# Tagged Questions

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### bounds of weighted sum of exponentials (related to Baker's theorem)

Given $n$ integers $a_1, \cdots, a_n$ and $n$ algebraic numbers $b_1, \cdots, b_n$, consider
$S=\sum_{i=0}^n a_i e^{b_i}$, the question is how to give a lower bound of $|S|$ assuming that $S\neq 0$, ...

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### Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to:
If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...

**6**

votes

**2**answers

457 views

### Is the infimum of Salem numbers > 1?

BACKGROUND
A Salem number is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their ...

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178 views

### Sharpenings of Liouville's inequality

The norm of an algebraic number $\alpha$ is the product of its conjugates, $N(\alpha)$.
Suppose that I have an inequality of the form $|x-\alpha*y| > c X^{n-\gamma}$ where $X=max{|x|,|y|}$ and c ...

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255 views

### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

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votes

**3**answers

491 views

### For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...

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**2**answers

1k views

### How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...