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1
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1answer
134 views

Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
4
votes
1answer
240 views

Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...
17
votes
1answer
845 views

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
1answer
168 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
0answers
182 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
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
1answer
155 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
1answer
150 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
1answer
319 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
1answer
165 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
votes
1answer
188 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 ...
0
votes
0answers
150 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 ...
7
votes
1answer
353 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
1answer
255 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
1answer
174 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 ...
0
votes
0answers
128 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
votes
1answer
367 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
1answer
272 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 $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ ...
1
vote
0answers
118 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 ...
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 - ...
5
votes
3answers
370 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 ...
6
votes
2answers
459 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 ...
4
votes
1answer
246 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 ...
1
vote
1answer
179 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 ...
12
votes
0answers
375 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
2answers
519 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
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 > $ ...
1
vote
2answers
204 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 $b_i,d_i \in \mathbb{N}$. ...
3
votes
1answer
199 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 - ...
17
votes
1answer
521 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 ...
1
vote
0answers
110 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
2answers
770 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 ...
4
votes
2answers
613 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
1answer
214 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 ...
5
votes
2answers
402 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
3answers
600 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 ...
5
votes
1answer
423 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
1answer
107 views
7
votes
2answers
429 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
1answer
419 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 ...
1
vote
2answers
497 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
0answers
451 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
1answer
271 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
votes
0answers
192 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
179 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 ...
5
votes
0answers
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 ...
0
votes
0answers
124 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
votes
1answer
221 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
2answers
356 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 ...
3
votes
2answers
269 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 ...