The diophantine-approximation tag has no wiki summary.
30
votes
3answers
811 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:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
21
votes
2answers
723 views
Is there any real number except 1 which is equal to its own irrationality measure?
Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
16
votes
3answers
960 views
Proving that $m+n\sqrt{2}$ is dense in R
I am having trouble proving the statement:
Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
14
votes
1answer
227 views
Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?
Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be ...
12
votes
1answer
433 views
Is there a 'far' irrational number?
I recently learned of so-called 'far' numbers at a talk. In the talk, it was proven that there is a dense subset of the interval $[0,1]$ of far numbers (however, far numbers were only a minor point of ...
11
votes
3answers
620 views
Calculate $\lim_{n \to \infty} \sqrt[n]{|\sin n|}$
I am having trouble calculating the following limit:
$$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
11
votes
3answers
449 views
When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?
Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if
$$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$
for ever non-zero integer ...
10
votes
1answer
364 views
For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?
Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it.
I have an always-nonnegative (on the ...
10
votes
0answers
165 views
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
Is is true that ...
8
votes
1answer
103 views
What is the sum of the squares of the differences of consecutive element of a Farey Sequence
A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$.
For example $F_6= ...
8
votes
1answer
83 views
Do best lower approximations of a quadratic irrational always form a linear recurrence sequence?
Let $\theta$ be an irrational number and let
$$
{\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
127 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
444 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
123 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
99 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 , ...
5
votes
1answer
157 views
Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$
Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put
$$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$
If $\alpha$ is irrational, can I find some constant $K$ such that ...
5
votes
1answer
121 views
badly approximated numbers on the real line form a meagre set
Let $S$ be the set of real numbers $x$ such that there exist infinitely many (reduced) rational numbers $p/q$ such that $$\left\vert x-\frac{p}{q}\right\vert <\frac{1}{q^8}.$$
I would like to prove ...
4
votes
1answer
145 views
Discrete rational rotations on the two dimensional torus
It is well known (Kronecker's Theorem) that "irrational rotations" are dense on $[0,1)$. That is, the set
$$
\{ x+nr\mod 1 : n \in \mathbb{N} \}
$$
is dense on $[0,1)$, provided that $r$ is ...
3
votes
1answer
42 views
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
$$
...
3
votes
2answers
165 views
Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error
I seen this equation at math.stack exchange
The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are (1,3) (11, 4), (19, 5), (29, 6), (61, 8) and other one is I don't know. This ...
3
votes
0answers
110 views
Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?
Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
2
votes
2answers
175 views
An experiment was performed n times. The exact success rate rounds to 85.8%. What is the minimum value of n?
An experiment was performed n times.
The success rate was given as 85.8%
Clearly, if there were 858 successes out of 1000 trials it would give that percentage.
However, the percentages are rounded to ...
2
votes
2answers
329 views
Is $\sin^2 n$ bounded away from zero?
Can one find a number $m$ such that $\sin^2 n \geq m > 0 $ for all integers $n$?
By continuity of $\sin x$, it is enough to say that $|n - k \pi| \geq m^{\prime} > 0$ for all integers $n,k$.
...
2
votes
1answer
32 views
Diophantine approximation problem
Denote $||x||=Min(x-[x],1-(x-[x]))$,it means the minimum distance between x and an integer.
Can we find a fast algorithm to get a natural number $n$ that satisfies
$$||na||<p,||nb||<p,...$$
...
2
votes
0answers
38 views
How accurately can a irrational algebraic number represent a different irrational number compared to a rational approximation?
If I am trying to approximate $x = \sqrt D$ such that D is a square free integer I can use Diophantine approximation and the Fundamental Recurrence Formulas to find a rational approximation ...
2
votes
0answers
95 views
Diophantine equations/Diophantine Geometry
I am very knew to this site and I am eagerly waiting for solutions of:
(1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
1
vote
1answer
117 views
For what $\lambda$ does the sequence $\left\{ \frac{k}{2\lambda} \right\}$ converge?
I am reading some materials on hypergeometric functions and their relation to general L-functions and the authors make use of the following "well-known elementary result from diophantine ...
1
vote
2answers
124 views
Constructive proof need to know the solutions of the equations
Observe the following equations:
$2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$
$x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$
$7x^2 + 11= 2 \cdot 3^n$ has ...
1
vote
2answers
208 views
Solutions of some Diophantine equations
Respected Mathematicians,
The diophantine equation $2^x$ + $5^y$ = $z^2$ has solutions $x = 3, y = 0, z = 3$ and $x = 2, y = 1, z = 3$. I got these solutions by trial and error method. To be honest, ...
1
vote
1answer
214 views
Non trivial upper bound for an exponential sum
Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for
$$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
1
vote
1answer
31 views
Measure of the set of real numbers that can be approximated in this way
Let $$A = \{x \in \mathbb{R}\mid \exists\,\text{infinitely many pairs of integers $p,q$ such that $|x-p/q| \leq 1/q^3$}\}.$$ Is the measure of $A$ equal to $0$? Any ideas?
1
vote
1answer
128 views
Why does the set of real numbers with irrationality measure $\gt 2$ have zero measure?
Recall the irrationality measure of a real number $r$ is $$
\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} ...
1
vote
1answer
122 views
How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?
I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products:
$A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
1
vote
0answers
34 views
+50
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 ...
1
vote
1answer
83 views
prove existence of integers $a,q$ which satisfy the following inequality
Let $x \in \mathbb{R}$ and integer $Q \geq 1$.
Prove: there exist integers $a$ and $1 \leq q \leq Q$ such that $|x - \frac aq | < \frac 1{qQ} $
any help would be appreciated!
0
votes
0answers
32 views
Solving system of equations with mixed variable types
I'm looking for solutions to the non-linear system of equations
$$
n_1x + (n_1 - 1)y = a_1 \\
n_2x + (n_2 - 1)y = a_2 \\
n_3x + (n_3 - 1)y = a_3 \\
n_4x + (n_4 - 1)y = a_4
$$
where $x$ and $y$ are ...
0
votes
1answer
159 views
Ordered triplet query
$$x^2 + y^2 + z^2 = 3xyz$$
How many ordered triples $(x,y,z)$ are there that satisfy the above equation.
are the only solutions $x=y=z=0$ and $1$?
Are there non trivial solutions?
I saw this ...
0
votes
0answers
24 views
Prove that $\log a$ is not a Liouville number, for integer $a>1$.
I've found a few results on bounds for the irrationality measures of specific logarithms, e.g. $\log 2$ and $\log 3$ (see, for example, An essay on irrationality measures of pi and other logarithms), ...
0
votes
0answers
93 views
integer solutions to $a^m+nx^2 = y^n$ with various conditions
I consider the following equation with conditions of obtaining solutions
$$a^m+nx^2 = y^n$$
This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...
