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3
votes
2answers
73 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 ...
2
votes
1answer
24 views

approximate a vector of complex numbers

Given a vector of complex number $\vec{z}=(z_1,\cdots, z_n)$ with $|z_i|=1$ and $z_i$ is not a root of unit, and a vector of complex numbers $\vec{r}=(r_1, \cdots, r_n)$ with $|r_i|=1$. Is it the case ...
0
votes
1answer
26 views

Yet another diophantine approximation question

Let $x=(x_1,\ldots,x_n)$ be a real vector in general position and let's say it is normalized: $\lVert x\rVert=1$. Let $y$ be a real number that can be arbitrarily large and $\epsilon>0$ can be ...
11
votes
0answers
302 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 ...
10
votes
0answers
204 views

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

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 ...
8
votes
0answers
261 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
5
votes
0answers
62 views

finding very small values of $r + s\sqrt p + t\sqrt q$

For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$. In particular, I'd like to generate approximations ...
3
votes
0answers
131 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
0answers
40 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
2
votes
0answers
33 views

A generalisation of Roth's result on Diophantine approximation?

It is a celebrated result of Roth that algebraic numbers cannot be approximated by rationals too accurately: $\newcommand{\norm}[1]{\left\lVert #1 \right\rVert_{\mathbb{R}/\mathbb{T}}}$ Theorem ...
2
votes
0answers
46 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
0answers
91 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 ...
2
votes
0answers
63 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
49 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
0answers
72 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
167 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
0answers
21 views

Modification of Kinchin's Theorem

I have question regarding Kinchin's Theorem. Part of the Theorems says the following **[Kinchin's Theorem] ** Let $p,q\in \mathbb{N}$ and $a \in \mathbb{R}$. Then if $\sum_{z_x=1}^\infty qf(q)$ ...
1
vote
0answers
47 views

Rational approximation bound for real numbers in (0,1)

I am working on a problem that related to rational approximation of real numbers. I am looking for a bound of the form: Given a positive real number, $\alpha \in (0,1)$, there exist positive ...
0
votes
0answers
24 views

Spatial Basis Functions…any practical hints on choosing them?

I am not coming from a pure mathematical background, and now in my daily life as an engineering student, I am getting faced to spatial basis functions to approximates variables. The problem is: Are ...
0
votes
0answers
37 views

Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
0
votes
0answers
34 views

Relation of n-dimensional Hausdorff measure to n-dimensional Lebesgue measure

Let $X \subseteq R^n$ then, for $s \in \mathbb{R}_{\geq 0}$ and $\rho \in \mathbb{R}^+$, we define \begin{equation} \mathcal{H}_\rho ^s (X) = \inf \left\{\sum_{i \in I} r(B_i)^s: (B_i)_{i \in I} ...
0
votes
0answers
58 views

can the power of a complex number be arbitrary close to a number?

Given a complex number $z$ with $|z|=1$ and $z$ is not a root of unit, and a complex number $r$ with $|r|=1$, and a natural number $N>0$. Is it the case that for any $\epsilon>0$, there exists ...
0
votes
0answers
27 views

Exponential Diophantine Inequality

how would one go about solving inequality of the form $|a2^n-b2^k|>1 $ for $a,b \in R$ and $n,k \in Z$. Assume that $|a|>|b|$. Any help will be appreciated. Thank you
0
votes
0answers
72 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
0answers
111 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) = ...