The tag has no usage guidance.

learn more… | top users | synonyms

2
votes
0answers
217 views

Simultaneous diophantine approximation with polynomial bound

For a given number $\alpha$ continued fractions expansion $(p_n, q_n)$ of $\alpha$ has the remarkable property that not only $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q_n^2}$, but the converse holds - ...
3
votes
3answers
513 views

For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer. Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
4
votes
0answers
325 views

Are these terms consisting of logarithms of primes rationally independent?

I expected it to be basic, but seem unable to find a proof of the following: Let $p_0, p_1, .., p_m$ be distinct primes. Then the $m+1$ terms $\dfrac{\log p_0}{\log p_j}$, are rationally independent. ...
1
vote
0answers
313 views

Diophantine approximation

Say absolute values of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$. Given positive integer $n$ that is reasonably large, we cannot always find integers $a,b,c$ such that $|a{b^{c}} - n|$ ...
0
votes
0answers
587 views

How quickly is $n\pi$ getting close to integers

Is there an understandable function $A(\epsilon)$ such that if $q < A(\epsilon)$ then $| q\pi - p| > \epsilon$ for all $p$? I want to know how quickly $n\pi$ is getting close to integers, e.g., ...
4
votes
1answer
816 views

Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...
102
votes
6answers
8k views

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 ...
4
votes
0answers
306 views

A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...
3
votes
1answer
356 views

Work exploring application of probability to metric number theory problems

I am interested in studying the application of probabilistic tools to study metric number theoretic problems, specifically the Duffin-Schaeffer conjecture ...
8
votes
2answers
436 views

Efficient computation of the least fraction with square denominator greater than the square root of 2.

The least rational number greater than $\sqrt{2}$ that can be written as a ratio of integers $x/y$ with $y\le10^{100}$ can be found in a moment using a little Python program. Can anyone write a ...
3
votes
1answer
671 views

Special case of Duffin-Schaeffer conjecture

The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: ...
3
votes
1answer
253 views

Weakening the hypotheses in the Duffin-Schaeffer conjecture?

The Duffin-Schaeffer conjecture is a conjecture in metric number theory, which asks for a given function $f : \mathbb{R} \rightarrow \mathbb{R}^+$ the measure of the set of real numbers $\alpha$ such ...
14
votes
1answer
504 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 ...
5
votes
5answers
548 views

Can the partial sums of a series be uniformly distributed modulo 1?

Earlier I asked in Is a sequence of the following type uniformly distributed modulo 1? whether the partial sums of the harmonic series is uniformly distributed modulo 1. Here I ask for necessary and ...
2
votes
0answers
108 views

Lower bounds on sums of S-units

Let $S$ be a fixed finite set of valuations on $\mathbb Q$ containing the archimedean one. A $S$-unit is $x\in\mathbb Q$ such that $|x|_v =1$ for all $v\notin S$. For any $S$-units $x_0, \dots, x_n$ ...
8
votes
1answer
1k views

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 ...
8
votes
1answer
484 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 | ...
11
votes
0answers
590 views

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| ...
2
votes
1answer
319 views

Best rational approximation in a special sense

Let $\alpha$ be an irrational number, $n\geq 1$ and $ X_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$ Now let $(x_n,y_n)$ minimize the quantity $x+y\alpha$ on $X_n$. ...
12
votes
5answers
1k views

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 ...
6
votes
2answers
704 views

Question related to Diophantine approximations and Roth's theorem

The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there ...
7
votes
3answers
1k views

Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer. ...
8
votes
0answers
608 views

Counting Lattice Points in Real Polytopes

Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...
0
votes
2answers
547 views

A simple question regarding the sum-of-divisors function

A good day to everyone. Consider the following "Conjecture": If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if ...
4
votes
2answers
1k views

Fast series for pi

A quick perusal of the wikipedia page for $\pi$ yields a large collection of known series for $\pi$. In particular, these series are hypergeometric in nature, and have large (but finite) radius of ...
3
votes
3answers
395 views

Is there a set of criteria to determine whether a number is transcendental for a subset of the reals with positive Lebesgue measure?

I am reading A. van der Poorten's 1978 paper on Apery's constant, and it cited the Thue-Siegel-Roth Theorem (that if $\beta$ is algebraic, then for all $\epsilon > 0$ the inequality $|\beta - p/q| ...
26
votes
1answer
1k views

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 ...
2
votes
1answer
331 views

Computational results related to Khinchin-Levy constants

Khinchin's Theorem in Diophantine approximations state that for a 'typical' real number $\alpha$ (all but a set of Lebesgue measure 0), the continued fraction $\alpha = [a_0, a_1, \cdots]$ where $a_0 ...
8
votes
1answer
215 views

inferring the slope of a digitized line

Given real numbers $a$ and $b$, and an integer $n \geq 2$, let $f(n,a,b)$ be the minimum of $(nint(ja+b)-nint(ia+b)+1)/(j-i)$ (for $1 \leq i < j \leq n$) minus the maximum of ...
15
votes
2answers
1k views

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 ...
1
vote
0answers
319 views

Diophantine Approximation in Higher Dimensions

Let $\mathbf{x} \in \mathbb{R}^K$ be an irrational vector. Assume that $\|\mathbf{x}\|^2 \leq 1$. Is is known that for all $N > 1$, there exists an $p_1 \in \mathbb{N}, \mathbf{q}_1 \in ...
1
vote
0answers
530 views

$\liminf n|\sin n|$ [closed]

$\liminf n|\sin n|$ Has this problem been solved? I looked up in some literature and didn't find the answer.
1
vote
1answer
366 views

number of solutions of diophantine approximation

Let $x$ be a real number and $N$ a positive integer. Define $E(N,\delta) = \{(p,q) \in \mathbb{Z}^2: |p - q x| \leq \frac{\delta}{N}, |p|, |q| \leq N \}$, i.e., the set of solutions to rational ...
6
votes
1answer
284 views

Bounding the growth of rational bivariate polynomials from below

The following question is an attempt to find a lower bound for the value of a polynomial at integer points. It is something that I originally thought about while trying to understand how it would be ...
1
vote
1answer
212 views

multiplicity under specialization

Let $C$ be a projective non-singular curve defined over a field $K$ with the characteristic zero. Let $y,z$ be non-constant rational functions defined $K$ such that $y$ is defined at all poles and ...
4
votes
0answers
1k views

Is there a limit of $\cos (n!)$? [closed]

I encountered a problem today to prove that $\cos (n!)$ does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are ...
8
votes
1answer
416 views

is there any way to bound the number of CM points by height functions?

It is known that if $X$ is a curve over a number field $F$ equipped with a flat regular model over $O_F$ the integer ring, one can define, using a suitable ample line bundle with an Hermitian metric, ...
10
votes
2answers
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| < ...
0
votes
1answer
399 views

An asymptotic expression for the solution to the squares problem suggested by statistical mechanics

The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or ...
6
votes
0answers
688 views

Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
8
votes
2answers
850 views

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 $\Vert x \Vert$ ...
14
votes
3answers
1k views

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 ...
4
votes
1answer
798 views

Strongest known version of Baker's theorem

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question: What is the the strongest known lower bound for ...
19
votes
3answers
2k views

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 ...
6
votes
1answer
334 views

Numbers characterized by extremal properties

The golden ratio $\phi=\frac{1+\sqrt5}2$ is sometimes said to be one of the most difficult numbers to approximate with rational numbers, because its continued fraction development $$\phi = 1 + ...
24
votes
2answers
2k views

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 ...
92
votes
4answers
6k 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 ...