The diophantine-approximation tag has no wiki summary.
30
votes
3answers
793 views
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration.
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}.
$$
I can see that:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
21
votes
2answers
712 views
Is there any real number except 1 which is equal to its own irrationality measure?
Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
16
votes
3answers
932 views
Proving that $m+n\sqrt{2}$ is dense in R
I am having trouble proving the statement:
Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
13
votes
1answer
217 views
Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?
Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be ...
12
votes
1answer
429 views
Is there a 'far' irrational number?
I recently learned of so-called 'far' numbers at a talk. In the talk, it was proven that there is a dense subset of the interval $[0,1]$ of far numbers (however, far numbers were only a minor point of ...
11
votes
3answers
617 views
Calculate $\lim_{n \to \infty} \sqrt[n]{|\sin n|}$
I am having trouble calculating the following limit:
$$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
11
votes
3answers
446 views
When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?
Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if
$$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$
for ever non-zero integer ...
10
votes
1answer
364 views
For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?
Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it.
I have an always-nonnegative (on the ...
10
votes
0answers
149 views
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
Is is true that ...
8
votes
1answer
102 views
What is the sum of the squares of the differences of consecutive element of a Farey Sequence
A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$.
For example $F_6= ...
8
votes
1answer
83 views
Do best lower approximations of a quadratic irrational always form a linear recurrence sequence?
Let $\theta$ be an irrational number and let
$$
{\cal L}= \bigg\lbrace (a,b) \in {\mathbb Z} \times {\mathbb N}^{*} \bigg| \frac{a}{b} \leq \theta \bigg\rbrace
$$
and
$$
{\cal B}= \bigg\lbrace ...
7
votes
1answer
124 views
Diophantine equation $x^3+x+y^3+y = z^3 + z$ for $x,y,z>0$
Consider the Diophantine equation $x^3+x+y^3+y = z^3 + z$ for positive integer $x,y,z$.
I tried small values and got some near equalities : $(5,6,7)$ and $(12,16,18)$ are true up to value $2$. $( 5^3 ...
6
votes
2answers
440 views
Applying the Thue-Siegel Theorem
Let $p(n)$ be the greatest prime divisor of $n$. Chowla proved here that $p(n^2+1) > C \ln \ln n $ for some $C$ and all $n > 1$.
At the beginning of the paper, he mentions briefly that the ...
6
votes
1answer
117 views
Finding near-integers in a range
I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
5
votes
2answers
92 views
The real part of $z^n$
Prove that $${\displaystyle\lim \limits_{n \to +\infty}{|r^ncos(nθ)|}}=+\infty,$$
where $n$ is integer, $r>1$, $θ/π$ is irrational.
I got this problem from here $1+x+\ldots+x^n$ perfect square , ...
