# Questions tagged [diophantine-approximation]

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### Diophantine approximation for polynomials in a single variable

Let $a_0, a_1, \cdots, a_n$ be non-zero algebraic numbers. Consider the polynomial $$\displaystyle f(x) = a_n x^n + \cdots + a_1 x + a_0.$$ For a real number $\alpha$, put $\lVert \alpha \rVert$ for ...
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### How to show the geodesic orbit of a badly approximable number are/are not homogeneously equidistributed on its orbit closure?

Let $x\in \mathbb R$ be a badly approximable number. By Dani's correspondence, $g_t u_x \mathbb Z^2$ is bounded away from the cusp in $X=\text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$, identified with ...
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### Enquiry on irrationality measures

Suppose $a,b$ are positive irrational numbers and let $a_0, b_0$ be their corresponding irrationality measures. Define $c=a/b$ and $d=a-b$. Are the respective irrationality measures of $c$ and $d$ ...
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### Well known applications of Roth's theorem

Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places. It is an ...
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### Improving Diophantine approximation by rescaling

Let $\lambda\in(0,1)$ be an irrational number such that its continued fraction expansion is bounded (for example, an irrational quadratic number, whose continued fraction is periodic). It is known ...
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### The growth of certain continued fractions

I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
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### Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves

Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
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### Equidistribution modulo 1

We know that the time spent by the sequence $na \mod 1$, $n$ ranging from $1$ up to $x$ and $a$ irrational, at any interval of length $\delta$ is approximately $\delta x$. There are known results when ...
### Markov constant of $\pi$
Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$ According to this document, if ...