Questions tagged [diophantine-approximation]

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70 views

Working with truncation of inverse of integers (number of necessary digits)

Ideally I would like to find exact value of $Mr'$ where $r'=\frac{1}{M'}$ where $M'$ ranges from $1$ to $T+1$ and $1\leq M\leq T$ holds. However in real world approximations have finite precision. If ...
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238 views

Chebyshev rational approximation of $e^{x}, x >0$: does it exist?

It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form ...
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319 views

Arithmetic-geometric mean for rationals?

Let $\operatorname{AGM}(x,y)$ be the arithmetic-geometric mean of $x$ and $y$. Given an error $\varepsilon>0$, a bound $b\in\mathbb R_+$ and a function $f:\mathbb R\rightarrow\mathbb R$ with $f(x)=...
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188 views

On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
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Double Diophantine approximation

Let $0 < \alpha < 1$. For any $n$ there is a closest lower Diophantine approximation $\max p / q \leq \alpha$ with integer $0 \leq p < q \leq n$. It can be found efficiently, e.g., with Stern-...
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For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
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65 views

Bounds on convergence of two orbits in the limit set of a Schottky group

Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...
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1answer
166 views

Lindemann theorem for Artin-Hasse exponential

Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
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167 views

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 ...
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197 views

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_{...
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151 views

Action on the upper half plane of the double coset of an irrational matrix in $\mathrm{SL}_2(\mathbb{R})$

Consider the action of $\mathrm{SL}_2(\mathbb{R})$ on the upper half plane $\mathbb{H}$ by Möbius transformations. Denote by $\Gamma$ the group $\mathrm{SL}_2(\mathbb{Z})$. It is known that if $z \in ...
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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 ...
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372 views

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 $$ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}. $$ ...
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204 views

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 ...
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126 views

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 ...
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1answer
100 views

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}.$...
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281 views

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 ...
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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 ...
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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 ...
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439 views

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)<...
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221 views

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-\...
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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 ...
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1answer
84 views

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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76 views

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 ...
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72 views

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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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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1answer
89 views

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 $\...
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147 views

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 ...
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271 views

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 (...
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38 views

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 ...
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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 ...
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95 views

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 $...
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195 views

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 ...
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565 views

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 ...
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1answer
406 views

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 ...
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119 views

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"? ...
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82 views

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, ...
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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 $A\in\mathbb Z^{m\times n}$ ...
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1answer
158 views

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 \\...
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1answer
211 views

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 $c$) ...
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84 views

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^...
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62 views

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 ...
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71 views

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 ...
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195 views

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 $\xi$: $$ \...
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50 views

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^...
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44 views

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(...
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183 views

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^\...
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0answers
127 views

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}\...
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1answer
102 views

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 ...

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