Questions tagged [diophantine-approximation]
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334
questions
142
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Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?
Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...
141
votes
4
answers
14k
views
If $2^x $and $3^x$ are integers, must $x$ be as well?
I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.
If $n^x$ is an integer for ...
114
votes
4
answers
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Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
39
votes
1
answer
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Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
35
votes
4
answers
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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 $n$ ...
28
votes
3
answers
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Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
27
votes
2
answers
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How often are irrational numbers well-approximated by rationals?
Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...
26
votes
4
answers
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For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?
For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. ...
26
votes
1
answer
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An inequality for cosine of n
Can anyone provide a proof of the following inequality?
If $n$ is a positive integer, $n\geq2$, then $$\cos(n) \leq 1 - 2^{-n}.$$
This is satisfied if $n$ is not within about $2^{-n/2}$ of a multiple ...
24
votes
1
answer
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Irrationality measure of log(2)/log(6)
As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity
$...
24
votes
1
answer
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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 ...
22
votes
4
answers
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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 ...
21
votes
3
answers
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When is $n/\ln(n)$ close to an integer?
As usual I expect to be critisised for "duplicating"
this question. But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my ...
20
votes
0
answers
871
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 ...
19
votes
4
answers
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Striking applications of Baker's theorem
I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...
18
votes
1
answer
646
views
Can the expansion of a large integer in all bases consist of almost all zeroes?
Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...
17
votes
1
answer
693
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{...
16
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6
answers
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Advances and difficulties in effective version of Thue-Roth-Siegel Theorem
A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...
16
votes
2
answers
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Is there an explicit example of such a real number with the following property?
In Diophantine approximation, for a given positive real number $\alpha$ let $[a_0, a_1, \cdots]$ denote its continued fraction expansion and let $p_n/q_n = [a_1, \cdots, a_n]$. Then it is known that $...
16
votes
0
answers
362
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Average value of j-invariant at infinity
Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$:
$$
\...
14
votes
1
answer
557
views
$\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for $n\geq2$
I stumbled upon the following claim online: $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for all integers $n\in \mathbb{N}$, $n\geq2$. Checking with the computer, the claim ...
14
votes
2
answers
703
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 ...
14
votes
1
answer
581
views
Rational approximations on the circle
The well-known Liouville theorem asserts that an irrational algebraic number $\alpha$ cannot have too good rational approximations, namely $|\alpha-p/q|\ge C(\alpha)/q^k$ where $k$ is the degree of $\...
13
votes
2
answers
712
views
Lindemann theorem for Artin-Hasse exponential
Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
13
votes
0
answers
305
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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 ...
13
votes
0
answers
582
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Should the number of small solutions in Roth's theorem be bounded uniformly, assuming the target is an algebraic integer?
Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible ...
12
votes
2
answers
804
views
Are there any solutions to the diophantine equation $x^n-2y^n=1$ with $x>1$ and $n>2$?
This problem arose when considering storage of cannonballs in n-dimensional pirate ships, as explained in this MSE post. This MO question can also be reduced to the $n=3$ case. If $x,y$ is a solution ...
12
votes
0
answers
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Roth's theorem, Lang's conjecture and beyond
Lang conjectured that for an irrational algebraic number $\alpha$ and $\epsilon > 0$, there exist
only finitely may rationals $p/q$ such that
$$ \left| \alpha - \frac{p}{q} \right| <\frac{1}{q^2(...
11
votes
2
answers
467
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:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
\quad\...
11
votes
4
answers
446
views
Sequential addition of points on a circle, optimizing asymptotic packing radius
Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
11
votes
1
answer
717
views
A weakening of the Littlewood conjecture
For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by
$$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|...
11
votes
0
answers
324
views
Given $2^n - 1 \mid 3^m - 1$, how large must $m$ be compared to $n$?
Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ ...
10
votes
1
answer
468
views
Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$
By using the LLL algorithm, I tried to find the best simultaneous Diophantine approximation of the three numbers $\sqrt{2} $ and $ \sqrt{2 \pm \sqrt{3}} $. I was expecting that to get a precision of $\...
10
votes
2
answers
925
views
Estimate number of solutions in the Roth's theorem
There is a fundamental theorem in Diophantine approximation :
For all algebraic irrational $\alpha$
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \...
10
votes
1
answer
602
views
Baker's theorem for integer combinations of logarithms of integers?
Baker's theorem in transcendental number theory states that
$$
\left|\beta_0 + \sum_{i=1}^n \beta_i \log \alpha_i\right| > H^{-C}
$$
where
$\beta_0, \ldots, \beta_n$ are algebraic numbers, not ...
10
votes
2
answers
3k
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The diophantine equation X^2 - Y^2 - Z^2 = +- 1
Hi everybody. I'd like to know if the diophantine equation
(1) $$X^2 - Y^2 - Z^2 = \pm 1$$
has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you ...
10
votes
2
answers
1k
views
What numbers can be approximated "pretty well" by rationals?
More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of integers such that
$$\left| \frac{p}{q} - r \right| < \...
10
votes
1
answer
705
views
Continuous variant of the Chinese remainder theorem
Let $x_1,x_2,\ldots, x_k \in [0,1]$ be irrational numbers.
I'm interested in what, if anything, can be said about the values $\{nx_i \bmod 1: n\in \mathbb{N}\}$.
Specifically, I'm interested if there ...
10
votes
1
answer
501
views
Distribution mod 1 of exponential growth sequences
Let $t_n$ be a sequence of real numbers and $C,r>1.$ Suppose that for every $n\geq 1$ we have $\frac{1}{C}r^n\leq t_n \leq Cr^n.$ Does there exist a real number $\xi$ and an $\varepsilon>0$ ...
10
votes
2
answers
1k
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Simultaneous rational approximation of two reals using their continued fractions
Littlewood's well-known conjecture about simultaneous rational approximation is that for all $x, y \in \mathbb{R}$, $\liminf_{n \to \infty} n \Vert nx \Vert \Vert ny \Vert = 0$ (where $\Vert x \Vert$ ...
10
votes
1
answer
558
views
How far away can we get by multiple rounding and unit change?
This question is inspired by xkcd #2585 (Rounding):
Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”.
Consider the following directed graph: its vertices ...
10
votes
1
answer
226
views
Distribution of good diophantine approximations
Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
9
votes
2
answers
713
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 ...
9
votes
3
answers
3k
views
Simultaneous diophantine approximation
Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor.
Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}...
9
votes
3
answers
926
views
Is the infimum of Salem numbers > 1?
BACKGROUND
A Salem number is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their ...
9
votes
1
answer
2k
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 ...
9
votes
1
answer
683
views
Square-free diophantine approximation
Given an irrational algebraic number $\alpha$ (and maybe I want to add: of degree greater than $2$?), do there exist infinitely many relatively prime and square-free $p$,$q$ with
$$|\alpha - p/q | <...
9
votes
3
answers
553
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 ...
9
votes
1
answer
2k
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A question related to the abc conjecture
The abc conjecture asserts that whenever $a,b,c$ are pairwise coprime positive integers such that $a + b = c$ and $\epsilon > 0$, there exists a constant $C_\epsilon > 0$ (which depends on $\...
9
votes
1
answer
250
views
Distribution of $\{cn^a\}$
Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...