Questions tagged [diophantine-approximation]
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338
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Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$
Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as
$$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$
Moreover, $\alpha$ is rational if and only if its ...
- 25.5k
8
votes
1
answer
262
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How often a random walk with irrational increments is close to 0?
Let $\omega$ be an irrational number, and $X$ a random variable taking values $1,-1,\omega,-\omega$ each with probability $1/4$. Let then $X_i$ be iid variables with the same law as $X$ and $S_n=\sum_{...
- 1,268
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0
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125
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Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
- 1,451
6
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1
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433
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Sign in Dirichlet's approximation theorem
Fix $\alpha \in \mathbf{R}$. The classical Dirichlet's approximation theorem states there exist infinitely many rationals $p/q$ such that
$$
\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}.
$$
...
- 1,451
2
votes
2
answers
245
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Approximation of a square with an irrational arithmetic progression
Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ be irrational. Does the arithmetic progression $(n\alpha )_{n\in\mathbb{N}}$ becomes arbitrarily close to squares?
More precisely, what can be said ...
- 3,554
4
votes
1
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161
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On the set of good approximators in the sense of Dirichlet's theorem
This question came up when thinking about an older question that hasn't been answered as of now.
Let $\mathbb{N}$ be the set of positive integers. If $\alpha\in\mathbb{R}$, we say $q\in\mathbb{N}$ is ...
- 45.5k
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1
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161
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Density of the set of numbers that are "good approximators" to a given real in the sense of Dirichlet's approximation theorem
Let $\mathbb{N}$ be the set of positive integers. Given a set $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$...
- 45.5k
6
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1
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352
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Rational approximation of an integer combination of two irrationals
Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$,
$$d(nx,\mathbb{Z})>C n^{-\beta}.$$
It is ...
- 1,268
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votes
1
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438
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Algorithm for comparing $x + y \cdot \log(z)$ for $x,y,z$ rational?
I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures ...
- 13.1k
14
votes
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558
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$\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for $n\geq2$
I stumbled upon the following claim online: $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for all integers $n\in \mathbb{N}$, $n\geq2$. Checking with the computer, the claim ...
2
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1
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460
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Is there a number for which we know precisely the approximation by rationals?
The question is about the existence of a number $x$ for which we know the existence of $c>0$ such that for all $u>0,n\in\mathbb{N}^*$ that $$
\frac {1}{nu}\sum_{j=1}^{n}1_{d(jx,\mathbb Z)<...
- 1,268
2
votes
1
answer
762
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Very badly approximable numbers
Roth's theorem states that for an algbraic number $a$, $a$ is badly approximated by rationals: for every $\alpha>0$ there is $C>0$ such that for $l\in \mathbb Z$, $$d(la,\mathbb Z)>Cl^{-1-\...
- 1,268
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1
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146
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Well distributed sequence uniformly over small intervals
Let $a$ an irrational number. Can we say that there is $c>0$ such that for all integer $k,n$ and $1>u>0$,
$$ \frac {1}{n}\sum_{i=k}^{k+n} 1_{( (ia) \in [0,u])}< cu ? $$
where (x) is the ...
- 1,268
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1
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95
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The growth of a sequence related to Liouville numbers [closed]
I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends ...
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Are there infinitely many primes $p$, positive integers $ k, n $ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?
Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
- 1,086
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Algebraic integers whose matrix representations have singular values in an interval
Let $K$ be a finite extension of $\mathbb{Q}$. Let $\mathcal{O}(K)$ be the ring of integers of $K$. Let $\omega_1,\ldots,\omega_n$ be an integral basis for $K$ over $\mathbb{Q}$.
For each $a \in K$...
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95
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Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator
This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
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The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1
We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...
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1
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98
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Estimating volume of a simple object
Volume computation is $\#P$ hard.
Take the $[0,1]^n$ polytope.
Slice it by an half space inequality with $poly(n)$ bit rational coefficients into two unequal halves.
Volume of bigger section is $\...
- 13.7k
3
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1
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220
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Linear independence of approximate square roots
From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
- 263
3
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4
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280
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Approximately satisfying simultaneous vector linear diophantine equations?
Pick three $a,b,c$ vectors in $\mathbb Z^n$ uniformly with $\max(\|a\|_\infty,\|b\|_\infty)<T$ and $\|c\|_\infty<T^2$ and an $\epsilon>0$.
Assume $a$ and $b$ are coordinatewise coprime (...
- 13.7k
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0
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220
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Examples of essentially sub-linear functions
A dimension function is an increasing, continuous function $%
f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$.
Say that a dimension function $f$ is essentially sub-linear ...
- 21
3
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0
answers
105
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Optimal Roth-type result in diophantine approximation
Let $\alpha$ be a real algebraic number. It is easy to see that if $\deg(\alpha) = 2$, that for there exists a number $c(D(\alpha))$, where $D(\alpha)$ is the discriminant of the primitive integral ...
- 25.5k
2
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1
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154
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infinite set of mutually irrational numbers which odd linear combinations approximate 0 badly
I'm looking for a set of real numbers $\{\lambda_i;i\geq 1\}$ such that for each odd $n$, one can control $\delta_n:=\inf| \sum_i \pm n_i \lambda_i|$ where the $n_i$ are natural integers that sum to $...
- 1,268
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0
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233
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Transcendental functions generating almost integers
Informally speaking, an "almost integer" is a real number very close to an integer.
There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
- 19
7
votes
1
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636
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Digits in an algebraic irrational number
I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture).
I know that by using Ridout theorem or Schmidt subspace theorem ...
- 515
1
vote
1
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661
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Closest area of research to Transcendental Number Theory or/and Geometry of Numbers?
I am highly interested in doing research in either of
1- Transcendental Number Theory and Algebraic Independence;
2- Diophantine Approximation and Geometry of Numbers.
There is no person working ...
user135767
2
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0
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135
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numbers independent over $\mathbb{Q}$ but not BA? numbers that aren't a basis for a number field but are BA?
Has anyone discovered a vector of algebraic real numbers $(a_1,...,a_k)$ such that $1,a_1,...,a_k$ are linearly independent over $\mathbb{Q}$ and such that $(a_1,...,a_k)$ is not "badly approximable"?
...
- 21
4
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0
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97
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Weighted distribution of irrational rotation
Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
- 615
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0
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95
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Smallest integer lattice point by box measure in a polytope?
Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
- 13.7k
8
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1
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Limit of quotients of elements of special Fibonacci matrices
Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices
$$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$
Let
$$\pmatrix{\alpha_n & \beta_n \\...
- 81
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1
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251
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Distribution of $\{cn^a\}$
Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...
- 103k
3
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0
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100
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Independence of number fields generated by roots of Littlewood polynomials
Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and
$$
c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
- 13.7k
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0
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81
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The Hausdorff dimensions of variations of Jarnik sets
For $\alpha, \beta>3,$ define
$$\{(x,y)\in[0,1]\times [0,1]: \|qx\|\le q^{1-\alpha}, \|qy\|\le q^{1-\beta} \quad \text{for infinitely many $ q\in \mathbb{N}$}\}.$$
This set can be regarded as a two ...
- 175
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108
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Approximation of an irrational point from a given direction
Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
- 233
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0
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362
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Average value of j-invariant at infinity
Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$:
$$
\...
- 507
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0
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67
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Discrepancy related independent vector from tensor product?
Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of
the point set $P$ of $N$ points in $\mathbb Z^...
- 13.7k
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0
answers
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Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?
Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
- 13.7k
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1
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Are there partially algebraic Hecke characters?
$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$
Let $F$ be a number field.
Let $\chi\colon \mathbb{A}_F^\...
- 4,933
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0
answers
139
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Combination of irrationals
Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is,
$$
S=\{(m_1,m_2)\in\mathbb{Z}\...
- 1,096
1
vote
1
answer
150
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Simultaneous rational approximation to transcendental and algebraically independent numbers
I’m interested in the problem of simultaneous rational approximation to $k\geq 2$ numbers $\alpha_1,…,\alpha_k$ in the generic case where the $\alpha_j$ are transcendental and algebraically ...
- 115
2
votes
1
answer
180
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Variations of Dirichlet's theorem on Diophantine approximation
Dirichlet's theorem on Diophantine approximation:
For any real number $x$, for integer $N>0$, there exists integers $a$ and $b>0$ with $(a,b)=1$ such that $b\leq N$ and $$|x-\frac a b|<\frac{...
- 3,764
2
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0
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130
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Quantitative Khintchin's theorem
The quantitative version of Kchintchin's theorem proved by W.Schmidt states that for a.e. $x\in[0,1]$ and every positive integer $K$, if we denote the number of solutions $(p,q)$ to the inequality
$$|...
- 539
4
votes
1
answer
440
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Explicit numbers with square root cancellation in Weyl's exponential sum
I'm interested in examples of real numbers $\alpha$ where we have
$$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$
or perhaps with the weaker estimate with the right side replaced ...
- 41
0
votes
1
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177
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Best possibe bound for the number of solutions of diophantine approximation?
I am interested in an explicit - best possible - bound for the number of solutions of the simple diophantine equation
$$\tag{1}|\mu_1 n - \mu_2 m| <1,$$
where $m,n \in \mathbb{Z}$ with $|n| \leq |\...
- 196
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0
answers
326
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Transcendental Continued Fractions
Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...
- 680
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3
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515
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Lower bound for the fractional part of $(4/3)^n$
My son, who is 16, is doing some independent research. A lower bound depending on $n$ for $\left\{ \left( \frac{4}{3} \right)^n \right\}=\left( \frac{4}{3} \right)^n-\left\lfloor \left(\frac{4}{3} \...
- 5,200
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2
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939
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Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$
Note: this question was updated (2) after GNiklasch's answer was posted, and taking Gro-Tsen's comment into account. The initial question (1) dealt with $\mathbb{Q}_3$.
Original post (1). Let's try ...
- 22.6k
1
vote
2
answers
244
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Does the set of Diophantine $m$-tuples has full measure?
We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
- 758
1
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1
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189
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Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators
Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...
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