All Questions
Tagged with approximation-theory nt.number-theory
16 questions
4
votes
2
answers
320
views
Approximating a fraction with a given denominator
Let $M$, $N$ be large natural numbers (say ~200 bits). Let $L$ be a smaller number, (say ~100 bits).
I want to approximate the fraction:
$$\frac{M}{N} \sim \frac{k}{L+r}$$
where $r$ is at most $L$. In ...
- 591
0
votes
0
answers
107
views
$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 48.9k
25
votes
2
answers
4k
views
A "better" rational approximation of pi?
$355/113$ is a good fractional approximation of $\pi$, because we use six digits to produce seven correct digits of $\pi$.
$$\frac{355}{113} = 3.1415929\ldots$$
Let $R$ be the ratio of the number of ...
- 251
1
vote
2
answers
333
views
Chebyshev rational approximation of $e^{x}, x >0$: does it exist?
It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form ...
- 111
1
vote
0
answers
80
views
Approximating the partial sum of remainders function
This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask.
Let $R_{k,N}$ denote the remainder of ...
- 111
17
votes
4
answers
3k
views
Using Quotient of Prime Numbers to Approximation Reals
We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime.
Question ...
- 273
7
votes
2
answers
2k
views
Variant of Fermat's last theorem
By Fermat's last theorem, the equation $u^3+v^3=w^3$ has no solutions
in positive integers $u,v,w$. Now consider the following variant : call $\rho(x)$
the distance between $x$ and the nearest integer,...
- 3,595
4
votes
1
answer
182
views
On the set of good approximators in the sense of Dirichlet's theorem
This question came up when thinking about an older question that hasn't been answered as of now.
Let $\mathbb{N}$ be the set of positive integers. If $\alpha\in\mathbb{R}$, we say $q\in\mathbb{N}$ is ...
- 48.9k
1
vote
0
answers
89
views
Smooth function approximating pi(x)
We can define the prime number function as $$\pi(x) = \int_{-\infty}^x \sum_{p}\delta(p-x).dx$$ That is, we include each prime p as a delta function $\delta_p(x) = \delta(p-x)$, set $P(x) = \sum_{p}\...
- 595
1
vote
1
answer
175
views
Density of the set of numbers that are "good approximators" to a given real in the sense of Dirichlet's approximation theorem
Let $\mathbb{N}$ be the set of positive integers. Given a set $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$...
- 48.9k
3
votes
0
answers
243
views
Interlacing sequences by polynomials?
Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...
- 13.9k
4
votes
2
answers
2k
views
Estimate on sum of squares of multinomial coefficients
I am interested in approximating the sum of the squares of the multinomial coefficients, i.e.
$a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2$
or more general,
$a_\...
- 145
3
votes
2
answers
306
views
Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
- 128k
3
votes
1
answer
289
views
Approximation of the form $\frac{1}{u}\pm\frac{1}{v}$
Given positive integers $m<n\in\mathbb{N}$ is there an algorithm to find integers $z_1, z_2\in \mathbb{Z}\setminus\{0\}$ such that $\frac{m}{n}$ is best approximated by $\frac{1}{z_1}+\frac{1}{z_2}$...
- 48.9k
1
vote
1
answer
159
views
Twisted Padé approximants
Let $f$ be a continuous function defined on $\mathbb Z_p$. By Mahler theorem, there exists a sequence $(\gamma_k)_{k\in\mathbb N}$ of $\mathbb Z_p$ such that for every $z\in\mathbb Z_p$
$$f(z)=\sum_{k\...
- 3,998
0
votes
1
answer
638
views
Rational solutions of homogeneous equations
Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\...
- 10.5k