Questions tagged [diophantine-approximation]
For questions about approximating real numbers by rational numbers.
471
questions
3
votes
0
answers
27
views
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{\...
1
vote
0
answers
28
views
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 $g \leq d − 3$...
2
votes
1
answer
31
views
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 $(0,1)$, ...
1
vote
0
answers
23
views
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 ...
1
vote
1
answer
39
views
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 ...
1
vote
2
answers
43
views
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
$(p,q)\in\mathbb Z^2$ ...
3
votes
1
answer
141
views
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 ...
2
votes
1
answer
64
views
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 $\
\...
0
votes
1
answer
30
views
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 $...
1
vote
0
answers
22
views
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, ...
0
votes
0
answers
12
views
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 ...
3
votes
1
answer
107
views
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
$$
\min_{1\leq k\leq n}\| k \alpha \| \leq \frac{1}{...
5
votes
2
answers
99
views
How is Siegel's lemma applied in number theory?
In the Wikipedia page, Siegel's lemma is stated as follows:
Consider the system $$ \begin{cases} \sum_{i=1}^Na_{1i}X_i=0\\
\vdots\\ \sum_{i=1}^Na_{Mi}X_i=0 \end{cases}, $$ where the
coefficients $a_{...
0
votes
0
answers
21
views
Does a limit exist for the fraction of $p\alpha_i$?
With $p$ an integer and $\alpha_i$ a real number, does $\{p\alpha_i\}\leq\epsilon$ hold for some predefined $\epsilon$, maybe depending on the number of $\alpha_i$ values? I looked at the simultaneous ...
0
votes
1
answer
107
views
Can we scale the elements of a set of real numbers so that they are arbitrarily close to integers?
Let $S$ be a finite set of real numbers. For any $\varepsilon > 0,$ is it always possible to choose $\alpha > 0$ such that every element of $S$ multiplied by $\alpha$ is withing $\varepsilon$ of ...
1
vote
1
answer
140
views
Do three arithmetic sequences of real numbers that include $0$ always get arbitrarily close to each other at nonzero numbers?
I know that two arithmetic sequences of real numbers that both contain the number $0$ have terms that are arbitrarily close together, that is, for any $\varepsilon>0$ there exists a real number ...
3
votes
0
answers
61
views
Absolute convergence of $\sum_{n=1}^\infty n^{-5} \csc(2n\pi \sqrt[3]{3})$
I want to prove the absolute convergence of this serie. There are two hints:
Firstly, prove that $|(\sin(\pi x)| \geq |x|$ for $|x| \leq \frac{1}{2}$.
Then use the fact that $2\sqrt[3]{3}$ is ...
0
votes
0
answers
35
views
How does Roth's Theorem show a number is transcendental
[Roth's theorem][1] says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions $\frac{p}{q}$to this:
$$\displaystyle \left|\alpha-\frac pq\right|<\...
0
votes
0
answers
25
views
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 $\left\{x\right\}$ ...
2
votes
0
answers
42
views
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 ...
7
votes
1
answer
158
views
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 ...
1
vote
1
answer
55
views
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 ...
2
votes
0
answers
59
views
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 ...
4
votes
0
answers
53
views
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 ...
3
votes
1
answer
64
views
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" ...
1
vote
0
answers
50
views
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
$$ \alpha \in\left[\frac{k}{m}-\frac{1}{m^2}, \frac{k}{m} + \frac{1}{...
3
votes
0
answers
64
views
How well approximated by rationals are almost all real numbers?
Let $\psi:\Bbb{N}\to\Bbb{R}_{\geq 0}$ be such that $q\mapsto q\psi(q)$ is a (weakly) decreasing function. We say a real number $x$ is $\psi$-approximable if $|qx-p|<\psi(q)$ has solutions for ...
0
votes
0
answers
19
views
Dyson's lemma implies index is small (in proving Roth's theorem)
I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
1
vote
1
answer
61
views
Growth rate of $|a^n+b^n|$ where $a,b=(1\pm\sqrt{-7})/2$.
For the real sequence
$$V_n = |a^n+b^n| \quad\text{ with }\quad a=\frac12(1+\sqrt{-7}),\ \ b=\frac12(1-\sqrt{-7}) \tag1$$
I'd like to derive some estimates on the growth rate of $V_n$ and $V_{2^n}$.
...
0
votes
1
answer
80
views
Reduction step in the proof of Roth's theorem
I am following Hindry-Silverman's book "Diophantine Geometry - An Introduction". In the proof of Roth's theorem, there is a reduction step showing that the following two theorems are ...
2
votes
0
answers
52
views
Reference request: A. Schinzel on digital sums of powers
I am searching for the earliest published proof of the following result:
$$\lim_{k\to\infty} s(2^k) = \infty$$
where $s(n)$ denotes the sum of the decimal digits of $n$.
This problem has been ...
2
votes
1
answer
76
views
Least upper bound on $x_n = \min{|a - b\sqrt{3}| : a + b = n}$.
For all integers $n \geq 1$, define $x_n = \min{|a - b\sqrt{3}| : a + b = n}$ where $a$ and $b$ are positive integers.
Find the smallest positive real number p such that $x_n \leq p$ for all $n \geq 1$...
3
votes
1
answer
47
views
Why does $|x-p/q|\ge\frac{C(x)}{q^a}$ for all $p/q$ imply: "$|x-p/q|\lt\frac{1}{q^a}$ has only finitely many solutions"?
This might be quite a trivial question but this has been bugging me, I don't know what the resolution is. Context: the irrationality measure $\mu=\mu(x)$ of some real number $x$ is defined to be that ...
1
vote
0
answers
32
views
Showing that linear fractional equivalences preserve the irrationality measure
Throughout this post, let $(a,b,c,d)$ refer to the entries of some unspecified element: $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{GL}_2(\Bbb Z)$$By a “linear fractional equivalence” (...
3
votes
1
answer
56
views
Approximating a number by using only a given set of prime-factors for the approximant.
Let $n \in\mathbb{N},$ and $S$ be a (finite) set of prime numbers.
I'm looking for an efficient algorithm to find the greatest $m\leq n$ such that $m$'s prime factors are of $S$?
For $S=\{p\}$ the ...
3
votes
1
answer
104
views
Simple estimation of separation of powers of 2 and powers of 3?
An accepted answer is in the cross post at https://mathoverflow.net/questions/428396/simple-estimation-of-difference-of-powers-of-2-and-powers-of-3 .
1. Question
How to get from the formulas
$$ \left| ...
1
vote
1
answer
60
views
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 $...
2
votes
1
answer
106
views
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 ...
0
votes
0
answers
30
views
$|e^{inr}-1| \geq C>0$, $\forall n \in \mathbb{Z}$
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 ...
0
votes
1
answer
50
views
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 $0<y_n<x_n$ ...
5
votes
1
answer
135
views
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 ...
1
vote
1
answer
49
views
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 $\mathbb{N}$ ...
-1
votes
2
answers
101
views
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 + ...
5
votes
1
answer
77
views
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^{...
3
votes
3
answers
89
views
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 ...
1
vote
1
answer
118
views
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 ...
1
vote
0
answers
26
views
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 ...
0
votes
1
answer
69
views
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 ...
1
vote
1
answer
50
views
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 ...
2
votes
0
answers
37
views
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-...