Diophantine approximation on Veech surfaces

Pascal Hubert; Thomas A. Schmidt

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Displaying similar documents to “Diophantine approximation on Veech surfaces”

The best Diophantine approximation functions by continued fractions

Jing Cheng Tong (1996)

Mathematica Bohemica

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Let $ξ = [a_{0}; a_{1}, a_{2}, \dots, a_{i}, \dots]$ be an irrational number in simple continued fraction expansion, $p_{i} / q_{i} = [a_{0}; a_{1}, a_{2}, \dots, a_{i}]$ , $M_{i} = q_{i}^{2} | ξ - p_{i} / q_{i} |$ . In this note we find a function $G (R, r)$ such that $\begin{matrix} M_{n + 1} < R and M_{n - 1} < r imply M_{n} > G (R, r), & M_{n + 1} > R and M_{n - 1} > r imply M_{n} < G (R, r) . \end{matrix}$ Together with a result the author obtained, this shows that to find two best approximation functions $\tilde{H} (R, r)$ and $\tilde{L} (R, r)$ is a well-posed problem. This problem has not been solved yet.

Explicit lower bounds for ${| | (3 / 2)}^{k} | |$

Laurent Habsieger (2003)

Acta Arithmetica

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On the Diophantine equation $X^{2} - (2^{2 m} + 1) Y^{4} = - 2^{2 m}$

Michael Stoll, P. G. Walsh, Pingzhi Yuan (2009)

Acta Arithmetica

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On metric theory of Diophantine approximation for complex numbers

Zhengyu Chen (2015)

Acta Arithmetica

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In 1941, R. J. Duffin and A. C. Schaeffer conjectured that for the inequality |α - m/n| < ψ(n)/n with g.c.d.(m,n) = 1, there are infinitely many solutions in positive integers m and n for almost all α ∈ ℝ if and only if $\sum_{n = 2}^{\infty} ϕ (n) ψ (n) / n = \infty$ . As one of partial results, in 1978, J. D. Vaaler proved this conjecture under the additional condition $ψ (n) = (n^{- 1})$ . In this paper, we discuss the metric theory of Diophantine approximation over the imaginary quadratic field ℚ(√d) with a square-free integer d < 0, and show...

On the Diophantine equation $x^{p} + y^{2 p} = z^{2}$

A. Rotkiewicz, A. Schinzel (1987)

Colloquium Mathematicae

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The diophantine equation $x^{2} + C = y^{n}$ , II

J. H. E. Cohn (2003)

Acta Arithmetica

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On the Diophantine equation $x^{2} + q^{2 m} = 2 y^{p}$

Sz. Tengely (2007)

Acta Arithmetica

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A note on the Diophantine equation $(a^{n} x^{m} \pm 1) / (a^{n} x \pm 1) = y^{n} + 1$

Jiagui Luo (2001)

Acta Arithmetica

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A note on the Diophantine equation $x^{2} + q^{m} = y^{3}$

Hui Lin Zhu (2011)

Acta Arithmetica

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On the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$

Florian Luca, Alain Togbé (2009)

Colloquium Mathematicae

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We find all the solutions of the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$ in positive integers x,y,α,β,n ≥ 3 with x and y coprime.

The Diophantine equation ${(b n)}^{x} + {(2 n)}^{y} = {((b + 2) n)}^{z}$

Min Tang, Quan-Hui Yang (2013)

Colloquium Mathematicae

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Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation $b^{x} + 2^{y} = {(b + 2)}^{z}$ has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

On some generalizations of the diophantine equation $s (1^{k} + 2^{k} + \dots + x^{k}) + r = d y^{n}$

Csaba Rakaczki (2012)

Acta Arithmetica

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On $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$

Susil Kumar Jena (2015)

Communications in Mathematics

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The two related Diophantine equations: $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$ , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.

Further remarks on Diophantine quintuples

Mihai Cipu (2015)

Acta Arithmetica

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A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < e $s a t i s f i e s$ d < 1.55·1072 $a n d$ b < 6.21·1035 $w h e n 4 a < b, w h i l e f o r b < 4 a o n e h a s e i t h e r$ c = a + b + 2√(ab+1)...