Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

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.

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A NOTE ON WEIGHTED BADLY APPROXIMABLE LINEAR FORMS

Glasgow Mathematical Journal ◽

10.1017/s0017089516000203 ◽

2016 ◽

Vol 59 (2) ◽

pp. 349-357 ◽

Cited By ~ 2

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.

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Gaps problems and frequencies of patches in cut and project sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116000128 ◽

2016 ◽

Vol 161 (1) ◽

pp. 65-85 ◽

Cited By ~ 6

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.

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Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation.

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1992.432.69 ◽

1992 ◽

Vol 1992 (432) ◽

pp. 69-76 ◽

Cited By ~ 3

Keyword(s):

Hausdorff Dimension ◽

Diophantine Approximation ◽

Lower Order ◽

Metric Diophantine Approximation

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Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091514000169 ◽

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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