The diophantine-approximation tag has no usage guidance.

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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 ...

**85**

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### 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 ...

**28**

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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 $n$ ...

**26**

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**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 ...

**23**

votes

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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 ...

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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 ...

**18**

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**1**answer

945 views

### 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
...

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**1**answer

559 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 ...

**15**

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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 ...

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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 ...

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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 ...

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**1**answer

361 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 ...

**13**

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**1**answer

481 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 ...

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373 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 ...

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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 ...

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412 views

### 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 ...

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499 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 ...

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988 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| < ...

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391 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 $\varepsilon>0$ ...

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85 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 ...

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555 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| ...

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458 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
...

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335 views

### Distribution of polynomials mod 1 using co-prime integers

If $a$ is any real irrational, then the set of numbers of the form $ax+y$ with $x$ and $y$ co-prime integers is dense in $\mathbb{R}$. I managed to prove this, in what I suspect is an overly ...

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321 views

### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

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499 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 ...

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### 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 ...

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463 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 | ...

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397 views

### On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$

Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. ...

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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 ...

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288 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 ...

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352 views

### Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background
Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by
$$
x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}.
$$
...

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427 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 ...

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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 ...

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396 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, ...

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548 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 ...

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### 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.
...

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192 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 ...

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838 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 $\Vert x \Vert$ ...

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### Expected symmetry in the diophantine approximations of an irrational number

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and ...

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432 views

### Diophantine elements in SU(2)

Following notions from [1], call a set of elements $g_1, \dots, g_k \in G = SU(2)$ Diophantine if it satisfies the following property: there exists a constant $D$ such that for every word $W_m$ of ...

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433 views

### Computing the measure of the projection on the torus of a semialgebraic set

Let $V \subseteq \mathbb{R}^n$ be a set cut out by a system of finitely many polynomial equations and inequalities with integer coefficients. Let $W$ be the set of all points in the box $[0,1]^n$ that ...

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### 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 + ...

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### Simultaneous Powers Far From 1

I'm looking for a reference or proof of the following. Let $K/\mathbb{Q}$ be a finite Galois extension of degree $n$. Let $a_1,\ldots,a_n$ be Galois conjugate elements in the ring of integers of $K$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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536 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 ...