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1
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0answers
92 views

Analog of Baker's theorem on linear combination of $\log a \log b$

Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination $$\sum m_i\log a_i$$ to cancel, then it is ...
7
votes
0answers
119 views

Efficient Dirichlet approximation (continued fractions?) over a number field

Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...
1
vote
0answers
148 views

Optimal Diophantine approximation of $\pi$

If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is the maximum value of $M$?
0
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0answers
41 views

Criterion for irrational numbers of constant type 2

From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...
1
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0answers
39 views

Representing sparse set as set of extremely good approximation

For $\alpha \in \mathbb R$ and $\varepsilon(n) > 0$, consider the set $$ N(\alpha,\varepsilon) = \{ n \in \mathbb{N} \ : \ \lVert \alpha n \rVert < \varepsilon(n) \} $$ (where $ \lVert x \rVert$...
6
votes
0answers
164 views

Solving polynomial equations modulo $1$

Let $P\in \mathbb{R}\lbrack x\rbrack$ be given. (In practice, the coefficients could be given as, say, decimals to sufficient precision.) Let $M\geq 1$, and let $I$ be an interval in $\mathbb{R}/\...
6
votes
1answer
227 views

On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have $$ \textrm{inf}(f(x)) > 0 \implies \textrm{inf}(f(x)) \geq \frac{3}{4} . $$ Could we generalize this (for ...
2
votes
1answer
114 views

On the construction of a certain sequence of integers

Let $\alpha \in \mathbb{R}$ be a fixed (positive) number. For each $k \in \mathbb{N}$ we choose $\varepsilon_k >0$ with the property that $\lim_k \varepsilon_k =0$. If $\alpha \in \mathbb{R} \...
1
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0answers
54 views

On a sequence of integers

I recall a well-known theorem due to Minkowski. Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...
2
votes
1answer
74 views

On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$ \xi_n = 2n \{n\xi\}-n, $$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
-2
votes
2answers
170 views

Precise asymptotic of diophantine approximation

I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that $$ \left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2} $$ for infinitely many choices of $p$ and $q$...
7
votes
2answers
225 views

On the density of the sequence $\{n \{n \xi \} \}_n$

I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory. Let us say that we have an irrational number $\xi$. The ...
11
votes
2answers
340 views

Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations: $$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\ -\frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{array}\right] \quad\...
4
votes
2answers
176 views

A particular Diophantine approximation of $\pi/2$

I have asked this question in math.stackexchange without any answer, so I have decided to post it here too. Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$ ...
8
votes
2answers
1k views

Approximating any integer by multiples of 2 and 3

Given any integer $n$ sufficiently large, I want to prove (or disprove) that there exists another integer $m\ge n$ with the form $m=2^a3^b$ ($a,b$ are no negative integers) such that $m-n=o(n)$, i.e., ...
12
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0answers
178 views

Diophantine approximation in the Julia set

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...
0
votes
1answer
92 views

approximation of products of polynomials

I am wondering whether the following can be proved: Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...
5
votes
0answers
98 views

Jarník-Besicovitch and outer measure

The set $A_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate $$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$ for infinitely many rationals $p/q$, ...
0
votes
1answer
102 views

Sizes and shapes of Dedekind cuts

My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra. If we define a ...
4
votes
2answers
384 views

Dirichlet's approximation only using prime power as denominator

I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...
8
votes
0answers
191 views

Attractors of arithmetically small points

Consider the "points" to be in $\mathbb{G}_m(\mathbb{C}) = \mathbb{C}^{\times}$. By a sequence of "arithmetically small points" is meant a sequence of pairwise different algebraic points $\beta$ that, ...
7
votes
0answers
239 views

Question on some coverings of the euclidean space

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...
8
votes
2answers
557 views

Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
6
votes
1answer
186 views

Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in \mathbb{R},...
2
votes
1answer
220 views

Central binomials and irrationals

I suppose this is une cause perdue, but it would be nice if the following held. Let $\theta$ be an irrational, and let $c_m = {{2m}\choose m}$, the central binomial. For a real number $r$, let $d(r)$ ...
9
votes
3answers
381 views

“Most Similar Vector Problem” on an Integer Lattice?

I am currently working on problem that I think could be expressed as an integer lattice problem. Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...
2
votes
0answers
149 views

binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
16
votes
1answer
1k views

Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number $\alpha$, there are infinitely ...
9
votes
1answer
509 views

A weakening of the Littlewood conjecture

For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by $$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|...
15
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0answers
311 views

Looking for an effective irrationality measure of $\pi$

Most standard summaries of the literature on irrationality measure simply say, e.g., that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$ for all sufficiently large $q$, without giving ...
1
vote
0answers
103 views

Rate of convergence of an algebraic irrational rotation

Let $\alpha \in \mathbb{S}^1$ be an algebraic number with $\mathop{\mathrm{arg}}(\alpha)/\pi$ irrational. Is it possible for the rotation by $\alpha$ to converge exponentially fast to a $\xi \in \...
1
vote
1answer
145 views

Positive rational numbers as sum of unit fractions [duplicate]

Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see ...
3
votes
1answer
133 views

Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...
1
vote
0answers
58 views

Big Omega result about number of totally positive integers with fixed trace

There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field. That number has a natural geometric approximation $G_a$, and we ...
11
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2answers
522 views

For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
7
votes
2answers
210 views

approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers: Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and $...
22
votes
4answers
1k views

Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
2
votes
0answers
161 views

Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...
2
votes
0answers
122 views

A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...
0
votes
2answers
135 views

Comparing the Rational Approximability of Infinite Continued Fractions

It is known, that $\phi := \frac{sqrt(5)-1}{2}$, is the number, that is hardest to approximate by rationals (cf e.g. the section properties of the golden ratio $\phi$ here: http://en.wikipedia.org/...
29
votes
4answers
3k views

Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...
16
votes
1answer
413 views

Is there a bounded sequence of points in the plane with pairwise distances at least $1/\sqrt{|i-j|}$?

Previously I have mentioned the following problem in an addition to the list of Contest problems with connections to deeper mathematics.. Is there an infinite bounded sequence $(P_n) \subset \mathbb{...
4
votes
2answers
395 views

Rate of convergence of an irrational rotation

Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$. If we assume that $\alpha$ is irrational, then there exists an increasing ...
2
votes
1answer
252 views

Computing all “suboptimal” rational approximations to $\pi/2$

I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy $$ n \epsilon(n)^2 \leq \tau $$ where $\tau$ is a known ...
7
votes
0answers
361 views

Is simultaneous diophantine approximation (in a weaker sense) NP hard?

The traditional problem of simultaneous diophantine approximation is: Given a set of rational numbers $g_1,\ldots,g_d$, an integer $N$, and a rational $\gamma>0$, is there an integer $W$ with $1\...
6
votes
0answers
91 views

Diophantine approximation in $\mathbb{G}_m^r$ with approximants restricted to a finiteley generated subgroup

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and ...
4
votes
1answer
159 views

Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality $$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$ and compares with the bound due to Minkowski that ...
3
votes
0answers
101 views

Conjectural growth rate for ergodic sums of logarithms

Let $\theta, \phi \in [0,1)$, and consider the sums $$ S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|. $$ The possible boundedness from above of such sums plays a key role in ...
1
vote
1answer
186 views

Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
5
votes
1answer
625 views

Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...