ◽  
Author(s):  
V. V. Peller ◽  
S. R. Treil
Keyword(s):  
Matrix Functions ◽  
Badly Approximable

scholarly journals Decaying and non-decaying badly approximable numbers

2017 ◽  
Vol 177 (2) ◽  
pp. 143-152 ◽  
Author(s):  
Ryan Broderick ◽  
Lior Fishman ◽  
David Simmons
Keyword(s):  
Badly Approximable

On Simultaneously Badly Approximable Numbers

2002 ◽  
Vol 66 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Andrew Pollington ◽  
Sanju Velani
Keyword(s):  
Badly Approximable

On the multiples of a badly approximable vector

2015 ◽  
Vol 168 (1) ◽  
pp. 71-81 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Badly Approximable

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 ON GENERALIZED THUE–MORSE FUNCTIONS AND THEIR VALUES

2019 ◽  
Vol 108 (2) ◽  
pp. 177-201
Author(s):  
DZMITRY BADZIAHIN ◽  
EVGENIY ZORIN
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Laurent Series ◽  
Regular Structure ◽  
Continued Fraction Expansion ◽  
Infinite Products ◽  
Morse Functions ◽  
Morse Number ◽  
Badly Approximable

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products $f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$, $d\in \mathbb{N}$, $d\geq 2$, which generalize the generating function $f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers $f_{d}(a)\in \mathbb{R}$, $a,d\in \mathbb{N}$, $a,d\geq 2$, are badly approximable.

Sign in / Sign up

Export Citation Format

Share Document