The diophantine-approximation tag has no usage guidance.

**15**

votes

**0**answers

311 views

### Looking for an effective irrationality measure of $\pi$

Most standard summaries of the literature on irrationality measure simply say, e.g., that
$$
\left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}}
$$
for all sufficiently large $q$, without giving ...

**12**

votes

**0**answers

178 views

### Diophantine approximation in the Julia set

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...

**12**

votes

**0**answers

440 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 ...

**11**

votes

**0**answers

600 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| <\frac{1}{q^2(...

**9**

votes

**0**answers

334 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 ...

**8**

votes

**0**answers

191 views

### Attractors of arithmetically small points

Consider the "points" to be in $\mathbb{G}_m(\mathbb{C}) = \mathbb{C}^{\times}$. By a sequence of "arithmetically small points" is meant a sequence of pairwise different algebraic points $\beta$ that, ...

**8**

votes

**0**answers

616 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 ...

**7**

votes

**0**answers

120 views

### Efficient Dirichlet approximation (continued fractions?) over a number field

Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...

**7**

votes

**0**answers

239 views

### Question on some coverings of the euclidean space

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...

**7**

votes

**0**answers

361 views

### Is simultaneous diophantine approximation (in a weaker sense) NP hard?

The traditional problem of simultaneous diophantine approximation is: Given a set of rational numbers $g_1,\ldots,g_d$, an integer $N$, and a rational $\gamma>0$, is there an integer $W$ with $1\...

**6**

votes

**0**answers

164 views

### Solving polynomial equations modulo $1$

Let $P\in \mathbb{R}\lbrack x\rbrack$ be given. (In practice, the coefficients could be given as, say, decimals to sufficient precision.) Let $M\geq 1$, and let $I$ be an interval in $\mathbb{R}/\...

**6**

votes

**0**answers

91 views

### Diophantine approximation in $\mathbb{G}_m^r$ with approximants restricted to a finiteley generated subgroup

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and ...

**6**

votes

**0**answers

691 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

**0**answers

98 views

### Jarník-Besicovitch and outer measure

The set $A_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate
$$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$
for infinitely many rationals $p/q$, ...

**4**

votes

**0**answers

328 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

**0**answers

308 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

**0**answers

101 views

### Conjectural growth rate for ergodic sums of logarithms

Let $\theta, \phi \in [0,1)$, and consider the sums
$$
S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|.
$$
The possible boundedness from above of such sums plays a key role in ...

**3**

votes

**0**answers

203 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

**0**answers

135 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

**0**answers

553 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 ...

**2**

votes

**0**answers

149 views

### binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or
having a negative solution, arose from an ergodic theory question,
presumably itself currently intractible. I am not a number ...

**2**

votes

**0**answers

161 views

### Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...

**2**

votes

**0**answers

122 views

### A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...

**2**

votes

**0**answers

173 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 $$\pi=4\prod_{n=1}^\...

**2**

votes

**0**answers

126 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 (...

**2**

votes

**0**answers

218 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

**0**answers

108 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

**0**answers

93 views

### Analog of Baker's theorem on linear combination of $\log a \log b$

Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination
$$\sum m_i\log a_i$$
to cancel, then it is ...

**1**

vote

**0**answers

148 views

### Optimal Diophantine approximation of $\pi$

If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is the maximum value of $M$?

**1**

vote

**0**answers

39 views

### Representing sparse set as set of extremely good approximation

For $\alpha \in \mathbb R$ and $\varepsilon(n) > 0$, consider the set
$$
N(\alpha,\varepsilon) = \{ n \in \mathbb{N} \ : \ \lVert \alpha n \rVert < \varepsilon(n) \}
$$
(where $ \lVert x \rVert$...

**1**

vote

**0**answers

54 views

### On a sequence of integers

I recall a well-known theorem due to Minkowski.
Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...

**1**

vote

**0**answers

103 views

### Rate of convergence of an algebraic irrational rotation

Let $\alpha \in \mathbb{S}^1$ be an algebraic number with $\mathop{\mathrm{arg}}(\alpha)/\pi$ irrational. Is it possible for the rotation by $\alpha$ to converge exponentially fast to a $\xi \in \...

**1**

vote

**0**answers

58 views

### Big Omega result about number of totally positive integers with fixed trace

There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field.
That number has a natural geometric approximation $G_a$, and we ...

**1**

vote

**0**answers

188 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

**0**answers

306 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

**0**answers

313 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 $|a{b^{c}} - n|$ ...

**1**

vote

**0**answers

605 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., ...

**1**

vote

**0**answers

323 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 \mathbb{N}^{...

**0**

votes

**0**answers

41 views

### Criterion for irrational numbers of constant type 2

From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...

**0**

votes

**0**answers

179 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 $\gcd(a,...

**0**

votes

**0**answers

159 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

**0**answers

216 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

**0**answers

221 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

**0**answers

133 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?