Questions tagged [diophantine-approximation]
For questions about approximating real numbers by rational numbers.
471
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Calculating a lower bound of Lebesgue measure of some Borel set
I have the following set. I need to calculate a lower bound on its measure in order to prove something about some distribution.
$I \subseteq [0,1)$. Take the set $\mathcal{J}(I) = I \bigcap \underset{\...
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Finitely Many Solutions to Diophantine Equation
I am trying to prove the following claim:
Let $F \in \mathbb{Z}[X, Y]$ be a square-free binary form of degree $d \geq 4$, and
let $G \in \mathbb{Z}[X, Y]$ be a polynomial of total degree $g \leq d − 3$...
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1
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Proving an exponential function dominates over an erratic fractional part
Consider the function $3^x(1-2^{-\text{frac}(x\sqrt2)})$, where $\text{frac}()$ denotes the fractional part function. Now $\sqrt 2$ being irrational by kronecker, this is uniformly dense in $(0,1)$, ...
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Example of the real numbers $x$ appearing in the Jarnik's theorem: $||nx||\leq n^{-\beta}$ for infinitely many $n\in\mathbb{N}$ for fixed $\beta>1$
According to Falconer (Falconer, '85, The Geometry of Fractal Sets, theorem 8.16, p. 134), a part of the Jarnik's theorem is the following:
Take $\beta>1$. The set of real numbers $x$ for which ...
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Generalisation of Kronecker's Theorem
I am reading this [Six Lonely Runners]https://www.researchgate.net/publication/220343204_Six_Lonely_Runners, specifically Chapter 4. I do not understand their use of Kronecker's Theorem here
and the ...
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2
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Liouville numbers and numbers having infinite irrationality measure
Let's start with the definition of a Liouville number:
A Liouville number is a real number $\xi$ such that for any
$m\in\mathbb N_>0$ there exists a pair of coprime integers
$(p,q)\in\mathbb Z^2$ ...
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Help understanding steps to prove Dirichlet's approximation theorem
In my elementary number theory textbook, there is a problem that is meant to help understand the proof of Dirichlet's rational approximation theorem. There are two parts to the problem: the first part ...
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Let $x\in\mathbb{R}.$ For each $n,$ define $x_n:=\min_{k\in\mathbb{Z}}\lvert\frac{k}{n}-x \rvert.$ For which values of $x$ does $\sum x_n$ converge?
Let $\ x\in\mathbb{R}.\ $ For each $\ n\in\mathbb{N},\ $ define $\ x_n
:= \displaystyle\min_{k\in\mathbb{Z}}\left\lvert \frac{k}{n} - x
\right\rvert.\ $ For which values of $\ x\ $ does $\
\...
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Three or more points moving on a circle and a convergent subsequence
Three (or more difficult version: any number $n>2$) points move on a unit circle with pairwise different constant speeds. For which initial points and which speeds is the following true:
for every $...
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Orbits Of $SL_d(\mathbb{Z})$ on non rational and non - irrational points in the $d$ - Dimensional Torus
On the $d$ - dimensional torus, for every irrational point $\theta = (\theta_1,...,\theta_d)$ (i.e $\theta_i \notin \mathbb{Q}$), the orbits of $SL_d(\mathbb{Z})$ on this point are dense.
However, ...
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Find the Lagrange number of a given infinite simple continued fraction.
This question arises in Ch. 2 of Martin Aigner's beautiful book Markov's Theorem and 100 years of the uniqueness conjecture (page 36, Remark 2.4). These definitions are from Aigner's book summarized ...
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Equality case in Dirichlet's approximation theorem
In its strong form, Dirichlet's approximation theorem (for dimension one) states that for $\alpha\in{\mathbb R}$, and an integer $n\geq 1$, we have
$$
\min_{1\leq k\leq n}\| k \alpha \| \leq \frac{1}{...
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How is Siegel's lemma applied in number theory?
In the Wikipedia page, Siegel's lemma is stated as follows:
Consider the system $$ \begin{cases} \sum_{i=1}^Na_{1i}X_i=0\\
\vdots\\ \sum_{i=1}^Na_{Mi}X_i=0 \end{cases}, $$ where the
coefficients $a_{...
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Does a limit exist for the fraction of $p\alpha_i$?
With $p$ an integer and $\alpha_i$ a real number, does $\{p\alpha_i\}\leq\epsilon$ hold for some predefined $\epsilon$, maybe depending on the number of $\alpha_i$ values? I looked at the simultaneous ...
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Can we scale the elements of a set of real numbers so that they are arbitrarily close to integers?
Let $S$ be a finite set of real numbers. For any $\varepsilon > 0,$ is it always possible to choose $\alpha > 0$ such that every element of $S$ multiplied by $\alpha$ is withing $\varepsilon$ of ...
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Do three arithmetic sequences of real numbers that include $0$ always get arbitrarily close to each other at nonzero numbers?
I know that two arithmetic sequences of real numbers that both contain the number $0$ have terms that are arbitrarily close together, that is, for any $\varepsilon>0$ there exists a real number ...
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Absolute convergence of $\sum_{n=1}^\infty n^{-5} \csc(2n\pi \sqrt[3]{3})$
I want to prove the absolute convergence of this serie. There are two hints:
Firstly, prove that $|(\sin(\pi x)| \geq |x|$ for $|x| \leq \frac{1}{2}$.
Then use the fact that $2\sqrt[3]{3}$ is ...
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How does Roth's Theorem show a number is transcendental
[Roth's theorem][1] says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions $\frac{p}{q}$to this:
$$\displaystyle \left|\alpha-\frac pq\right|<\...
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The fractional part of a geometric sequence.
Prove or deny: for any given positive integer $ N $, there is a positive integer $ m\geq n $ such that $ \left\{ \frac{2^m}{\alpha} \right\} >1-\frac{1}{\alpha} $, where $\left\{x\right\}$ ...
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Rational approximations of logarithms of ratios of primes
I noticed that there are many pairs of primes $p_1, p_2$ such that $$\frac {\log(p_1/p_2)}{ \log(2)} \approx \frac{N}{69324}$$ for some natural N with an error of order $10^{-8}$ . I found around 20 ...
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Are there any perfect squares of the form 88...81 (in decimal, at least two 8's)?
I saw this problem recently and it is deceptively hard. The usual mod 4 trick won't work, and indeed there will be perfect squares whose last n digits will be 88...81, for any n.
I can show that if ...
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Reversing the coefficients of the continued fraction expansion of a rational number
On this Wikipedia page about the Markov constant, there's a surprising theorem. Basically if $\alpha$ is a real number with continued fraction expansion $[a_0;a_1,a_2,...]$, the claim is that the ...
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Measuring how good the best possible rational approximation of a real number is
The Markov constant of a real number $r$ is
$$
\limsup_{d \to \infty} \frac{1}{|r-n/d|d^2}
$$
where we choose the best possible $n$ for each corresponding $d$, e.g. $n = \text{round}(r\cdot d)$.
This ...
- 7,092
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Almost Diophantine approximation
We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$?
$$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$
Where ...
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Diophantine approximation theory and "logarithmic" rather than "linear" error
There is an enormous amount of literature on Diophantine approximations, including the general theory of continued fractions, the Stern-Brocot tree, the notion of "badly approximable number" ...
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Given $\alpha\in \mathbb{R},$ do there exist arbitrarily large $m,k$ such that $\vert \alpha - \frac{k}{m}\vert < \frac{1}{m^2}$? [duplicate]
Given $\alpha\in \mathbb{R},\ n\in\mathbb{N}$ does there exist $m\in\mathbb{N}$ with $m>n,\ $ and $\ k\in\mathbb{Z}, $ such that
$$ \alpha \in\left[\frac{k}{m}-\frac{1}{m^2}, \frac{k}{m} + \frac{1}{...
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How well approximated by rationals are almost all real numbers?
Let $\psi:\Bbb{N}\to\Bbb{R}_{\geq 0}$ be such that $q\mapsto q\psi(q)$ is a (weakly) decreasing function. We say a real number $x$ is $\psi$-approximable if $|qx-p|<\psi(q)$ has solutions for ...
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Dyson's lemma implies index is small (in proving Roth's theorem)
I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
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Growth rate of $|a^n+b^n|$ where $a,b=(1\pm\sqrt{-7})/2$.
For the real sequence
$$V_n = |a^n+b^n| \quad\text{ with }\quad a=\frac12(1+\sqrt{-7}),\ \ b=\frac12(1-\sqrt{-7}) \tag1$$
I'd like to derive some estimates on the growth rate of $V_n$ and $V_{2^n}$.
...
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Reduction step in the proof of Roth's theorem
I am following Hindry-Silverman's book "Diophantine Geometry - An Introduction". In the proof of Roth's theorem, there is a reduction step showing that the following two theorems are ...
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Reference request: A. Schinzel on digital sums of powers
I am searching for the earliest published proof of the following result:
$$\lim_{k\to\infty} s(2^k) = \infty$$
where $s(n)$ denotes the sum of the decimal digits of $n$.
This problem has been ...
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Least upper bound on $x_n = \min{|a - b\sqrt{3}| : a + b = n}$.
For all integers $n \geq 1$, define $x_n = \min{|a - b\sqrt{3}| : a + b = n}$ where $a$ and $b$ are positive integers.
Find the smallest positive real number p such that $x_n \leq p$ for all $n \geq 1$...
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Why does $|x-p/q|\ge\frac{C(x)}{q^a}$ for all $p/q$ imply: "$|x-p/q|\lt\frac{1}{q^a}$ has only finitely many solutions"?
This might be quite a trivial question but this has been bugging me, I don't know what the resolution is. Context: the irrationality measure $\mu=\mu(x)$ of some real number $x$ is defined to be that ...
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Showing that linear fractional equivalences preserve the irrationality measure
Throughout this post, let $(a,b,c,d)$ refer to the entries of some unspecified element: $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{GL}_2(\Bbb Z)$$By a “linear fractional equivalence” (...
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Approximating a number by using only a given set of prime-factors for the approximant.
Let $n \in\mathbb{N},$ and $S$ be a (finite) set of prime numbers.
I'm looking for an efficient algorithm to find the greatest $m\leq n$ such that $m$'s prime factors are of $S$?
For $S=\{p\}$ the ...
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Simple estimation of separation of powers of 2 and powers of 3?
An accepted answer is in the cross post at https://mathoverflow.net/questions/428396/simple-estimation-of-difference-of-powers-of-2-and-powers-of-3 .
1. Question
How to get from the formulas
$$ \left| ...
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Rational distances to 3 points in ${\bf R}^2$
Consider three points $(0,0)$, $(1,0)$, and $(0, {\alpha})$ in ${\bf R}^2$. If there exists a point $P =(x,y)$ such that $x^2 + y^2 =r^2$, $(x-1)^2 + y^2 =s^2$, and $x^2 + (y-{\alpha})^2 =t^2$ where $...
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Approximating reals by p-smooth ratios
Is there a techinque that allows us to approximate, with arbitrary precision, any positive real number $x$ by a sequence of ratios of $p$-smooth numbers, for given prime number $p>2$? If yes, is ...
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$|e^{inr}-1| \geq C>0$, $\forall n \in \mathbb{Z}$
I'm looking for an optimal assumption on $r$ such that $|e^{2inr}-1| \geq C>0$, for any $n$.
It is clear that $$|e^{2inr}-1|^2=2(1-\cos(2nr)).$$
If $r$ is irrational, $1-\cos(2nr)>0$ but it is ...
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How fast can a series of positive rationals converge if we know its limit is rational?
Question: Let $\{x_n\}_{n\ge 0}$ be a sequence of positive reals for which $\sum_{n\ge 0}x_n$ converges. Does there always exist a sequence $\{y_n\}_{n\ge 0}$ of rationals such that $0<y_n<x_n$ ...
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How does Wolfram Alpha find polynomial equation of given roots?
I am experimenting with a method which will converge hopefully to a real number, for which I suspect, that it is the root of a polynomial equation. How does Wolfram Alpha find its guess?
How does WA ...
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Can $c^2(a\cdot b)+c(a+b)=2^c-2$ be solved to find $\mathbb{N}$ solutions?
$c^2(a\cdot b)+c(a+b)=2^c-2$ is of the form $mx+ny=k$ and should open the door to Diophantine, but do the constraints $x=(a\cdot b), y=(a+b)$ make a difference when trying to solve for $\mathbb{N}$ ...
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2
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How can I find natural number solutions to $\frac{186-x}{11x+1}=y$? [duplicate]
I'm trying to find solutions to $\frac{186-x}{11x+1}=y$, where $x,y \in \mathbb{N}$. I've been researching Diophantine equations to try and solve this, but everything I've found is in the format $ax + ...
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Is $(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$ true for all integers $c>1$, when $n$ is an odd integer?
Let $n$ be an odd integer. Is $$(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$$ true for all integers $c>1$?
Notes:
$c=1$ has a counterexample $2^{31}-\lfloor2^{31/2}\rfloor^2>2^{...
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Does $\lfloor \tan (n)\rfloor =n$ have infinite roots?
Background
This post can be understood as the gap between $\tan n$ and $n$ has no upper limit.
Does the sequence $n+\tan(n), n \in\mathbb{N}$ have a lower bound?
I want to know how close $\tan n$ can ...
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If $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$ then $p/q$ is necessarily one of the convergents : Extend the proof to irrational $x$
Prove that, if $x$ is any irrational number, and if $p/q$ is a rational fraction in lowest terms, with $q\geq 1$, such that $$\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$$ then $p/q$ is necessarily one ...
- 7,277
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0
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Validity of Equality in the Condition for a Rational to be a Convergent
Let $x$ be an irrational number and the rational number $𝑎/𝑏$ satisfy the inequality:
$$
\bigg|𝑥−\frac{a}{b}\bigg|<\frac{1}{2b^2}
$$
Then $𝑎/𝑏$ is a convergent of $x$
A proof by contradiction ...
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1
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Rational solutions of the equation $2xy^2=kx^2+1$
I am looking for the rational solutions of the equation: $$2xy^2=kx^2+1$$
where $k$ is a fixed positional natural number.
For $k=6$, Maple shows it is irreducible and doesn't produces the rational ...
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1
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Does every line through the origin come arbitrarily close to some other lattice point?
Given positive real numbers $x$ and $\epsilon$, do there necessarily exist positive integers $p$ and $q$ such that $\left| qx - p \right| < \epsilon$?
Or geometrically: in a two-dimensional ...
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Range of Lagrange's 'best approximations' law
In my reference the theorem is stated as, convergents to an (irrational) number give a sequence of best approximations.
It is also given a proof by contradiction :
Assuming $|q\alpha-p|<|q_n\alpha-...
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