badly approximable points
Recently Published Documents


TOTAL DOCUMENTS

7
(FIVE YEARS 1)

H-INDEX

3
(FIVE YEARS 0)

2018 ◽  
Vol 324 ◽  
pp. 148-202 ◽  
Author(s):  
Jinpeng An ◽  
Victor Beresnevich ◽  
Sanju Velani

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI

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.


2015 ◽  
Vol 202 (3) ◽  
pp. 1199-1240 ◽  
Author(s):  
Victor Beresnevich

2014 ◽  
Vol 359 (3-4) ◽  
pp. 969-1023 ◽  
Author(s):  
Dzmitry Badziahin ◽  
Sanju Velani

Sign in / Sign up

Export Citation Format

Share Document