4
votes
1answer
123 views

Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality $$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$ and compares with the bound due to Minkowski that ...
4
votes
1answer
277 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
853 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
1answer
157 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 ...
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
168 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
192 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
151 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 ...
8
votes
1answer
358 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
256 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
132 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
371 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
278 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 $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ ...
5
votes
3answers
373 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
463 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 ...
1
vote
1answer
186 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
378 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
520 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 $K$ ...
17
votes
1answer
526 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
votes
0answers
113 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
773 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
643 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
216 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
405 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
613 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 ...
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 ...
2
votes
2answers
522 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
462 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 $\theta$ ...
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 ...
1
vote
1answer
149 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
1answer
379 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 $\lbrace\lfloor n\alpha+\beta\rfloor:n=1,2,\dots\rbrace$ ...
1
vote
0answers
173 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 ...
5
votes
1answer
435 views

Analogue of van der Corput sequence for prime numbers

A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and ...
3
votes
1answer
317 views

Hausdorff measure on product spaces of p-adic integers

This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
3
votes
4answers
472 views

Equidistribution Theorem: distance between solutions

Can please someone help me with the following problem. Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational. Now I need to solve the inequality $nx \; ...
4
votes
2answers
467 views

Primes in generalized fibonacci sequences

In C. McMullen's Uniformly Diophantine numbers in a fixed real quadratic field generalized Fibonacci sequence are defined as follows: $f_0=0,f_1=1,f_m=tf_{m-1}-nf_{m-2}$ where some fixed $t\in ...
3
votes
1answer
257 views

Repetitions of the totient

In a program I'm writing I'm using that the function: $rphi(1) = 0$ $rphi(n) = 1+rphi(phi(n))$ grows very slowly. Judging from https://oeis.org/A003434 it would seam like it is approximately ...
1
vote
0answers
285 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 ...
3
votes
3answers
493 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 ...
4
votes
0answers
266 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. ...
0
votes
0answers
285 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
474 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., ...
4
votes
1answer
784 views

Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...
4
votes
0answers
279 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
1answer
315 views

Work exploring application of probability to metric number theory problems

I am interested in studying the application of probabilistic tools to study metric number theoretic problems, specifically the Duffin-Schaeffer conjecture ...