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12
votes
0answers
362 views

Should the number of small solutions in Roth's theorem be bounded uniformly, assuming the target is an algebraic integer?

Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible ...
10
votes
0answers
467 views

Roth's theorem, Lang's conjecture and beyond

Lang conjectured that for an irrational algebraic number $\alpha$ and $\epsilon > 0$, there exist only finitely may rationals $p/q$ such that $$ \left| \alpha - \frac{p}{q} \right| ...
6
votes
0answers
459 views

Counting Lattice Points in Real Polytopes

Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...
6
votes
0answers
625 views

Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
5
votes
0answers
251 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 ...
5
votes
0answers
576 views

A question related to the abc conjecture

The abc conjecture asserts that whenever $a,b,c$ are pairwise coprime positive integers such that $a + b = c$ and $\epsilon > 0$, there exists a constant $C_\epsilon > 0$ (which depends on ...
4
votes
0answers
89 views

Khintchine theorem - necessity of monotonicity in divergence condition

Hi, I'm trying to get a hold of Khintchine theorem in metric Diophantine approximation. Right now I'm interested in the divergence condition, namely: If $\sum_{q=1}^\infty\psi(q) = \infty > $ ...
4
votes
0answers
263 views

Are these terms consisting of logarithms of primes rationally independent?

I expected it to be basic, but seem unable to find a proof of the following: Let $p_0, p_1, .., p_m$ be distinct primes. Then the $m+1$ terms $\dfrac{\log p_0}{\log p_j}$, are rationally independent. ...
4
votes
0answers
272 views

A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...
3
votes
0answers
178 views

Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes. For example, for $M=2$ and $N=4$ you get ...
3
votes
0answers
117 views

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 - ...
3
votes
0answers
449 views

Is $\pi$ well-approximable?

Is it known whether, for all $c > 0$, there always exist integers $p$ and $q$ such that $\left| \pi - \frac{p}{q}\right| < \frac{c}{q^2}$? This seems like a fundamental question but I couldn't ...
3
votes
0answers
1k views

Is there a limit of cos (n!)?

Hi, I encountered a problem today to prove that cos (n!) does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that i mean less complex theorems ...
2
votes
0answers
201 views

Simultaneous diophantine approximation with polynomial bound

For a given number $\alpha$ continued fractions expansion $(p_n, q_n)$ of $\alpha$ has the remarkable property that not only $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q_n^2}$, but the converse holds - ...
2
votes
0answers
100 views

Lower bounds on sums of S-units

Let $S$ be a fixed finite set of valuations on $\mathbb Q$ containing the archimedean one. A $S$-unit is $x\in\mathbb Q$ such that $|x|_v =1$ for all $v\notin S$. For any $S$-units $x_0, \dots, x_n$ ...
1
vote
0answers
109 views

Dalzel's integral for $\pi$ and the lemniscate constant

$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers $$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$ and, for example, the Wallis product formula ...
1
vote
0answers
107 views

linear forms in abelian logarithms and a conjecture of Lang

Consider the following conjecture, going back to Lang and restated (and proved) in the elliptic case in a 2009 Crelle paper by David and Hirata-Kohno (see Conjecture 1.2 in their paper). Conjecture ...
1
vote
0answers
172 views

Norm related to diophantine approximation?

I'm trying to read this paper: http://www.springerlink.com/content/g0046660260825x3/ or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.20.1813&rep=rep1&type=pdf But I don't ...
1
vote
0answers
282 views

What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
1
vote
0answers
275 views

Diophantine Approximation in Higher Dimensions

Let $\mathbf{x} \in \mathbb{R}^K$ be an irrational vector. Assume that $\|\mathbf{x}\|^2 \leq 1$. Is is known that for all $N > 1$, there exists an $p_1 \in \mathbb{N}, \mathbf{q}_1 \in ...
0
votes
0answers
38 views

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 $S\neq 0$, ...
0
votes
0answers
142 views

On the irrationality measure of generalized Stoneham numbers

Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and ...
0
votes
0answers
127 views

Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations

For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the ...
0
votes
0answers
191 views

Folklore Lemma (p-adic case)

Is there a p-adic version of the so called 'folklore lemma' which relates a sequence of diophnatine approximations to an algebraic number to the exponential of irrationality.
0
votes
0answers
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 ...
0
votes
0answers
122 views

The p-adic subspace Theorem

Could someone explain how the subspace theorem could be used to transfer results from archimedian valuatons to nonarchimedian ones?
0
votes
0answers
281 views

Diophantine approximation

Say absolute values of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$. Given positive integer $n$ that is reasonably large, we cannot always find integers $a,b,c$ such that $|a{b^{c}} - n|$ ...
0
votes
0answers
462 views

How quickly is $n\pi$ getting close to integers

Is there an understandable function $A(\epsilon)$ such that if $q < A(\epsilon)$ then $| q\pi - p| > \epsilon$ for all $p$? I want to know how quickly $n\pi$ is getting close to integers, e.g., ...