For questions about approximating real numbers by rational numbers.
Questions tagged [diophantine-approximation]
494 questions
11
votes
2 answers
Line through the origin mod $1$ visits every sub-cube in $\mathbb{R}^n$.
Let $v$ be the vector $[a_1,a_2,\ldots,a_n]$ where the $a_i$'s are positive integers with $\gcd(a_1,\ldots, a_n)=1$. Let $C$ be the hypercube $[0,1]\times [0,1]\times \cdots \times [0,1]$ in $\mathbb{R}^n.$ Imagine the line given by $vt$ as $t$…
11
votes
3 answers
Why does $29^2 : 31^2 : 41^2$ have a close integer approximation with small numbers?
"Everybody knows" that such coincidences as
$$2\times2\times\overbrace{41\times41} = 6724 \approx 6728 = 2\times2\times2\times\overbrace{29\times29}$$
(And why did I bother with the first two factors of $2$ on each side? Be patient.)
are…
4
votes
1 answer
Kronecker's theorem - converse
I don't know how to prove that Kronecker's theorem is false if $\alpha_{1}$, $\alpha_{2}$, $\dots$, $\alpha_{M}$
are not independent.
Kronecker's theorem
Suppose
that $\alpha_{1}$, $\alpha_{2}$, $\dots$, $\alpha_{M}$
are independent real…
porkramen
- 663
4
votes
1 answer
diophantine approximation
For which $\alpha \in \mathbb R$ can one say that $\forall \epsilon > 0$ there $\exists N \geq 1$ such that $\forall i \in \mathbb N$ one has that some $n\in \{1, \dots, N\}$ is a solution to
$$
\min_{m\in \mathbb Z} |\alpha^{in} - m| \leq…
Thomas
- 155
4
votes
1 answer
How fast do the denominators in Dirichlets approximation for irrationals grow?
For any irrations $a$, there are infinitely many $p,q$ in $N$ such that $|a-\frac{p}{q}|<\frac{1}{q^2}$, how fast do those $q$ grow?
What I mean is if we take the series of $q$ that satisfy the above (there exist p so that...) how fast does the…
Andy
- 1,874
3
votes
2 answers
Bound to the Number of Solutions of the Thue-Siegel-Roth Theorem
I have to prove for an assignment that if the partial quotients of a number $x$ are $\frac{p_n}{q_n}$, and
$$\limsup\frac{\log\log (q_n)\sqrt{\log n}}{n}=\infty$$
then $x$ is transcendental.
I do not want a proof for this claim, as I would like to…
David
- 31
3
votes
1 answer
Real part of complex root of $X^5-X-k^5+k+1$ is $\frac{1}{23k}$ away from any integer ($k\geq 2$)
Let $k\geq 2$ be an integer. The polynomial $P_k=X^5-X-k^5+k+1$ is easily
seen to have exactly one real root and two pair of conjugate non-real roots. Is it true
that if $\alpha_k + i \beta_k$ any non-real root of $P_k$, then $\lbrace \lbrace…
Ewan Delanoy
- 61,600
2
votes
0 answers
Finding a close integer point to a line with irrational slopes
Let $\alpha$, $\beta$ be two irrational numbers.
Is there a good way to find some integer $n,m$ that $|n\alpha-\beta-m|$ is sufficiently small?
For example, if $\beta=0$, we know that there exists $n,m$ that $|n\alpha-m|<\frac{1}{n}$ and this can be…
newbie
- 21
- 1
2
votes
1 answer
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 there a way to determine the "best" of such…
user3257624
- 361
2
votes
2 answers
Is there a number such that the sequence of its best rational approximations is strictly increasing?
The rational approximations of $\sqrt 2$ given by its continuous fraction are: 1.5, 1.333, 1.4, 1.417, 1.412, etc. which is not strictly increasing.
Similarly, this sequence for $\phi=(1+\sqrt5)\big/2$ is 2.0, 1.5, 1.667, 1.6, 1.625, etc.
Is there a…
user1001232
2
votes
0 answers
Approximating irrational numbers with numbers in $\mathbb{Q}(\sqrt{5})$
I am trying to evaluate an infinite series that involves Fibonacci numbers. By adding the first thousands of terms, I get an approximation of the sum: $S = 5.3598856662431775531720113029189271796889051337319684864955538153...$.
I guess that $S$ can…
Joseph
- 560
2
votes
1 answer
Dirichlet approximation theorem and BA numbers
I think some version of the following definition given in "Diophantine Approximations" by W.M. Schmidt:
Let $\sigma \in (0,1)$ and $x \in \mathbb{R}$. Then $x$ is
$\sigma$-improvement for Dirichlet theorem if there is $Q_0$ so
that for all $Q…
Pavel
- 1,522
2
votes
2 answers
Dirichlet's approximation theorem
A corollary of Dirichlet's approximation theorem is that for any irrational $\alpha$, there are infinitely many integer solutions of $$\left|\frac{p}{q}-\alpha\right|<\frac{1}{q^2}$$
Are there any similar results concerning the integer solutions of…
user502266
2
votes
2 answers
Why does the set of real numbers with irrationality measure $\gt 2$ have zero measure?
Recall the irrationality measure of a real number $r$ is $$
\mu(r)= \inf \left\{ \lambda\colon \left\lvert r-\frac{x}{y}\right\rvert\lt \frac{1}{y^{\lambda}} \text{ has only finitely many solutions} \right\}$$
Does anyone have a reference or proof?
hello
- 135
1
vote
2 answers
Exponential irrationality
I want to prove that there are only finitely many rational solutions to
$$\left|\frac{\log(5)}{\log(7)}-\frac{a}{b}\right|\le \frac{1}{7^b}$$
And once I have done this, I would like to put a bound on how many solutions there are. Any thoughts?…
Elie Bergman
- 3,917