The diophantine-approximation tag has no usage guidance.

**18**

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**1**answer

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### Irrationality measure of log(2)/log(6)

As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity
$...

**3**

votes

**1**answer

185 views

### Higher dimensional analogue of Thue's equation

The classical Thue equation is
$$\displaystyle F(x,y) = h,$$
for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ ...

**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 $...

**0**

votes

**1**answer

173 views

### Reference request: on sums of the form $ax^m + by^n = h$

I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...

**4**

votes

**1**answer

177 views

### Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals.
Anyway, let $E$ be the "constructible numbers," ...

**9**

votes

**1**answer

339 views

### Distribution of polynomials mod 1 using co-prime integers

If $a$ is any real irrational, then the set of numbers of the form $ax+y$ with $x$ and $y$ co-prime integers is dense in $\mathbb{R}$. I managed to prove this, in what I suspect is an overly ...

**3**

votes

**1**answer

196 views

### Numbers with balanced diophantine approximations

This is a follow-up to Question 146635, namely Expected symmetry in the diophantine approximations of an irrational number, which I will refer to for notation and terminology used here without ...

**7**

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**1**answer

229 views

### Expected symmetry in the diophantine approximations of an irrational number

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\|...

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**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,...

**8**

votes

**1**answer

430 views

### On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$

Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. ...

**2**

votes

**1**answer

261 views

### (efficient) method to test $\{n\alpha\}\not\in [A, B]\subset [0,1]$

Suppose $\alpha$ is a fixed given irrational number with $\alpha\in [A, B]\subset [0,1]$, are there any (efficient) methods to compute the least integer $n$ such that the decimal part of $n\alpha$ ...

**2**

votes

**1**answer

190 views

### Height on a semiabelian variety

Let $A$ be a semiabelian variety over $\bar{\mathbb{Q}}$ and $B$ a semiabelian subvariety of $A$. Let $\pi:A\to A/B$ be the canonical morphism. Let $h:A(\bar{\mathbb{Q}})\to\mathbb{R}$ and $h':(A/B)(\...

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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 (...

**10**

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**1**answer

412 views

### Distribution mod 1 of exponential growth sequences

Let $t_n$ be a sequence of real numbers and $C,r>1.$ Suppose that for every $n\geq 1$ we have $\frac{1}{C}r^n\leq t_n \leq Cr^n.$ Does there exist a real number $\xi$ and an $\varepsilon>0$ ...

**8**

votes

**1**answer

420 views

### Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background
Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by
$$
x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}.
$$
...

**2**

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**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}^\...

**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 - ...

**5**

votes

**3**answers

403 views

### Siegel's theorem with real coefficients

Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$?
When $h$ itself has integer coefficients, a ...

**9**

votes

**3**answers

664 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 ...

**5**

votes

**1**answer

539 views

### Winning sets of full measure (Schmidt's game)

A quick reminder of the definition of Schmidt's game:
Let ${X}$ be a metric space and ${S\subset X}$ be a subset. Let
${0<\alpha,\beta<1}$ be constants. Bob chooses any open ball
${B_0\...

**2**

votes

**1**answer

207 views

### Approximation by binary fractions

For general Diophantine approximation, the Thue–Siegel–Roth theorem states that for any irrational algebraic number $x$, and any $\varepsilon>0$, there exists a constant $c=c(x,\varepsilon)$ such ...

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**0**answers

441 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 ...

**6**

votes

**2**answers

535 views

### Simultaneous Powers Far From 1

I'm looking for a reference or proof of the following. Let $K/\mathbb{Q}$ be a finite Galois extension of degree $n$. Let $a_1,\ldots,a_n$ be Galois conjugate elements in the ring of integers of $K$ ...

**4**

votes

**1**answer

132 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
> $ ...

**1**

vote

**2**answers

211 views

### Existence of a solution to a system of Diophantine Inequalities

Does the solution to the following system of inequalities exist?
$$a-1\geq a\left( b_ic -d_i\right)\geq 1$$
where $a\in \mathbb{N}_{\geq3}$, $c\in \mathbb{R}$ and $b_i,d_i \in \mathbb{N}$. Moreover,...

**3**

votes

**1**answer

232 views

### Diophantine equation with primitive nth root of unity

Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\...

**18**

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**1**answer

573 views

### Can the expansion of a large integer in all bases consist of almost all zeroes?

Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...

**2**

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**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 (...

**3**

votes

**2**answers

1k views

### Solved cubic Thue equation

Hi everybody. I need to know if the cubic Thue equation $x^3 + x^2y + 3xy^2 - y^3 = \pm 1$ is completely solved. I know that there are effective algorithms to solve any cubic Thue equation and that ...

**5**

votes

**2**answers

1k views

### The diophantine equation X^2 - Y^2 - Z^2 = +- 1

Hi everybody. I'd like to know if the diophantine equation
(1) $$X^2 - Y^2 - Z^2 = \pm 1$$
has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you ...

**0**

votes

**1**answer

228 views

### Number theory question

Given $a$ and $b$ irrational numbers with $a/b$ also irrational, how do you prove that
$( \{ na\} , \{ nb \})$ is dense in $[0,1] * [0,1]$ , where $n$ ranges over the integers?
$\{x\}$ is the ...

**6**

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**2**answers

477 views

### Still more generalized Dirichlet Theorem

Dirichlet proved a classical theorem about approximating irrational real numbers with rational numbers, saying that for any irrational real number $\alpha$, you can find infinitely many rational ...

**5**

votes

**3**answers

985 views

### Simultaneous diophantine approximation

Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor.
Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}...

**5**

votes

**1**answer

605 views

### Rational approximation to a set of reals

Are there any well known algorithms for finding good rational approximations to sets of real numbers?
Given just two real numbers, I can use continued fractions to find a rational approximation to ...

**1**

vote

**1**answer

113 views

### are p-limits scales dense in the infinite musical scale of all rational frequencies?

In the wiki section on prime limit tuning, one reads:
...

**7**

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**2**answers

438 views

### Computing the measure of the projection on the torus of a semialgebraic set

Let $V \subseteq \mathbb{R}^n$ be a set cut out by a system of finitely many polynomial equations and inequalities with integer coefficients. Let $W$ be the set of all points in the box $[0,1]^n$ that ...

**0**

votes

**1**answer

451 views

### How small can {log p/q} be?

Denote by $||x||$ the distance between $x$ and the nearest integer.
Mahler conjectured that there is a constant $c > 0$ such that for any integer $n \geq 2$
$$
||\log n|| \geq n^{-c}
$$
and ...

**2**

votes

**2**answers

738 views

### Liouville's Theorem in Diophantine Approximation

Liouville's Theorem states that for any algebraic $\alpha \in \mathbb{R}$ of degree $n$, there exists a positive constant $c:=c(\alpha)$ such that $$|\alpha-\frac{p}{q}|>\frac{c}{q^n}$$ for any $p \...

**3**

votes

**0**answers

554 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 ...

**4**

votes

**1**answer

319 views

### Diophantine approximations of ratios of transcendental numbers

I am looking for good diophantine approximations for a specific class of irrational numbers.
Let $e^{2 \pi i \theta}$ be a complex algebraic number. I would like a result to the effect that $\theta$ ...

**0**

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**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**

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**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 ...

**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 ...

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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?

**2**

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**1**answer

307 views

### p-adic version of Liouville's approximation theorem

Does anyone know of a p-adic analogue of Liouville's approximation theorem http://mathworld.wolfram.com/LiouvillesApproximationTheorem.html with proof?
I'm aware of Roth's theorem and subspace ...

**1**

vote

**2**answers

430 views

### Chebyshev's Theorem

Hi,
I´m looking for Chebyshev´s theorem which says that the inequality $|x(k)-y|<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\...

**3**

votes

**2**answers

394 views

### Any relationship between Viswanath's constant and the Khinchine-Lévy constant?

It is well-known that if ${\{{F_n}\}}$ is a random Fibonacci sequence then we have almost certainly $\lim \limits_{n\to\infty}\sqrt[n]{|F_n|}=\tau$ where $\tau\approx 1.554682275$ is Viswanath's ...

**7**

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**1**answer

452 views

### Diophantine elements in SU(2)

Following notions from [1], call a set of elements $g_1, \dots, g_k \in G = SU(2)$ Diophantine if it satisfies the following property: there exists a constant $D$ such that for every word $W_m$ of ...

**1**

vote

**1**answer

152 views

### Is the closure of a semialgebraic set mod 1 also semialgebraic?

Let $p:\mathbb{R}^n\to[0,1)^n$ be the map defined by $p(x_1,\ldots,x_n)=(\{x_1\},\ldots,\{x_n\})$, where $\{\cdot\}$ is the fractional part operator. Experimentation suggests that if $S \subseteq \...

**5**

votes

**1**answer

552 views

### Generalizations of the Rayleigh(-Beatty) theorem

For a given irrational number $\alpha>0$ and a real number $\beta$,
the inhomogeneous Beatty sequence
sequence $S_{\alpha,\beta}$ is the set $\lbrace\lfloor n\alpha+\beta\rfloor:n=1,2,\dots\rbrace$
...