scholarly journals The Hausdorff Dimension of Sets Arising from Diophantine Approximation with a General Error Function

1998 ◽  
Vol 71 (2) ◽  
pp. 166-171 ◽  
Author(s):  
Bryan P. Rynne
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Error Function ◽  
General Error

scholarly journals A note on the theorem of Jarník-Besicovitch

1997 ◽  
Vol 39 (2) ◽  
pp. 233-236 ◽  
Author(s):  
H. Dickinson
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Error Function ◽  
Positive Function ◽  
Real Numbers ◽  
Linear Forms ◽  
Error Functions ◽  
General Error

This note draws together and extends two recent results on Diophantine approximation and Hausdorff dimension. The first, by Hinokuma and Shiga [12], considers the oscillating error function | sinq|/qτ rather than the strictly decreasing function qτ of Jarnik's theorem. The second is Rynne's extension [17] to systems of linear forms of Borosh and Fraenkel's paper [3] on restricted Diophantine approximation with real numbers. Rynne's result will be extended to a class of general error functions and applied to obtain a more general form of [12] in which the error function is any positive function.

scholarly journals Gaps problems and frequencies of patches in cut and project sets

2016 ◽  
Vol 161 (1) ◽  
pp. 65-85 ◽  
Author(s):  
ALAN HAYNES ◽  
HENNA KOIVUSALO ◽  
JAMES WALTON ◽  
LORENZO SADUN
Keyword(s):  
Hausdorff Dimension ◽  
Frequency Spectrum ◽  
Diophantine Approximation

AbstractWe establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

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.

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 Hausdorff dimension and p-adic diophantine approximation

1999 ◽  
Vol 10 (3) ◽  
pp. 337-347 ◽  
Author(s):  
H. Dickinson ◽  
M.M. Dodson ◽  
J. Yuan
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation
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