For questions about approximating real numbers by rational numbers.

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21
votes
1answer
437 views

Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
2
votes
1answer
63 views

Simultaneous Diophantine approximation

Is the following statement true? For all $\alpha,\beta\in \mathbb R$ and for all $\varepsilon \in \mathbb R_{>0}$, there exist $a,b,c\in \mathbb Z$ such that $|a-c\alpha|<\varepsilon$ and ...
2
votes
2answers
285 views

Does every normal number have irrationality measure $2$?

A normal number is a number whose digit expansion in any base is "uniform" in the sense that all finite digit strings occur with the "statistically expected" frequency. I read a sentence somewhere ...
7
votes
5answers
745 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
16
votes
1answer
245 views

Does $ \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit?

The entire question is essentially given in the title: For $ n $ a positive integer, does the sequence $ x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit as $ n \to \infty $? ...
2
votes
0answers
52 views

Improving a diophantine approximation

Let $x\in \mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $p/q\in\mathbb Q^d/\mathbb Z^d$ be such that $\|x-p/q\| \leqslant t$ (with $t$ small) and $q\leqslant Q$ can I say that for all $m\in\mathbb N^\star$ ...
2
votes
2answers
146 views

Criteria for irrationality of real numbers

Concerning the criteria for irrationality: Theorem says "A number $\alpha$ is irrational if and only if for every $\epsilon >0$ there exist integers $h$ and $q$ such that $0 < | q\alpha - h | ...
3
votes
1answer
50 views

Approximating the square root of two with fractions

I would like to prove that there exist only finitely many $m, n \in \mathbb{N}$ satisfying $$\left | \sqrt{2} - \frac{m}{n} \right | < \frac{1}{4n^2}.$$ Any thoughts? Thank you for your help.
16
votes
5answers
4k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
0
votes
1answer
96 views

How is the Borel-Cantelli lemma used in this proof on $\psi$-approximable numbers?

I'm trying to understand a paper called "Almost no points on a Cantor set are very well approximable". In the proof the author uses the Borel-Cantelli Lemma (in the eighth line at the beginning of the ...
2
votes
0answers
108 views

Diophantine approximation and Lebesgue measure

Show that the set of $x\in\mathbb{R}$ such that there are infinitely many fractions p/q with p,q relatively prime integers and $|x-p/q|\le1/q^3$ has Lebesgue measure zero. I know how to show ...
3
votes
1answer
106 views

Approximate solution of a diophantine equation

Consider the Diophantine equation $P(x)=y^2$, where $P$ is a (nonconstant) polynomial with integer coefficients and $x$ and $y$ must be integers. For $\varepsilon \gt 0$, I say that an integer $x$ is ...
0
votes
1answer
43 views

Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
3
votes
2answers
91 views

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 ...
6
votes
1answer
508 views

Cube roots don't sum up to integer

My question looks quite obvious, but I'm looking for a strict proof for this. (At least, I assume it's true what I claim.) Why can't the sum of two cube roots of positive non-perfect cubes be an ...
2
votes
3answers
229 views

Sum of roots is integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
2
votes
0answers
83 views

Duffin-Schaeffer theorem/conjecture (counter)example

By the "easy" direction of Duffin-Schaeffer conjecture, it is known that if (*)$\sum_{q=1}^{\infty}\phi(q)f(q) < \infty $ (when $\phi(q)$ is euler totient function) then almost all numbers are not ...
2
votes
0answers
51 views

Least upper bound for a positive real sequence satisfying $|x_u−x_v|\cdot|u−v|>1$

The starting point of this question is the: IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct ...
2
votes
1answer
186 views

Show that a rational number has no good rational approximations

This is homework question. The teacher proved that if $a$ is irrational, there are infinitely many rational numbers $\frac{x}{y}$ such that $|\frac{x}{y} - a|<\frac{1}{y^2}$. What we need to ...
3
votes
1answer
261 views

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 $\alpha_{1}$, ...
1
vote
1answer
99 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?
2
votes
1answer
353 views

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 ...
5
votes
2answers
181 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 , ...
6
votes
1answer
157 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 ...
2
votes
1answer
53 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 $$||na||<p,||nb||<p,...$$ ...
15
votes
1answer
323 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 ...
0
votes
0answers
99 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 ...
2
votes
0answers
80 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 ...
3
votes
1answer
92 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 $$ ...
2
votes
0answers
205 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
227 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 + ...
38
votes
3answers
1k 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 + ...
0
votes
1answer
307 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 ...
6
votes
2answers
629 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 ...
7
votes
1answer
175 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 ...
12
votes
0answers
342 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 ...
0
votes
0answers
117 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) = ...
1
vote
1answer
268 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
114 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!
10
votes
1answer
386 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 ...
4
votes
1answer
319 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 ...
1
vote
2answers
136 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 ...
3
votes
2answers
201 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 ...
8
votes
1answer
101 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 ...
13
votes
1answer
545 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 ...
4
votes
0answers
149 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 ...
7
votes
1answer
146 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 ...
23
votes
2answers
851 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 ...
12
votes
3answers
537 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 ...
25
votes
3answers
3k 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.