The diophantine-approximation tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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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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### 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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### Solved cubic Thue equation

Hi everybody. I need to know if the cubic Thue equation $x^3 + x^2y + 3xy^2 - y^3 = \pm 1$ is completely solved. I know that there are effective algorithms to solve any cubic Thue equation and that ...

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### 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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### How can vectors of a specific Diophantine type be computed?

We say that $\alpha \in \mathbb{R}^n$ is Diophantine of type $\kappa$ if there exists a constant $C$ such that
\begin{equation}
\max_{i \in \{1,\dots,n\}} | \, \alpha_i - \tfrac{m_i}{q} | \, > ...

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### 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 ...

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### 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 ...

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113 views

### Khintchine theorem - necessity of monotonicity in divergence condition

Hi,
I'm trying to get a hold of Khintchine theorem in metric Diophantine approximation. Right now I'm interested in the divergence condition, namely:
If $\sum_{q=1}^\infty\psi(q) = \infty
> $ ...

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### 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: ...

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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 $n$ ...

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### 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 ...

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### 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 ...

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208 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 ...

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### 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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### 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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### 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
...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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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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### 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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### Higher dimensional analogue of Thue's equation

The classical Thue equation is
$$\displaystyle F(x,y) = h,$$
for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ ...

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### Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes.
For example, for $M=2$ and $N=4$ you get ...

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### 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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### Reference request: on sums of the form $ax^m + by^n = h$

I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...

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### Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals.
Anyway, let $E$ be the "constructible numbers," ...

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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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### 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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### Height on a semiabelian variety

Let $A$ be a semiabelian variety over $\bar{\mathbb{Q}}$ and $B$ a semiabelian subvariety of $A$. Let $\pi:A\to A/B$ be the canonical morphism. Let $h:A(\bar{\mathbb{Q}})\to\mathbb{R}$ and ...

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### 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 $a$ ...

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### Numbers with balanced diophantine approximations

This is a follow-up to Question 146635, namely Expected symmetry in the diophantine approximations of an irrational number, which I will refer to for notation and terminology used here without ...

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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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### 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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### On the irrationality measure of generalized Stoneham numbers

Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and ...

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### 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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### (efficient) method to test $\{n\alpha\}\not\in [A, B]\subset [0,1]$

Suppose $\alpha$ is a fixed given irrational number with $\alpha\in [A, B]\subset [0,1]$, are there any (efficient) methods to compute the least integer $n$ such that the decimal part of $n\alpha$ ...

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### Generalizations of the Rayleigh(-Beatty) theorem

For a given irrational number $\alpha>0$ and a real number $\beta$,
the inhomogeneous Beatty sequence
sequence $S_{\alpha,\beta}$ is the set $\lbrace\lfloor n\alpha+\beta\rfloor:n=1,2,\dots\rbrace$
...

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### Existence of a solution to a system of Diophantine Inequalities

Does the solution to the following system of inequalities exist?
$$a-1\geq a\left( b_ic -d_i\right)\geq 1$$
where $a\in \mathbb{N}_{\geq3}$, $c\in \mathbb{R}$ and $b_i,d_i \in \mathbb{N}$. ...

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### Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations

For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the ...

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### 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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### Dalzel's integral for $\pi$ and the lemniscate constant

$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers
$$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$
and, for example, the Wallis product formula ...

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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 ...

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### Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to:
If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...

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### Siegel's theorem with real coefficients

Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$?
When $h$ itself has integer coefficients, a ...