# Questions tagged [diophantine-approximation]

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### Exponential of Liouville Numbers

By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that Any Liouville number is a $U$-number. $\log \alpha$ is either an $S$- or a $T$-...
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### Diophantine equations involving recurrence sequences

I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
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### Central limit theorem for irrational rotations

Let $\alpha$ be an algebraic integer of modulus 1, and $R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$? Birkhoff's ergodic ...
1 vote
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### Diophantine approximation away from $0$

Let $\alpha$ be a real irrational algebraic number. The now-classic Thue-Siegel-Roth theorem asserts that for any $\varepsilon > 0$ there exists a positive number $c = c(\alpha, \varepsilon)$ such ...
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### Simultaneously approximating all $x \in [0,1]$ with fractions of bounded denominator

Dirichlet's theorem says that all numbers $x\in [0,1]$ can be approximated by infinitely many fractions $p/q \in \mathbb{Q}$ with error $|x - p/q| \le 1/q^2$. I am interested in the following question:...
1 vote
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### The liminf of an expression involving an irrational rotation

Let $0 < a < 1$ be an irrational number. Is it true that $$\liminf_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$ Note: Here $\{\cdot\}$ denotes the fractional part.
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### Extreme elliptic curves from good $abc$-triples

It is a well-known fact that the $abc$-conjecture of Masser and Oesterle and Szpiro's conjecture are equivalent. For the convenience of the reader I will write down the statements for both: $abc$-...
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### Stability of successive minima with respect to the metric on the space of lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
1 vote
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### Uniform distribution mod $1$ vs independence of random variables

Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...
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### The range of each of successive minima for all unimodular lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
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### How far away can we get by multiple rounding and unit change?

This question is inspired by xkcd #2585 (Rounding): Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”. Consider the following directed graph: its vertices ...
### number of integers $n$ with $\|n \alpha \|$ small?
Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity $$D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \},$$ $\| x \|$ denotes the distance to the ...