The diophantine-approximation tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\theta$ ...

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### 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.

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### 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 ...

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### 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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### The p-adic subspace Theorem

Could someone explain how the subspace theorem could be used to transfer results from archimedian valuatons to nonarchimedian ones?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\lbrace\lfloor n\alpha+\beta\rfloor:n=1,2,\dots\rbrace$
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 \; ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 - ...

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### 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 ...

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### 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.
...

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

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### 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., ...

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### 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 $a$ ...

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### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...

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### 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 ...

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### 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 ...

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### Efficient computation of the least fraction with square denominator greater than the square root of 2.

The least rational number greater than $\sqrt{2}$ that can be written as a ratio of integers $x/y$ with $y\le10^{100}$ can be found in a moment using a little Python program. Can anyone write a ...

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### Special case of Duffin-Schaeffer conjecture

The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: ...

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### Weakening the hypotheses in the Duffin-Schaeffer conjecture?

The Duffin-Schaeffer conjecture is a conjecture in metric number theory, which asks for a given function $f : \mathbb{R} \rightarrow \mathbb{R}^+$ the measure of the set of real numbers $\alpha$ such ...

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### Rational approximations on the circle

The well-known Liouville theorem asserts that an irrational algebraic number $\alpha$ cannot have too good rational approximations, namely $|\alpha-p/q|\ge C(\alpha)/q^k$ where $k$ is the degree of ...

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### Can the partial sums of a series be uniformly distributed modulo 1?

Earlier I asked in Is a sequence of the following type uniformly distributed modulo 1? whether the partial sums of the harmonic series is uniformly distributed modulo 1. Here I ask for necessary and ...

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### 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 $x_0, \dots, x_n$ ...

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### 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 ...

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### Square-free diophantine approximation

Given an irrational algebraic number $\alpha$ (and maybe I want to add: of degree greater than $2$?), do there exist infinitely many relatively prime and square-free $p$,$q$ with
$$|\alpha - p/q | ...

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### 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| ...

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### Best rational approximation in a special sense

Let $\alpha$ be an irrational number, $n\geq 1$ and
$ X_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$
Now let $(x_n,y_n)$ minimize the quantity $x+y\alpha$ on $X_n$. ...

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### Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...

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### Question related to Diophantine approximations and Roth's theorem

The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there ...

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### Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.
...

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### 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 ...

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### A simple question regarding the sum-of-divisors function

A good day to everyone.
Consider the following "Conjecture":
If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if ...

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### Fast series for pi

A quick perusal of the wikipedia page for $\pi$ yields a large collection of known series for $\pi$. In particular, these series are hypergeometric in nature, and have large (but finite) radius of ...

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### Is there a set of criteria to determine whether a number is transcendental for a subset of the reals with positive Lebesgue measure?

I am reading A. van der Poorten's 1978 paper on Apery's constant, and it cited the Thue-Siegel-Roth Theorem (that if $\beta$ is algebraic, then for all $\epsilon > 0$ the inequality $|\beta - p/q| ...

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### An inequality for cosine of n

Can anyone provide a proof of the following inequality?
If $n$ is a positive integer, $n\geq2$, then $$\cos(n) \leq 1 - 2^{-n}.$$
This is satisfied if $n$ is not within about $2^{-n/2}$ of a multiple ...

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### Computational results related to Khinchin-Levy constants

Khinchin's Theorem in Diophantine approximations state that for a 'typical' real number $\alpha$ (all but a set of Lebesgue measure 0), the continued fraction $\alpha = [a_0, a_1, \cdots]$ where $a_0 ...

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### inferring the slope of a digitized line

Given real numbers $a$ and $b$, and an integer $n \geq 2$, let $f(n,a,b)$ be the minimum of $(nint(ja+b)-nint(ia+b)+1)/(j-i)$ (for $1 \leq i < j \leq n$) minus the maximum of ...